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Free Math Worksheets — Over 100k free practice problems on Khan Academy

Looking for free math worksheets.

You’ve found something even better!

That’s because Khan Academy has over 100,000 free practice questions. And they’re even better than traditional math worksheets – more instantaneous, more interactive, and more fun!

Just choose your grade level or topic to get access to 100% free practice questions:

Kindergarten, basic geometry, pre-algebra, algebra basics, high school geometry.

• Trigonometry

Statistics and probability

High school statistics, ap®︎/college statistics, precalculus, differential calculus, integral calculus, ap®︎/college calculus ab, ap®︎/college calculus bc, multivariable calculus, differential equations, linear algebra.

• Addition and subtraction
• Place value (tens and hundreds)
• Addition and subtraction within 20
• Addition and subtraction within 100
• Addition and subtraction within 1000
• Measurement and data
• Counting and place value
• Measurement and geometry
• Place value
• Measurement, data, and geometry
• Add and subtract within 20
• Add and subtract within 100
• Add and subtract within 1,000
• Money and time
• Measurement
• Intro to multiplication
• 1-digit multiplication
• Addition, subtraction, and estimation
• Intro to division
• Understand fractions
• Equivalent fractions and comparing fractions
• More with multiplication and division
• Arithmetic patterns and problem solving
• Quadrilaterals
• Represent and interpret data
• Multiply by 1-digit numbers
• Multiply by 2-digit numbers
• Factors, multiples and patterns
• Add and subtract fractions
• Multiply fractions
• Understand decimals
• Plane figures
• Measuring angles
• Area and perimeter
• Units of measurement
• Decimal place value
• Add decimals
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• Multi-digit multiplication and division
• Divide fractions
• Multiply decimals
• Divide decimals
• Powers of ten
• Coordinate plane
• Algebraic thinking
• Converting units of measure
• Properties of shapes
• Ratios, rates, & percentages
• Arithmetic operations
• Negative numbers
• Properties of numbers
• Variables & expressions
• Equations & inequalities introduction
• Data and statistics
• Negative numbers: addition and subtraction
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• Fractions, decimals, & percentages
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• Expressions, equations, & inequalities
• Numbers and operations
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• Linear equations and functions
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• Foundations
• Algebraic expressions
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• Expressions with exponents
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• Working with units
• Linear equations & graphs
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• Inequalities (systems & graphs)
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• Non-right triangles & trigonometry (Advanced)
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• Transformations of functions
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• Trigonometric equations and identities
• Analyzing categorical data
• Displaying and comparing quantitative data
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• Modeling data distributions
• Exploring bivariate numerical data
• Study design
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• Scatterplots
• Data distributions
• Two-way tables
• Binomial probability
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• Displaying and describing quantitative data
• Inference comparing two groups or populations
• Chi-square tests for categorical data
• More on regression
• Prepare for the 2020 AP®︎ Statistics Exam
• AP®︎ Statistics Standards mappings
• Polynomials
• Composite functions
• Probability and combinatorics
• Limits and continuity
• Derivatives: definition and basic rules
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• Applications of derivatives
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• Applications of integrals
• Differentiation: definition and basic derivative rules
• Differentiation: composite, implicit, and inverse functions
• Contextual applications of differentiation
• Applying derivatives to analyze functions
• Integration and accumulation of change
• Applications of integration
• AP Calculus AB solved free response questions from past exams
• AP®︎ Calculus AB Standards mappings
• Infinite sequences and series
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• Integrals review
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• Thinking about multivariable functions
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• Integrating multivariable functions
• Green’s, Stokes’, and the divergence theorems
• First order differential equations
• Second order linear equations
• Laplace transform
• Vectors and spaces
• Matrix transformations
• Alternate coordinate systems (bases)

Frequently Asked Questions about Khan Academy and Math Worksheets

Why is khan academy even better than traditional math worksheets.

Khan Academy’s 100,000+ free practice questions give instant feedback, don’t need to be graded, and don’t require a printer.

What do Khan Academy’s interactive math worksheets look like?

Here’s an example:

What are teachers saying about Khan Academy’s interactive math worksheets?

“My students love Khan Academy because they can immediately learn from their mistakes, unlike traditional worksheets.”

Is Khan Academy free?

Khan Academy’s practice questions are 100% free—with no ads or subscriptions.

What do Khan Academy’s interactive math worksheets cover?

Our 100,000+ practice questions cover every math topic from arithmetic to calculus, as well as ELA, Science, Social Studies, and more.

Is Khan Academy a company?

Khan Academy is a nonprofit with a mission to provide a free, world-class education to anyone, anywhere.

Want to get even more out of Khan Academy?

Then be sure to check out our teacher tools . They’ll help you assign the perfect practice for each student from our full math curriculum and track your students’ progress across the year. Plus, they’re also 100% free — with no subscriptions and no ads.

