Lesson 15

Putting All the Solids Together

  • Let’s calculate volumes of prisms, cylinders, cones, and pyramids.

Problem 1

A right cone has a base with diameter 10 units. The volume of the cone is \(100\pi\) cubic units. What is the length of a segment drawn from the apex to the edge of the circular base?

Problem 2

A right pyramid has a square base with sides of length 10 units. Each segment connecting the apex to a midpoint of a side of the base has length 13 units. What is the volume of the pyramid?

Problem 3

For each pair of solids, determine if their volumes are the same or different. If the volumes are different, identify the solid with the greatest volume. Explain your reasoning.

  1. A prism and a pyramid have the same height. The pyramid’s base has 3 times the area of the prism's base.
  2. A pyramid and a cylinder have bases with the same area. The cylinder’s height is 3 times that of the pyramid.
  3. A cone and a cylinder have the same height. The cone’s radius is 3 times the length of the cylinder’s radius.

Problem 4

A pyramid has a height of 8 inches and a volume of 120 cubic inches. Determine 2 possible shapes, with dimensions, for the base. 

(From Unit 5, Lesson 14.)

Problem 5

A toy company packages modeling clay in the shape of a rectangular prism with dimensions 6 inches by 1 inch by \(\frac12\) inch. They want to change the shape to a rectangular pyramid that uses the same amount of clay. Determine 2 sets of possible dimensions for the pyramid.

(From Unit 5, Lesson 14.)

Problem 6

These 3 congruent square pyramids can be assembled into a cube with side length 2 feet. What is the volume of each pyramid?

Rectangular pyramid 
Rectangular pyramid 
Rectangular pyramid 
(From Unit 5, Lesson 12.)

Problem 7

A monster truck wheel has an area of \(324\pi\) square inches. A toy company wants to create a scaled copy of the monster truck with a wheel area of \(9\pi\) square inches. What scale factor should the company use?

(From Unit 5, Lesson 7.)