unit 5 homework 2 geometry

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

Geometry (all content)

Unit 1: lines, unit 2: angles, unit 3: shapes, unit 4: triangles, unit 5: quadrilaterals, unit 6: coordinate plane, unit 7: area and perimeter, unit 8: volume and surface area, unit 9: pythagorean theorem, unit 10: transformations, unit 11: congruence, unit 12: similarity, unit 13: trigonometry, unit 14: circles, unit 15: analytic geometry, unit 16: geometric constructions, unit 17: miscellaneous.

• Precalculus
• Signup / Login

About This Course

Welcome to the Math Medic Geometry course! Here you will find a ready-to-be-taught lesson for every day of the school year, along with expert tips and questioning techniques to help the lesson be successful. Each lesson is designed to be taught in an Experience First, Formalize Later (EFFL) approach, in which students work in small groups on an engaging activity before the teacher formalizes the learning.

Our Geometry course develops reasoning, justification, and proof skills through an in-depth study of shapes and their properties, rigid transformations and congruence, and the relationship between similarity and right triangle trigonometry. Rich opportunities for problem solving culminate in the unit on surface area and volume. This course was created using the Common Core State Standards as a guide. The standards taught in each Math Medic Geometry lesson can be found here . Additionally, we've chosen to include a unit on Statistics and Probability that can be used as a stand-alone unit at any time during high school course work. The unit overviews and learning targets for the Math Medic Geometry course can be found here .

Math Medic Help

Slicing Solids

2.1: Slice This (5 minutes)

CCSS Standards

Building On

• HSG-GMD.B.4

Building Towards

• HSG-GMD.A.1

Routines and Materials

Instructional Routines

• Think Pair Share

Required Materials

• Cylindrical food items

The purpose of this activity is for students to visualize what a cross section might look like and then test the prediction by observing the result of slicing through a solid. Cylindrical food items, such as cheese or carrots, are convenient examples.

Arrange students in groups of 2. Tell students that a cross section is the intersection between a solid and a plane, or a two-dimensional figure that extends forever in all directions. Using a cylindrical food item such as cheese or carrots, or another cylindrical object, demonstrate that slicing a cylinder parallel to its base produces a circular cross section.

Then, give students quiet work time and then time to share their work with a partner.

Student Facing

Imagine slicing a cylinder with a straight cut. The flat surface you sliced along is called a cross section . Try to sketch all the possible kinds of cross sections of a cylinder.

Student Response

For access, consult one of our IM Certified Partners .

Anticipated Misconceptions

Students may not consider non-horizontal or non-vertical cross sections at first. Remind them that a cross section is the intersection of any plane with a solid—the plane doesn't have to be vertical or horizontal.

Activity Synthesis

Ask students to share their predictions of what the cross sections will look like. Demonstrate slicing each cylindrical food item according to student instructions to see several examples.

2.2: Slice That (20 minutes)

• MLR7: Compare and Connect
• Dental floss

In this activity, students continue to develop familiarity with three-dimensional solids and their cross sections. Students use spatial visualization to predict what cross sections might look like and then test their predictions.

This activity works best when each student has access to devices that can run the applet because students will benefit from seeing the relationship in a dynamic way. If students don’t have individual access, projecting the applet would be helpful during the synthesis.

Arrange students in groups of 3–4. Ask students to think about definitions of some geometric solids: spheres, prisms, pyramids, cones, and cylinders. Give students some quiet work time and then time to share their work with a partner. Follow with a whole-class discussion.

A sphere is the set of points in three-dimensional space the same distance from some center. A prism  has two congruent faces  (or sides) that are called bases. The bases are connected by quadrilaterals. A  cylinder  is like a prism except the bases are circles. A  pyramid  has one base. The remaining faces are triangles that all meet at a single vertex. A  cone is like a pyramid except the base is a circle.

The triangle is a cross section formed when the plane slices through the cube.

• Sketch predictions of all the kinds of cross sections that could be created as the plane moves through the cube.
• The 3 red points control the movement of the plane. Click on them to move them up and down or side to side. You will see one of these movement arrows appear. Sketch any new cross sections you find after slicing.

Are you ready for more?

Delete the cube and build another solid by following the directions in its Tooltip. Make predictions about the the kinds of cross sections that could be created if the plane moves through the solid. Move your plane to confirm.

