Building a Volume Formula for a Pyramid

ID: kasip-nojig
Illustrative Mathematics, CC BY 4.0
Subject: Geometry

14 questions

Building a Volume Formula for a Pyramid

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Problem 1

1) 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

A cylinder has radius 9 inches and height 15 inches. A cone has the same radius and height. Leave answers in exact form, in terms of ﻿$\pi$﻿ if necessary.

2) Find the volume of the cylinder.

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3) Find the volume of the cone.

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4) What fraction of the cylinder’s volume is the cone’s volume?

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Problem 3

Each solid in the image has height 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 scale factor ﻿$k = 0.25$﻿.

5) Calculate the area of each of the 2 cross sections.

6) Suppose a new cross section was created in each solid, both at the same height, using some scale factor ﻿$k$﻿. What would the area of these cross sections be?

Problem 4

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

a)

Triangular pyramid

b)

Regular prism

c)

Square prism

d)

Right triangular prism

Problem 5

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

a)

Rectangular pyramid

b)

Triangular pyramid

c)

Right triangular prism

d)

Rectangular prism

Problem 6

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

Problem 7

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.

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

12) 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?

13) Explain or show your reasoning.

Problem 8

14) 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?

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