MG1.3 Know and use the formulas for the volume of triangular prisms and cylinders (area of base × height); compare these formulas and explain the similarity between them and the formula for the volume of a rectangular solid. Also covered: AF3.1, AF3.2 California Standards

Vocabulary Volume Volume of Prisms Formula

Volume is the number of cubic units needed to fill a space.

You need 10, or 5 · 2, centimeter cubes to cover the bottom of this rectangular prism. You need 3 layers of 10 cubes each to fill the prism. It takes 30, or 5 · 2 · 3, cubes.

Volume is expressed in cubic units, so the volume of the prism is 5 cm · 2 cm · 3 cm = 30 cubic centimeters, or 30 cm3.

The volume of a rectangular prism is the area of its base times its height. This formula can be used to find the volume of any prism.

Teacher Example 1: Finding the Volume of a Rectangular Prism Find the volume of the rectangular prism. Step 1: Find the area of the base. 13 in. 11 in. 26 in. B = 26 · 11 The base is a rectangle. B = 286 Multiply.

Teacher Example 1 Continued: Find the volume of the rectangular prism. Step 2: Find the volume. 13 in. 11 in. 26 in. V = Bh Write the formula. V = 286 • 13 Substitute for B and h. V = 3,718 in3 Multiply. The volume of the prism is 3,718 in3.

Student Practice 1: Find the volume of the rectangular prism. Step 1: Find the area of the base. 16 in. 12 in. 29 in. B = 29 · 12 The base is a rectangle. B = 348 Multiply.

Student Practice 1 Continued: Find the volume of the rectangular prism. Step 2: Find the volume. 16 in. 12 in. 29 in. V = Bh Write the formula. V = 348 • 16 Substitute for B and h. V = 5,568 in3 Multiply. The volume of the prism is 5,568 in3.

You can also use the formula V= Bh to find the volume of a triangular prism. For triangular prisms, B represents the area of a triangle, rather than a rectangle.

Teacher Example 2: Finding the Volume of a Triangular Prism Find the volume of each triangular prism. A. Step 1: Find the area of the base. B = ( • 3.9 • 1.3) 1 2 __ The base is a triangle. B = 2.535 Multiply.

Teacher Example 2A: Finding the Volume of a Triangular Prism Continued Find the volume of each triangular prism. A. Step 2: Find the volume. V = Bh Write the formula. V = 2.535 • 4 Substitute for B and h. V = 10.14 m3 Multiply. The volume of the prism is 10.14 m3.

The height of a prism is the distance between its two bases. Caution!

Teacher Example 2B: Finding the Volume of a Triangular Prism Find the volume of each triangular prism. B. Step 1: Find the area of the base. B = ( • 6.5 • 7) 1 2 __ The base is a triangle. B = 22.75 Multiply.

Teacher Example 2B: Finding the Volume of a Triangular Prism Continued Find the volume of each triangular prism. B. Step 2: Find the volume. V = Bh Write the formula. V = 22.75 • 6 Substitute for B and h. V = 136.5 ft3 Multiply. The volume of the prism is 136.5 ft3.

Student Practice 2A: Find the volume of each triangular prism. A. Step 1: Find the area of the base. 7 m 1.6 m 4.2 m B = ( • 4.2 • 1.6) 1 2 __ The base is a triangle. B = 3.36 Multiply.

Find the volume of each triangular prism. Student Practice 2A Continued: Find the volume of each triangular prism. A. Step 2: Find the volume. 7 m 1.6 m 4.2 m V = Bh Write the formula. V = 3.36 • 7 Substitute for B and h. V = 23.52 m3 Multiply. The volume of the prism is 23.52 m3.

Student Practice 2B: Find the volume of each triangular prism. B. Step 1: Find the area of the base. 9 ft 5 ft 4.5 ft B = ( • 4.5 • 9) 1 2 __ The base is a triangle. B = 20.25 Multiply.

Student Practice 2B Continued: Find the volume of each triangular prism. B. Step 2: Find the volume. 9 ft 5 ft V = Bh 4.5 ft Write the formula. V = 20.25 • 5 Substitute for B and h. V = 101.25 ft3 Multiply. The volume of the prism is 101.25 ft3.

Teacher Example 3: Application An artist wants to make glass paper-weights with the dimensions shown. He estimates that he will need less than 20 cubic centimeters of glass for each paperweight. Is his estimate reasonable? Explain. 5.2 cm 3 cm Step 1: Find the area of the base. 6 cm B = • 6 • 5.2 = 15.6 1 2 __ The base is a triangle.

Teacher Example 3 Continued An artist wants to make glass paper-weights with the dimensions shown. He estimates that he will need less than 20 cubic centimeters of glass for each paperweight. Is his estimate reasonable? Explain. 5.2 cm 3 cm Step 2: Find the volume. 6 cm V = Bh Write the formula. V = 15.6 • 3 = 46.8 cm3 Substitute for B and h. No; each paperweight will require about 47 cm3 of glass.

Step 1: Find the area of the base. Student Practice 3: An architect wants to make a model building with the dimensions shown. He estimates that he will need more than 60 cubic centimeters of paper for each building. Is his estimate reasonable? Explain. 5.5 cm 5 cm Step 1: Find the area of the base. 6 cm B = • 6 • 5.5 = 16.5 1 2 __ The base is a triangle.

Student Practice 3 Continued: An architect wants to make a model building with the dimensions shown. He estimates that he will need more than 60 cubic centimeters of paper for each building. Is his estimate reasonable? Explain. 5.5 cm 5 cm Step 2: Find the volume. 6 cm V = Bh Write the formula. V = 16.5 • 5 = 82.5 cm3 Substitute for B and h. No; each building will require about 83 cm3 of paper.

10.8 Warm-Up Find the volume of each figure. 1. triangular prism with a height of 12 cm and a triangular base with base length 7.3 cm and height 3.5 cm 2. Find the volume of the figure shown.