A sphere is the set of points in three-dimensional space the same distance from some center. A prism  has two congruent faces  (or sides) that are called bases. The bases are connected by quadrilaterals. A  cylinder  is like a prism except the bases are circles. A  pyramid  has one base. The remaining faces are triangles that all meet at a single vertex.  A  cone is like a pyramid except the base is a circle.

Give each group clay or playdough formed into the shape of a three-dimensional solid (cube, sphere, cylinder, cone, or other solids), and dental floss to slice the clay. Tell students that to view multiple cross sections, they will slice the shape, then re-form the shape and slice again.

An alternative is to find food items with interesting cross sections or three-dimensional foam solids from a craft store and providing plastic knives to slice the solids. In this case, provide each group with several of the same solid so they can experiment with multiple slices.

Try to include a sphere, a cube, and a cone in the collection of solids.

Your teacher will give your group a three-dimensional solid to analyze.

• Sketch predictions of all the kinds of cross sections that could be created from your solid.
• Slice your solid to confirm your predictions. Sketch any new cross sections you find after slicing.

If using the paper and pencil version of this activity and students are stuck, suggest they slice their solids at different angles and locations to see if different cross sections are generated.

Invite groups of students with different solids to share their list of cross sections with the class. Ask students:

• “Were there any cross sections that caught you by surprise?” (It was surprising that a cube can have cross sections that are triangles, quadrilaterals, pentagons, and hexagons.)
• “Compare and contrast the different cross sections of a sphere.” (All the cross sections were circles, but they were different sizes.)
• “How are a cube’s cross sections different from a sphere’s?” (The cube has many differently-shaped cross sections, while the sphere’s cross sections are all circles.)

2.3: Stack ‘Em Up (10 minutes)

• MLR8: Discussion Supports

In the last activity, students started with solids and identified various cross sections. In this activity, students view three-dimensional slabs of a solid between parallel cross sections and try to determine what the original solid was. Being able to visualize the relationship between a solid and its cross sections is important to later work on Cavalieri’s Principle.

Ask students, “What solid would a stack of all the same coins create?” Display a stack of quarters and note that it creates the shape of a cylinder. Then display, in order, a quarter, a nickel, a penny, and a dime. Ask, “What solid would a stack of coins decreasing in size create?” Make a stack with a few of each type of coin to make a solid that resembles a cone.

Each question shows several parallel cross-sectional slabs of the same three-dimensional solid. Name each solid.

Description: <p>8 geometric shapes in this order. Small triangle, medium triangle, larger triangle with corners cut off, hexagon, an odd shaped hexagon, larger triangle with corners cut off, medium triangle, small triangle.</p>

3D-printers stack layers of material to make a three-dimensional shape. Computer software slices a digital model of an object into layers, and the printer stacks those layers one on top of another to replicate the digital model in the real world.

Attribution: Toy Rocket, by fernandozhiminaicela. Public Domain. Pixabay. Source .

• Draw 3 different horizontal cross sections from the object in the image.
• The layers can be printed in different thicknesses. How would the thickness of the layers affect the final appearance of the object?
• Suppose we printed a rectangular prism. How would the thickness of the layers affect the final appearance of the prism?

Ask students to share their predictions for what solids are formed. Then display these images for all to see.

Now focus students’ attention on cross sections that are taken parallel to a solid’s base (for those solids that have bases). Ask students how cross sections can be used to differentiate between prisms and pyramids. (The cross sections of prisms taken parallel to the base are congruent to each other. The cross sections of pyramids taken parallel to the base are similar to each other.)

Lesson Synthesis

In this lesson, students worked with three-dimensional solids and their cross sections . Here are questions for discussion:

• “How are the cross sections in this lesson different from the two-dimensional figures we looked at in the last lesson?” (In the last lesson, we rotated the two-dimensional figures to trace out a solid. The two-dimensional figures were usually an outline of half of the figure, and they had to have a relationship to the axis of rotation of the solid. Here, our cross sections cut through the entire solid, and they can come from anywhere in the solid.)
• “What kinds of applications of cross sections might we see in real life?” (There is a field of medicine called tomography that is about finding ways to get images of cross sections of people. Technologies like the CAT scan, the MRI, and the PET scan allow doctors to examine cross sections of a brain, a lung, or an injury and visualize what the three-dimensional body part looks like.)

2.4: Cool-down - Sketch It (5 minutes)

Student lesson summary.