Get Khanmigo

The best way to learn and teach with AI is here. Ace the school year with our AI-powered guide, Khanmigo. 

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Whiteboard App

Solve problems and explain your thinking in a digital math workspace. 

The Whiteboard app is a digital workspace for teachers and students to solve problems and explain their thinking. Math concepts can be explored in a variety of ways using a flexible set of tools to sketch, write, and build equations.

The Whiteboard app is an open-ended educational tool, ideal for elementary classrooms and other learning environments that use laptops, iPads, or Chromebooks.

App Features

• Use the Drawing tools to solve problems and show understanding.
• Add equations, expressions, and descriptions with the Math Text and Writing tools.
• Hide and reveal work with resizable covers to create your own problems and model strategies.
• Share your work by saving an image or creating an 8-character code and link to send to others.

Technical Support and Feedback

Please contact our team for app support and feedback.

© 2022 The Math Learning Center

This app was crafted by Clarity Innovations: “We improve teaching and learning by creating solutions that match promising technologies with the needs of education.” www.clarity-innovations.com .

Online Mathematics Editor a fast way to write and share mathematics 100k + users registered, 450k + documents created

Desktop (offline) version, built for your privacy

Test and Worksheet Generators for Math Teachers

Test and worksheet generators, learning management system, how it works, you choose the topic., you choose the mathematical properties of the questions., every topic has different options., it creates as many questions as you would like., distribute assignments to your students., print hard copies to give to students., post assignments to kuta works., students complete online., unlimited questions.

Once you have created an assignment, you can regenerate all of its questions with a single click. The new questions will conform to the same parameters as the original questions, but they will be completely new. This feature is at the heart of our software and is what makes it so powerful: you choose the properties of the questions, not the questions themselves. When a question is replaced, you get a new one that is similar to the original question. How it works. You can regenerate entire assignments, particular question groups, or individual questions.

Easy Spacing

Respace the entire assignment to the desired length with one click. Easily give your students enough room to show their work by increasing the spacing. Or you can save paper by decreasing the spacing.

Spacing can also be controlled manually.

Presentation Mode

Very useful as a teaching aid when used in combination with an LCD projector or other display system. One to four questions at a time are shown on the screen.

Use this feature while you teach. Prepare your examples with the software, and then use a projector to display the questions on the board. This saves time during planning and during the lesson, and it makes it very easy to present long questions or questions with graphs and diagrams. With one question displayed, you can:

• Change the zoom level -- so students in the back can read it
• Draw lines beside the question to help you organize your work if you solve the question
• Jump to another question -- useful while reviewing homework
• Reveal the answer
• Show / hide the question number and the directions.

Multiple-Version Printing

Print multiple versions of an assignment. You control how each new version is created: scramble the choices, scramble the questions, or make completely new questions. You can also save each new version after it is created.

Scale Assignment

Proportionally increase or decrease the number of questions in the assignment. This is very useful when planning a lesson. You can create a few questions to use as examples, and then scale up the number of question to create a homework assignment. The questions on the homework will be completely new, yet follow precisely from the lesson--and you don't need to design the questions again.

Export Questions

Export questions as bitmap images and paste them into your favorite word processing software. Questions created with our products can be added to existing assignments you have created with other programs. Or you can freshen old assignments by replacing old questions with new ones.

All questions are available for export.

Good Multiple-Choice Questions

Every question you create can be toggled between free-response and multiple-choice format. Multiple-choice questions come with smart, potentially misleading choices. Some are based on common mistakes students make while others are just random but near the correct answer.

You control the number of choices each question has, from two to five.

Merge Assignments

Merge two or more assignments into one. Easily create quizzes, tests, and reviews by merging the assignments from the unit and then scaling the total to an appropriate length. The questions will be new while following exactly from what you taught.

Diagrams Drawn to Scale

Diagrams are all accurately drawn, except if the answer would be given away. If an angle is labeled as 30°, then it really is 30°. If a triangle's sides are labeled 3, 4, and 5, then its lengths truly are in a 3:4:5 ratio. Seeing accurate diagrams helps students gain an intuitive understanding of angles and measurements.

Answer Format

When you print an assignment, you choose how the answers are reported:

• On an answer sheet
• On an answer sheet with just the odds
• In context (next to or within the question)
• No answer sheet

Graphing and Graph Paper Utility

Supplement your lessons with high-quality graphs and graph paper of any size. Each graph can have zero to two functions graphed on it. Graphs can be of any logical and physical size. You can also tile graphs across the page to maximize your paper use.

Custom Directions and Custom Questions

Enter your own directions to create new types of problems. Shown on the left was a standard order of operations question that has been modified to be more analytical. You can alter the directions on any question type.

