Lesson 3.7

Core Focus on Shapes & Angles Volume of Pyramids

Warm-Up Find the volume of each prism. 1.

2 yd

2. 15 cm

5 yd 14 yd

7 cm 11 cm

1,155 cm3

70 yd 3

Lesson 3.7

Volume of Pyramids Calculate the volume of pyramids.

Explore!

Pyramid Vs Prism

Make or find a pyramid and prism that have congruent bases and are the same height to use in this activity. Step 1 Estimate how many times larger the volume of the prism is compared to the pyramid. Step 2 Fill the pyramid with rice, beans or popcorn kernels. Pour the contents of the pyramid into the prism. Repeat until the prism is full. Step 3 How many times did you need to empty the pyramid to fill the prism? Step 4 What fraction is the volume of the pyramid compared to the prism? Step 5 The volume of a prism is found using the formula V = Bh. How could you modify this formula to find the volume of a pyramid? Write the formula for the volume of a pyramid. Step 6 Use the formula to find the volume of each pyramid.

Volume of a Pyramid The volume of a pyramid is equal to one-third of the product of the base area (B) and the height (h).

1 V  (area of the base)(height) 3 1 Bh V  Bh or V  3 3

Example 1 A 15-foot tall pentagonal pyramid has a base area of 38 square feet. Find the volume of the pyramid.

Write the formula needed.

V  13 Bh

Substitute known values.

V  13 (38)(15)

Multiply.

V  190

The volume of the pyramid is 190 cubic feet.

Extra Example 1 A square pyramid is 8 yards on each side. The pyramid is 11 yards tall. Find the volume of the pyramid.

Example 2 Crystal received an award for being an outstanding hospital patient. The award was shaped like a square pyramid. The base of the pyramid was 8 cm on each side. It was 10 cm tall. Find the volume of the award. Round to the nearest whole number. Find the area of the base.

B = 8(8) = 64

Use the volume formula for a pyramid.

V  13 Bh

Substitute all known values for the variables.

V  13 (64)(10)

Multiply.

V  213.3

Round to the nearest whole number.

 213

The volume of Crystal’s award was a little more than 213 cm3.

Extra Example 2 The base of a pyramid is a triangle with sides that are 9 meters long and a height of 7.8 meters. The pyramid stands 17 meters tall. What is the volume of the pyramid?

198.9 m³

Communication Prompts 1. Your best friend was absent from math class today. Write them a note about the volume of a pyramid so they will be able to complete the assignment on their own.

2. Why do you think foods are not often packaged in pyramidal packages?

Exit Problems 1. Find the volume of a pyramid with a base area of 297 square inches and a height of 19 inches. 1,881 cubic inches

2. Find the volume of the pyramid shown below. 51.2 m3 9.6 m

4m 4m