7.6B Volume and Surface Area Lesson Tutorials

7.6b Volume and Surface Area Lesson Tutorials

To ﬁ nd the volume of a rectangular prism, multiply the number of cubes needed to ﬁ ll the prism by the volume of one of the cubes.

EXAMPLE 1 Finding the Volume of a Rectangular Prism

Use cubes to ﬁ nd the volume of the rectangular prism. 3 in. 1 4 1 in. Use cubes with an edge length of — inch. 4 4 1 1 in. 1 in. 1 in. 4 4 1 in. 2 2 The bottom layer is 6 cubes long and 2 cubes wide. So, you need 6 × 2, or Check 12 cubes to cover the bottom layer. = = 2 cubes V Bh ℓwh 6 cubes 1 1 3 = 1 — — — ( 2)( 2)( 4) To ﬁ ll the prism, you need three layers of 3 1 3 12 cubes. So, you need 3 × 12, or 36 cubes. = — — — (2)(2)(4) 1 3 1 The volume of each cube is — = — cubic inch. 9 3 4 64 = — in. ✓ ( ) 36 cubes 16 1 9 So, the volume of the prism is 36 × — = — cubic inch. 64 16

Use cubes with the given edge length to ﬁ nd the volume of the rectangular prism. Check your answer using the volume formula. 1 1 1. Edge length: — cm 2. Edge length: — ft 5 6 1 ft 1 cm 3 5

4 1 cm ft 5 5 2 1 ft 1 cm 6 5

3. SINK A sink is shaped like a rectangular prism. How much water can the sink hold? 1 1 ft 3

1 1 ft 2 1 2 ft 4

317A Chapter 7 Equations

MMSCC6PE2_0706b.inddSCC6PE2_0706b.indd 4 111/17/101/17/10 9:32:539:32:53 AMAM The surface area S of a three-dimensional fi gure is the sum of the areas of its faces. Surface area is measured in square units. You can fi nd the surface area of a three-dimensional fi gure by using a two-dimensional representation of the fi gure called a net.

Net of a Rectangular Prism

top w w w

h Unfold. Flatten. sideback side front h w

bottom

EXAMPLE 2 Finding the Surface Area of a Rectangular Prism

Find the surface area of the rectangular prism. Use a net to ﬁ nd the area of each face. 3 in. 7 in. 4 in. 7 in. Top: 7 ⋅ 4 = 28 top 4 in. = Bottom: 7 ⋅ 4 28 4 in. 4 in. 7 in. Front: 7 ⋅ 3 = 21 side back side front 3 in. Back: 7 ⋅ 3 = 21 Side: 4 ⋅ 3 = 12 bottom Side: 4 ⋅ 3 = 12

Find the sum of the areas of the faces. Surface Area of Area of Area of Area of Area of Area of = + + + + + area top bottom front back a side a side S = 28 + 28 + 21 + 21 + 12 + 12 = 122 The surface area is 122 square inches.

Find the surface area of the rectangular prism. 4. 5. 6. 3.5 ft

1 5 m 6 in. 10 ft 2 6 m 9 m 8 ft 5 in. 5 in.

Volume and Surface Area 317B

MMSCC6PE2_0706b.inddSCC6PE2_0706b.indd 5 111/17/101/17/10 9:32:569:32:56 AMAM Net of a Triangular Prism A triangular prism is a three-dimensional ﬁ gure that has two triangular faces and three rectangular faces. Remember back The area A of a triangle with base b and height 1 Unfold. Flatten. side bottom side h is A = — bh. 2 front

EXAMPLE 3 Finding the Surface Area of a Triangular Prism

Find the surface area of the triangular prism. 13 cm 5 cm Use a net to ﬁ nd the area of each face.

8 cm Bottom: 12 ⋅ 8 = 96 13 cm 12 cm 5 cm 1 13 cm back Front: — ⋅ 12 ⋅ 5 = 30 2 12 cm 1 Back: — ⋅ 12 ⋅ 5 = 30 side bottom side 8 cm 2 Side: 13 ⋅ 8 = 104 front Side: 8 ⋅ 5 = 40

Find the sum of the areas of the faces.

Surface Area of Area of Area of Area of Area of = + + + + area bottom front back a side a side

S = 96 + 30 + 30 + 104 + 40 = 300 The surface area is 300 square centimeters.

Find the surface area of the triangular prism. 7. 8. 9. 5 yd 10 m 7.6 ft 10 m 6.3 ft

3 yd 6 m 9 m 4 yd 16 m 4 yd 10.3 ft 8.6 ft

317C Chapter 7 Equations

MMSCC6PE2_0706b.inddSCC6PE2_0706b.indd 6 111/17/101/17/10 9:32:579:32:57 AMAM Net of a Square Pyramid A square pyramid is a three-dimensional ﬁ gure that has one square face and four identical triangular faces.

side

Unfold. Flatten. sidebottom side

side

EXAMPLE 4 Finding the Surface Area of a Square Pyramid

Find the surface area of the square pyramid. 10 m Use a net to ﬁ nd the area of each face.

Bottom: 7 ⋅ 7 = 49 sides —1 = 7 m Side: ⋅ 7 ⋅ 10 35 7 m 2 10 m 1 Side: — ⋅ 7 ⋅ 10 = 35 7 m bottom 2 1 Side: — ⋅ 7 ⋅ 10 = 35 2 sides 1 Side: — ⋅ 7 ⋅ 10 = 35 2 Find the sum of the areas of the faces. Surface Area of Area of Area of Area of Area of = + + + + area bottom a side a side a side a side S = 49 + 35 + 35 + 35 + 35 = 189 The surface area is 189 square meters.

Find the surface area of the square pyramid. 10. 11. 12. 5 cm 3 ft 3.6 in.

2 ft 5 cm 4 in.

Volume and Surface Area 317D

MMSCC6PE2_0706b.inddSCC6PE2_0706b.indd 7 111/17/101/17/10 9:32:599:32:59 AMAM