From time to time, you will need to enter your own question. That's what custom questions are for. They can be either free response or multiple-choice and can contain math formatted text (equations, expressions, etc).

Modify Automatically-Generated Questions

Most automatically-generated questions can be modified manually. If there is a choice you don't like, you can change it. If you wish a question was slightly different, you can change it.

Paper Size and Margins

Print assignments on any sized paper that your printer supports. If you decide to print an assignment on legal-sized paper, no problem. The questions will automatically be repositioned for you--no cutting and pasting the assignment back together just to use a different paper size. You also have control over the margins, page numbering, and paper orientation.

Maths with David

Problem solving. draw diagram.

In mathematics, diagrams are often a useful way of organising information and help us to see relationships. A diagram can be a rough sketch, a number line, a tree diagram or two-way table, a Venn diagram, or any other drawing which helps us to tackle a problem.

Labels (e.g. letters for vertices of a polygon) are useful in a diagram to help us be able to refer to items of interest.

A diagram can be updated as we find out new information.

Examples of using a diagram to tackle a problem

First we will read all three examples and have a quick think about them and then we will look at how a diagram can help us with each one:

Restaurant Example

A restaurant offers a “business lunch” where people can choose either fish or chicken or vegetables for their main course, accompanied by a side portion of rice, chips, noodles or salad. How many different combined meals can they choose between?

Rectangle Area Example

To the nearest centimetre, the length and width of a rectangle is 10cm and 8cm.

• What are the limits of accuracy for the area of the rectangle?
• the lengths of the sides?

Prime Numbers Example

Masha says that if she writes out numbers in rows of six then all of the prime numbers will either be in the column that has 1 at the top, or they will be in the column that has 5 at the top. How can you find out if she is correct?

Worked Solutions to Examples

One way to tackle this would be to write out a list, being systematic to ensure that all combinations are considered.

Another is to draw out a diagram like the one below. As shown, you actually don’t need to finish the diagram in order to conclude how many combinations there are:

You could also use a 2-way table as shown below:

Drawing a rough sketch of the rectangle labelled with the boundaries of its side lengths can really help us to visualise the situation here:

It can then be helpful to draw sketches of the smallest possible rectangle and the largest possible rectangle:

We can now answer the questions, so (a) the smallest possible area is 7.5 x 9.5 = 71.25cm 2 and the largest “possible” area is 8.5 x 10.5 = 89.25cm 2 . So the limits of accuracy are [71.25,89.25) cm 2 .

For (b), we can see from the sketches that the difference between the minimum and the maximum values is 1cm in the case of both the width and the lenght. For part (ii) we simply subtract the numbers above to give 89.25-71.25 = 18cm 2 .

Here, listing out numbers, especially for the first few is going to be helpful. We should list them as specified in the question, and we can highlight the prime numbers:

Because we know that no even numbers other than 2 are prime, we know that further prime numbers cannot be in the second, fourth or sixth column. The third column keeps adding 6s, so it is adding multiples of 3 to multiples of 3, so the numbers will always be divisible by 3, so further numbers in this column cannot be prime. So she is correct that the prime numbers must be in the first or the fifth column.

31 Questions of increasing difficulty

1.) In a cement factory, cement bags are placed on pallets made of planks of wood and bricks. The number of bricks needed to make a pallet is calculated as being one more than the length of the plank in metres (as shown below):

a.) What length of pallet uses five bricks?

b.) If the pallet is 7m long, how many bricks are used in it?

The factory needs pallets with a total length of 15m for the next batch of cement. It has planks of wood that are 4m long and 3m long.

c.) What combinations of planks can they have?

d.) How many bricks would be needed for each combination?

2.) Sonia wants to plant an apple tree in her garden. She needs to make sure that there is a circular area of lawn with diameter 3m around the base of the tree, so that all of the fruit will fall onto the lawn area.

Below is a (not to scale) sketch of Sonia’s garden:

Where could the tree be placed to meet her requirements?

3.) The diagram below represents towns A and B in a mountainous region:

The mountain rescue helicopters from both towns will always be sent to rescue any casualty within a 25km radius of town A or town B. The fire and rescue team from town B will travel to any accident scene closer to town B than town A.

Shade the region that the helicopters and town B’s fire an rescue team will both cover.

4.) A rectangle has length (2x+3) and width (x-1).

a.) Write an expression for the perimeter of the rectangle.

b.) Write an expression for the area of the rectangle.

The area of the rectangle is 250cm 2 .

c.) How long is the longest side?

d.) What is the perimeter of the rectangle?

5.) The probability that Hannah catches the 6.30am train to the city is 0.7.

If she misses the train, she will be late for work.

The probability that the train will be late is 0.15.

If the train is late, she will be late for work.

What is the probability that Hannah will be on time for work on a particular day?

