Unit 5, Lesson 13

Building a Volume Formula for a Pyramid

Geometry

Practice

Problem 1
Student Task Statement
Find the volume of a pyramid whose base is a square with side lengths of 6 units and height of 8 units.

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Problem 2
Student Task Statement
A cylinder has radius 9 inches and height 15 inches. A cone has the same radius and height.

Find the volume of the cylinder.

Find the volume of the cone.

What fraction of the cylinder’s volume is the cone’s volume?
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Problem 3
Student Task Statement
Each solid in the image has a height of 4 units. The area of each solid’s base is 8 square units. A cross-section has been created in each by dilating the base using the apex as a center with a scale factor of .

Calculate the area of each of the 2 cross-sections.

Suppose a new cross-section was created in each solid, both at the same height, using some scale factor . How would the areas of these 2 cross-sections compare? Explain your reasoning.
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Problem 4
ForLesson 12: Prisms and Pyramids
Student Task Statement
Select the most specific and accurate name for the solid in the image.

triangular pyramid

regular prism

square prism

right triangular prism
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Problem 5
ForLesson 12: Prisms and Pyramids
Student Task Statement
A solid can be constructed with 4 triangles and 1 rectangle. What is the name for this solid?

rectangular pyramid

triangular pyramid

right triangular prism

rectangular prism
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Problem 6
ForLesson 11: Prisms Practice
Student Task Statement
Find the volume of the solid produced by rotating this two-dimensional shape around the axis shown.

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Problem 7
ForLesson 10: Cross-Sections and Volume
Student Task Statement
This zigzag crystal vase has a height of 20 centimeters. The cross-sections parallel to the base are always rectangles that are 12 centimeters wide by 6 centimeters long.

If we assume the crystal itself has no thickness, what would be the volume of the vase?

The crystal is actually 1 centimeter thick on each of the sides and on the bottom. Approximately how much space is contained within the vase? Explain or show your reasoning.
to access Practice Problem Solutions.
Problem 8
ForLesson 5: Scaling and Unscaling
Student Task Statement
A trapezoid has an area of 10 square units. What scale factor would be required to dilate the trapezoid to have an area of 90 square units?

9

6

3
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Practice

Problem 1

Student Task Statement

Find the volume of a pyramid whose base is a square with side lengths of 6 units and height of 8 units.

to access Practice Problem Solutions.

Problem 2

Student Task Statement

A cylinder has radius 9 inches and height 15 inches. A cone has the same radius and height.

Find the volume of the cylinder.
Find the volume of the cone.
What fraction of the cylinder’s volume is the cone’s volume?

to access Practice Problem Solutions.

Problem 3

Student Task Statement

Each solid in the image has a height of 4 units. The area of each solid’s base is 8 square units. A cross-section has been created in each by dilating the base using the apex as a center with a scale factor of .

<p>A pyramid with a triangular base. Another pyramid with a rectangle base.</p>

Calculate the area of each of the 2 cross-sections.
Suppose a new cross-section was created in each solid, both at the same height, using some scale factor . How would the areas of these 2 cross-sections compare? Explain your reasoning.

to access Practice Problem Solutions.

Problem 4

ForLesson 12: Prisms and Pyramids

Student Task Statement

Select the most specific and accurate name for the solid in the image.

<p>5 sided solid. Two right triangles and 3 rectangles.<br>
</p>

triangular pyramid
regular prism
square prism
right triangular prism

to access Practice Problem Solutions.

Problem 5

ForLesson 12: Prisms and Pyramids

Student Task Statement

A solid can be constructed with 4 triangles and 1 rectangle. What is the name for this solid?

rectangular pyramid
triangular pyramid
right triangular prism
rectangular prism

to access Practice Problem Solutions.

Problem 6

ForLesson 11: Prisms Practice

Student Task Statement

Find the volume of the solid produced by rotating this two-dimensional shape around the axis shown.

<p>Grid, horizontal axis 1 block above the bottom. Rectangle 1 block above the axis, 2 blocks tall and 6 blocks wide.</p>

to access Practice Problem Solutions.

Problem 7

ForLesson 10: Cross-Sections and Volume

Student Task Statement

This zigzag crystal vase has a height of 20 centimeters. The cross-sections parallel to the base are always rectangles that are 12 centimeters wide by 6 centimeters long.

<p>Photograph of a zig zag crystal vase </p>

If we assume the crystal itself has no thickness, what would be the volume of the vase?
The crystal is actually 1 centimeter thick on each of the sides and on the bottom. Approximately how much space is contained within the vase? Explain or show your reasoning.

to access Practice Problem Solutions.

Problem 8

ForLesson 5: Scaling and Unscaling

Student Task Statement

A trapezoid has an area of 10 square units. What scale factor would be required to dilate the trapezoid to have an area of 90 square units?

to access Practice Problem Solutions.

Building a Volume Formula for a Pyramid

Practice

Problem 1

Student Task Statement

Problem 2

Student Task Statement

Problem 3

Student Task Statement

Problem 4

Student Task Statement

Problem 5

Student Task Statement

Problem 6

Student Task Statement

Problem 7

Student Task Statement

Problem 8

Student Task Statement

Lesson 13 Resources

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Building a Volume Formula for a Pyramid

Practice

Problem 1

Student Task Statement

Problem 2

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Problem 3

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Problem 4

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Problem 5

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Problem 6

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Problem 7

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Problem 8

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Lesson 13 Resources