In earlier grades, you learned some vocabulary terms about solid geometry: A sphere is the set of points in three-dimensional space the same distance from some center. A prism has two congruent faces (or sides) that are called bases. The bases are connected by parallelograms. A cylinder is like a prism except the bases are circles. A pyramid has one base. The remaining faces are triangles that all meet at a single vertex. A cone is like a pyramid except the base is a circle.

We often analyze cross sections of solids. A cross section is the intersection of a solid with a plane , or a two-dimensional figure that extends forever in all directions. For example, some cheese is sold in cylindrical blocks. If you stand the cheese on end and slice vertically, you will get a rectangle, as shown. This rectangle is a cross section of the cylinder.

Here are 3 more examples of cross sections created by intersecting a plane and a cylinder.

If you wanted to serve your cylindrical cheese at a party, you might cut it into several pieces, like this. The pieces are thin cylinders. They are like cross sections, but they are three-dimensional. All the cuts were made parallel to one another. By looking at the slices, or by stacking them up, you could figure out that the original shape of the cheese was a cylinder.

What if another cheese plate contained slices whose radii got bigger to a maximum size and then got smaller again? The cheese was probably in the shape of a sphere. A sphere has circular cross sections. The size of the circular cross sections increases as you get closer to the center of the sphere, then decreases past the center.

Slicing Solids

Lesson Narrative

In grade 7, students described the two-dimensional figures that result from slicing three-dimensional figures. Here, these concepts are revisited with some added complexity. Students analyze cross sections , or the intersections between planes and solids, by slicing three-dimensional objects. Next, they identify three-dimensional solids given parallel cross-sectional slices. In addition, they revisit solid geometry vocabulary terms from earlier grades: sphere , prism , cylinder , cone , pyramid , and faces .

Spatial visualization in three dimensions is an important skill in mathematics. Understanding the relationship between solids and their parallel cross sections will be critical to understanding Cavalieri’s Principle in later lessons. Cavalieri’s Principle will be applied to the development of the formula for the volume of pyramids and cones. Students use spatial visualization to make sense of three-dimensional figures and their cross sections throughout the lesson (MP1).

Learning Goals

Teacher Facing

• Generate multiple cross sections of three-dimensional figures.
• Identify the three-dimensional shape resulting from combining a set of cross sections.

Student Facing

• Let’s analyze cross sections by slicing three-dimensional solids.

Required Materials

• Cylindrical food items
• Dental floss

Required Preparation

Obtain several cylindrical food items to cut with a plastic knife.

Devices are required for the digital version of the activity Slice That. If using the paper and pencil version, prepare various solids from clay or play dough, such as cubes, spheres, cones, and cylinders. Each group of 3-4 students should have access to a three-dimensional solid to analyze.

Alternatively, you might consider getting food items from the grocery store with interesting cross sections or three-dimensional foam solids from a craft store, and plastic knives to slice the solids.

Learning Targets

• I can identify the three-dimensional shape that generates a set of cross sections.
• I can visualize and draw multiple cross sections of a three-dimensional figure.

CCSS Standards

Building On

• HSG-GMD.B.4

Building Towards

• HSG-GMD.A.1

Glossary Entries

A cone is a three-dimensional figure with a circular base and a point not in the plane of the base called the apex. Each point on the base is connected to the apex by a line segment.

The figure formed by intersecting a solid with a plane.

A cylinder is a three-dimensional figure with two parallel, congruent, circular bases, formed by translating one base to the other. Each pair of corresponding points on the bases is connected by a line segment.

Any flat surface on a three-dimensional figure is a face.

A cube has 6 faces.

A prism is a solid figure composed of two parallel, congruent faces (called bases) connected by parallelograms. A prism is named for the shape of its bases. For example, if a prism’s bases are pentagons, it is called a “pentagonal prism.”

rectangular prism

triangular prism

pentagonal prism

A pyramid is a solid figure that has one special face called the base. All of the other faces are triangles that meet at a single vertex called the apex. A pyramid is named for the shape of its base. For example, if a pyramid’s base is a hexagon, it is called a “hexagonal pyramid.”

square pyramid

pentagonal pyramid

A sphere is a three-dimensional figure in which all cross-sections in every direction are circles.

Print Formatted Materials

Teachers with a valid work email address can  click here to register or sign in for free access to Cool Down, Teacher Guide, and PowerPoint materials.

Additional Resources