6.) Two five-sided spinners are numbered 1 to 5. When the arrows are spun, your total score is calculated by adding the two numbers that the spinners land on.

a.) Draw a suitable diagram to show all possible outcomes when spinning these spinners.

b.) What is the highest score you could get?

c.) What is the probability of getting a total score of 8?

Worked Solutions to Questions

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Geometry Worksheets

Welcome to the geometry worksheets page at Math-Drills.com where we believe that there is nothing wrong with being square! This page includes Geometry Worksheets on angles, coordinate geometry, triangles, quadrilaterals, transformations and three-dimensional geometry worksheets.

Get out those rulers, protractors and compasses because we've got some great worksheets for geometry! The quadrilaterals are meant to be cut out, measured, folded, compared, and even written upon. They can be quite useful in teaching all sorts of concepts related to quadrilaterals. Just below them, you'll find worksheets meant for angle geometry. Also see the measurement page for more angle worksheets. The bulk of this page is devoted to transformations. Transformational geometry is one of those topics that can be really interesting for students and we've got enough worksheets for that geometry topic to keep your students busy for hours.

Don't miss the challenging, but interesting world of connecting cubes at the bottom of this page. You might encounter a few future artists when you use these worksheets with students.

Most Popular Geometry Worksheets this Week

Lines and Angles

In this section, there are worksheets for two of the basic concepts of geometry: lines and angles.

Lines (or straight lines to be precise) in geometry are continuous and extend in both directions to infinity. They have no width, depth or curvature. In math activities, they are often represented by a drawn straight path with some width. To show that they are lines, arrows are drawn on each end to show they extend to infinity. A line segment is a finite section of a line. Line segments are often represented with points at each end of a drawn straight path. Rays start at a point and extend in a straight line to infinity. This is shown with a point at one end of a drawn straight path and an arrow at the other end.

• Identifying Lines, Line Segments and Rays Identify Lines, Segments and Rays

Angles can be classified into six different types. Acute angles are greater than 0 degrees but less than 90 degrees. Right angles are exactly 90 degrees. Obtuse angles are greater than 90 degrees but less than 180 degrees. Straight angles are exactly 180 degrees. Reflex angles are greater than 180 degrees but less than 360 degrees. Complete/Full angles are exactly 360 degrees.

• Identifying Angle Types Worksheets Identifying Acute and Obtuse Angles Identifying Acute and Obtuse Angles (No Angle Marks) Identifying Acute, Obtuse and Right Angles Identifying Acute, Obtuse and Right Angles (No Angle Marks) Identifying Acute, Obtuse, Right and Straight Angles Identifying Acute, Obtuse, Right and Straight Angles (No Angle Marks) Identifying Acute, Obtuse, Right, Straight and Reflex Angles Identifying Acute, Obtuse, Right, Straight, Reflex and Complete/Full Angles

There are several angle relationships of which students should be aware. Complementary angles are two angles that together form a 90 degree angle; supplementary angles are two angles that together form a 180 degree angle; and explementary angles are two angles that together form a 360 degree angle. Vertical angles are found at line intersections; angles opposite each other are equal. Students can practice determining and/or calculating the unknown angle(s) in the following angle relationships worksheets.

• Angle Relationships Worksheets Complementary Angles Complementary Angles (Diagrams Rotated) Supplementary Angles Supplementary Angles (Diagrams Rotated) Mixed Complementary and Supplementary Angles Questions (Diagrams Rotated) Explementary Angles Explementary Angles (Diagrams Rotated) Mixed Adjacent Angles Questions (Diagrams Rotated) Vertical/Opposite Angles Vertical/Opposite Angles (Diagrams Rotated) Mixed Angle Relationships Questions(Diagrams Rotated)
• Angles of Transversals Intersecting Parallel Lines Interior Alternate Angles Exterior Alternate Angles Alternate Angles Corresponding Angles Co-Interior Angles Transversals

Measuring angles worksheets, can be found on the Measurement Page

Triangles, Quadrilaterals and Other Shapes

The quadrilaterals set can be used for a number of activities that involve classifying and recognizing quadrilaterals or for finding the properties of quadrilaterals (e.g. that the interior angles add up to 360 degrees). The tangram printables are useful in tangram activities. There are several options available for the tangram printables depending on your printer, and each option includes a large version and smaller versions. If you know someone with a suitable saw, you can use the tangram printable as a template on material such as quarter inch plywood; then simply sand and paint the pieces.

• Shape Sets Quadrilaterals Set Tangrams
• Identifying Regular Polygons Identifying Regular Shapes from Triangles to Octagons

Worksheets for classifying triangles by side and angle properties and for working with Pythagorean theorem.

If you are interested in students measuring angles and sides for themselves, it is best to use the versions with no marks. The marked versions will indicate the right and obtuse angles and the equal sides.

• Classifying Triangles Worksheets Classifying Triangles by Side Properties Classifying Triangles by Angle Properties Classifying Triangles by Side and Angle Properties Classifying Triangles by Side Properties (No Marks) Classifying Triangles by Angle Properties (No Marks) Classifying Triangles by Side and Angle Properties (No Marks)

A cathetus (plural catheti) refers to a side of a right-angle triangle other than the hypotenuse.

• Calculating Triangle Dimensions Using Pythagorean Theorem Calculate the Hypotenuse Using Pythagorean Theorem (No Rotation) Calculate the Hypotenuse Using Pythagorean Theorem Calculate a Cathetus Using Pythagorean Theorem (No Rotation) Calculate a Cathetus Using Pythagorean Theorem Calculate any Side Using Pythagorean Theorem (No Rotation) Calculate any Side Using Pythagorean Theorem

Trigonometric ratios are useful in determining the dimensions of right-angled triangles. The three basic ratios are summarized by the acronym SOHCAHTOA. The SOH part refers to the ratio: sin(α) = O/H where α is an angle measurement; O refers the length of the side (O)pposite the angle measurement and H refers to the length of the (H)ypotenuse of the right-angled triangle. The CAH part refers to the ratio: cos(α) = A/H where A refers to the length of the (A)djacent side to the angle. The TOA refers to the ratio: tan(α) = O/A.

• Calculating Angles and Sides Using Trigonometric Ratios Calculating Angles Using the Sine Ratio Calculating Sides Using the Sine Ratio Calculating Angles and Sides Using the Sine Ratio Calculating Angles Using the Cosine Ratio Calculating Sides Using the Cosine Ratio Calculating Angles and Sides Using the Cosine Ratio Calculating Angles Using the Tangent Ratio Calculating Sides Using the Tangent Ratio Calculating Angles and Sides Using the Tangent Ratio Calculating Angles Using Trigonometric Ratios Calculating Sides Using Trigonometric Ratios Calculating Angles and Sides Using Trigonometric Ratios

Quadrilaterals are interesting shapes to classify. Their classification relies on a few attributes and most quadrilaterals can be classified as more than one shape. A square, for example, is also a parallelogram, rhombus, rectangle and kite. A quick summary of all quadrilaterals is as follows: quadrilaterals have four sides. A square has 90 degree corners and equal length sides. A rectangle has 90 degree corners, but the side lengths don't have to be equal. A rhombus has equal length sides, but the angles don't have to be 90 degrees. A parallelogram has both pairs of opposite sides equal and parallel and both pairs of opposite angles are equal. A trapezoid only needs to have one pair of opposite sides parallel. A kite has two pairs of equal length sides where each pair is joined/adjacent rather than opposite to one other. A bowtie is sometimes included which is a complex quadrilateral with two sides that crossover one another, but they are readily recognizable. Any other four-sided polygon can safely be called a quadrilateral if it doesn't meet any of the criteria for a more specific classification.

• Classifying Quadrilaterals Classifying Simple Quadrilaterals Classifying All Quadrilaterals Classifying All Quadrilaterals (+ Rotation)

Coordinate Plane Worksheets

Coordinate point geometry worksheets to help students learn about the Cartesian plane.

• Plotting Random Coordinate Points Plotting Coordinate Points in All Quadrants Plotting Coordinate Points in Positive x Quadrants Plotting Coordinate Points in Positive y Quadrants

There are many other Cartesian Art plots scattered around the Math-Drills website as many of them are associated with a holiday. To find them quickly, use the search box.

• Cartesian Art Cartesian Art Maple Leaf
• Coordinate Plane Distance and Area Calculating Pythagorean Distances of Coordinate Points Calculating Perimeter and Area of Triangles on Coordinate Planes Calculating Perimeter and Area of Quadrilaterals on Coordinate Planes Calculating Perimeter and Area of Triangles and Quadrilaterals on Coordinate Planes

Transformations Worksheets

Transformations worksheets for translations, reflections, rotations and dilations practice.

Here are two quick and easy ways to check students' answers on the transformational geometry worksheets below. First, you can line up the student's page and the answer page and hold it up to the light. Moving/sliding the pages slightly will show you if the student's answers are correct. Keep the student's page on top and mark it or give feedback as necessary. The second way is to photocopy the answer page onto an overhead transparency. Overlay the transparency on the student's page and flip it up as necessary to mark or give feedback.

Also known as sliding, translations are a way to mathematically describe how something moves on a Cartesian plane. In translations, every vertex and line segment moves the same, so the resulting shape is congruent to the original.

• Translations Worksheets Translation of 3 vertices by up to 3 units. Translation of 3 vertices by up to 6 units. Translation of 3 vertices by up to 25 units. Translation of 4 vertices by up to 6 units. Translation of 5 vertices by up to 6 units.
• Translations Worksheets (Multi-Step) Two-Step Translation of 3 vertices by up to 6 units. Two-Step Translation of 4 vertices by up to 6 units. Three-Step Translation of 3 vertices by up to 6 units. Three-Step Translation of 4 vertices by up to 6 units.

Reflect on this: reflecting shapes over horizontal or vertical lines is actually quite straight-forward, especially if there is a grid involved. Start at one of the original points/vertices and measure the distance to the reflecting line. Note that you should measure perpendicularly or 90 degrees toward the line which is why it is easier with vertical or horizontal reflecting lines than with diagonal lines. Measure out 90 degrees on the other side of the reflecting line, the same distance of course, and make a point to represent the reflected vertex. Once you've done this for all of the vertices, you simply draw in the line segments and your reflected shape will be finished.

Reflecting can also be as simple as paper-folding. Fold the paper on the reflecting line and hold the paper up to the light. On a window is best because you will also have a surface on which to write. Only mark the vertices, don't try to draw the entire shape. Unfold the paper and use a pencil and ruler to draw the line segments between the vertices.

• Reflections Worksheets Reflection of 3 Vertices Over x = 0 and y = 0 Reflection of 4 Vertices Over x = 0 and y = 0 Reflection of 5 Vertices Over x = 0 and y = 0 Reflection of 3 Vertices Over Various Lines Reflection of 4 Vertices Over Various Lines Reflection of 5 Vertices Over Various Lines
• Reflections Worksheets (Multi-Step) Two-Step Reflection of 3 Vertices Over Various Lines Two-Step Reflection of 4 Vertices Over Various Lines Three-Step Reflection of 3 Vertices Over Various Lines Three-Step Reflection of 4 Vertices Over Various Lines

Here's an idea on how to complete rotations without measuring. It works best on a grid and with 90 or 180 degree rotations. You will need a blank overhead projector sheet or other suitable clear plastic sheet and a pen that will work on the page. Non-permanent pens are best because the plastic sheet can be washed and reused. Place the sheet over top of the coordinate axes with the figure to be rotated. With the pen, make a small cross to show the x and y axes being as precise as possible. Also mark the vertices of the shape to be rotated. Using the plastic sheet, perform the rotation, lining up the cross again with the axes. Choose one vertex and mark it on the paper by holding the plastic sheet in place, but flipping it up enough to get a mark on the paper. Do this for the other vertices, then remove the plastic sheet and join the vertices with line segments using a ruler.

• Rotations Worksheets Rotation of 3 Vertices around the Origin Starting in Quadrant I Rotation of 4 Vertices around the Origin Starting in Quadrant I Rotation of 5 Vertices around the Origin Starting in Quadrant I Rotation of 3 Vertices around the Origin Rotation of 4 Vertices around the Origin Rotation of 5 Vertices around the Origin Rotation of 3 Vertices around Any Point Rotation of 4 Vertices around Any Point Rotation of 5 Vertices around Any Point
• Rotations Worksheets (Multi-Step) Two-Step Rotations of 3 Vertices around Any Point Two-Step Rotations of 4 Vertices around Any Point Two-Step Rotations of 5 Vertices around Any Point Three-Step Rotations of 3 Vertices around Any Point Three-Step Rotations of 4 Vertices around Any Point Three-Step Rotations of 5 Vertices around Any Point
• Dilations Worksheets Dilations Using Center (0, 0) Dilations Using Various Centers
• Determining Scale Factors Worksheets Determine Scale Factors of Rectangles (Whole Numbers) Determine Scale Factors of Rectangles (0.5 Intervals) Determine Scale Factors of Rectangles (0.1 Intervals) Determine Scale Factors of Triangles (Whole Numbers) Determine Scale Factors of Triangles (0.5 Intervals) Determine Scale Factors of Triangles (0.1 Intervals) Determine Scale Factors of Rectangles and Triangles (Whole Numbers) Determine Scale Factors of Rectangles and Triangles (0.5 Intervals) Determine Scale Factors of Rectangles Triangles (0.1 Intervals)
• Mixed Transformations Worksheets (Multi-Step) Two-Step Transformations Three-Step Transformations

Constructions Worksheets

Constructions worksheets for constructing bisectors, perpendicular lines and triangle centers.

It is amazing what one can accomplish with a compass, a straight-edge and a pencil. In this section, students will do math like Euclid did over 2000 years ago. Not only will this be a lesson in history, but students will gain valuable skills that they can use in later math studies.

• Constructing Midpoints And Bisectors On Line Segments And Angles Midpoints on Horizontal Line Segments Perpendicular Bisectors on Horizontal Line Segments Perpendicular Bisectors on Rotated Line Segments Angle Bisectors (Angles not Rotated) Angle Bisectors (Angles Randomly Rotated)
• Constructing Perpendicular Lines Construct Perpendicular Lines Through Points on a Line Segment Construct Perpendicular Lines Through Points Not on Line Segment Construct Perpendicular Lines Through Points on Line Segment (Segments are randomly rotated) Construct Perpendicular Lines Through Points Not on Line Segment (Segments are randomly rotated)
• Constructing Triangle Centers Centroids for Acute Triangles Centroids for Mixed Acute and Obtuse Triangles Orthocenters for Acute Triangles Orthocenters for Mixed Acute and Obtuse Triangles Incenters for Acute Triangles Incenters for Mixed Acute and Obtuse Triangles Circumcenters for Acute Triangles Circumcenters for Mixed Acute and Obtuse Triangles All Centers for Acute Triangles All Centers for Mixed Acute and Obtuse Triangles

Three-Dimensional Geometry

Three-dimensional geometry worksheets that are based on connecting cubes and worksheets for classifying three-dimensional figures.

Connecting cubes can be a powerful tool for developing spatial sense in students. The first two worksheets below are difficult to do even for adults, but with a little practice, students will be creating structures much more complex than the ones below. Use isometric grid paper and square graph paper or dot paper to help students create three-dimensional sketches of connecting cubes and side views of structures.

• Connecting Cube Structures Side Views of Connecting Cube Structures Build Connecting Cube Structures
• Classifying Three-Dimensional Figures Classify Prisms Classify Pyramids Classify Prisms and Pyramids

This section includes a number of nets that students can use to build the associated 3D solids. All of the Platonic solids and many of the Archimedean solids are included. A pair of scissors, a little tape and some dexterity are all that are needed. For something a little more substantial, copy or print the nets onto cardstock first. You may also want to check your print settings to make sure you print in "actual size" rather than fitting to the page, so there is no distortion.

• Nets of Three-Dimensional Figures Nets of Platonic and Archimedean Solids Nets of All Platonic Solids Nets of Some Archimedean Solids Net of a Tetrahedron Net of a Cube Net of an Octahedron Net of a Dodecahedron (Version 1) Net of a Dodecahedron (Version 2) Net of an Icosahedron Net of a Truncated Tetrahedron Net of a Cuboctahedron Net of a Truncated Cube Net of a Truncated Octahedron Net of a Rhombicuboctahedron Net of a Truncated Cuboctahedron Net of a Snub Cube Net of an Icosidodecahedron

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Constructions

Geometric Constructions ... Animated!

"Construction" in Geometry means to draw shapes, angles or lines accurately.

These constructions use only compass, straightedge (a ruler, but not using the numbers) and a pencil.

This is the "pure" form of geometric construction: no numbers involved!

Learn these two first, they are used a lot:

Points and Lines

And it is useful to know how to do 30°, 45° and 60° angles. We can use the angle bisector method (above) to create other angles such as 15°, etc.

Triangle Basics

Triangle centers, circle basics, circles and tangents.

And for the "Master Class":

(Note: You can also see how to Use the Drafting Triangle and Ruler , Use the Protractor , and How to construct a Triangle With 3 Known Sides , but they are not "pure" geometric constructions.)

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Scale Drawing

Here we will learn about scale drawings, including creating scale drawings, using scale factors, and word problems.

There are also scale diagrams and drawings worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is a scale drawing?

A scale drawing is an enlargement of an object.

An enlargement changes the size of an object by multiplying each of the lengths by a scale factor to make it larger or smaller .

The scale of a drawing is usually stated as a ratio .

This is said as “ 1 centimetres to 5 metres” and means every 1 centimetre on the diagram represents 5 metres in real life.

In order to interpret and produce scale drawings we need to know the scale factor and the actual lengths of the object.

Below is a scale drawing of a pool with a scale of \bf{1cm:2m} or 1:200.

This means that every centimetre on the diagram represents 2 metres in real life. 

So the 6cm width of the diagram represents a 12m width on the real pool.

The real pool has been been enlarged by a scale factor of \frac{1}{200} to give the scale drawing.

The majority of scale drawing questions will involve polygons or 2D drawings such as floor plans or using a map although there is overlap with topics such as enlargement, construction of triangles , and loci . 

How to calculate the actual / real life distance from a scale

In order to calculate the actual/real life distance from a scale:

State the scale of the enlargement as a ratio in the form \bf{1:n} .

Multiply \bf{n} by the length given from the scale drawing.

Write the units.

Explain how to calculate the actual / real life distance from a scale

Scale drawing worksheet

Get your free scale drawing worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Related lessons on scale

Scale drawing is part of our series of lessons to support revision on scale . You may find it helpful to start with the main scale lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

• Scale maths
• Scale diagram

Scale drawing examples

Example 1: map.

A map has a scale of 1cm:2km. Find the actual distance represented by 5cm on the map.

2 Multiply \bf{n} by the length given from the scale drawing.

3 Write the units.

Example 2: plan of a building

A plan of a block of flats has the scale 2.5cm:800m. What is the real distance represented by 5.7cm on the plan?

Dividing both sides by 2.5, we get 1cm:320m

Example 3: atlas

A world atlas has a scale of 2cm:5 \ 000km. Calculate the real world distance that is represented by 23mm on the atlas.

As the length required is in millimetres, we first need to change the 2cm to millimetres.

2cm:5 \ 000km = 20mm:5 \ 000km

The we can reduce the ratio so we have 1 \ mm:n \ km. Dividing both sides by 20, we get:

How to calculate the scale drawing distance from a scale

In order to calculate the scale drawing distance from a scale:

Divide the real life distance by the scale ratio.

Calculate scale drawing distance examples

Example 4: model object.

A model car is made using the model to an actual distance of 1cm:40cm. The height of the car is 170cm. Calculate the height of the model car.

The ratio given is 1:40

Example 5: floor plan

A plan of a kitchen uses the scale 1cm:0.2m. Calculate the distance on the plan for the actual distance of 5.62m.

The ratio is already in the form 1:n .

Example 6: map of the UK

A map of the UK is drawn using the scale 50cm:1400km. Calculate how far 86.8km would be on the map.

Let’s convert the ratio to the form 1 \ cm:n \ km by dividing both sides of the ratio by 50.

50cm:1400km=1cm:28km

Now we can calculate the length on the map

86.8 \div 28=3.1

Common misconceptions

• Multiplying the real life distance by the ratio scale

The real life distance is multiplied by the ratio scale, giving an incorrect distance on the plan.

• Dividing the model / plan / map distance by the ratio scale

The distance on the model / plan / map is divided by the ratio scale meaning that the real life distance is incorrect.

• Ratio scale not simplified

If given the ratio 2cm:5km, it is easier to calculate when the ratio is in the form 1:n and so we must find an equivalent ratio before using the scale.

Here the ratio would be 1cm:2.5km so 1cm on the map would be equal to 2.5km in real life.

• Incorrect units in the solution

The units for the model are mixed up with the units for the real life distance. For example if we were calculating the distance of 10cm on a map with the scale ratio of 1cm:5km, \ 10 \times 5=50cm is stated whereas the correct solution would be 10 \times 5=50km.

• Converting units

Sometimes the units need to be converted so that you are working with the same units, making calculations much easier to comprehend. Make sure you know how to convert between different metric units. For example, the map scale is given as 1:25 \ 000 which means that 1cm on the map is equivalent to 25 \ 000cm in real life. If the answer is asked to be written in kilometres, the real life value in centimetres must be divided by 100 \ 000 to get the same measurement in kilometres. 1:25 \ 000 = 1cm:0.25km

Practice scale drawing questions

1. A map has a scale of 1cm:3km. How far is the actual distance of 3.5cm on the map?

2. An atlas has the scale 2cm:500km. Calculate the real distance for a length of 15cm on the map.

3. A plan of a bridge uses the scale 10cm:18.2m. The height of the bridge on the plan is 5.9cm. Calculate the real life height of the bridge.

4. A model elephant is created using a scale of 10cm:2.8m. A distance of 138.8cm is taken from the real life elephant. How long is this length on the model?

5. A map of the UK uses the scale 8m:1400km. Calculate the distance on the map that represents the actual length of 132km.

6. The plan of a garage is drawn using the drawing to actual scale of 0.9cm:20cm. What is the length of the garage on the plan, if it is 5.6m in real life?

Model to actual ratio is therefore 1cm:22.\dot{2}cm

Scale drawing GCSE questions

1. A map has a scale of 1cm:0.5km. A dog walker walks around 18km per day. Calculate how far they will have walked on a map.

2. (a) The distance between two cities in an American State is 146.8km. The distance between the two cities on a map is 29.36cm. Calculate the scale ratio of the map to the actual distance in the form 1cm:nkm.

(b) The river that travels through the state is measured on the plan to be 45cm. Calculate the actual length of the river. State the units in your solution.

3. Below is the map of a desert island.

Using the scale provided, calculate the distance between the Palm Trees and the Waterfall.

Distance [6.3-6.7cm]

Their 6.5 \times 50

Learning checklist

You have now learned how to:

• Multiplying and dividing by powers of 10 in scale drawings or by multiplying and dividing by powers of a 1,000 in converting between units such as kilometres and metres
• Solve problems involving similar shapes where the scale factor is known or can be found

The next lessons are

• Units of measurement
• Best buy maths

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