Spectrum Math Grade 8 Chapter 5 Lesson 11 Answer Key Volume: Cones

Students can use the Spectrum Math Grade 8 Answer Key Chapter 5 Lesson 5.11 Volume: Cones as a quick guide to resolve any of their doubts.

Spectrum Math Grade 8 Chapter 5 Lesson 5.11 Volume: Cones Answers Key

Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².

V = \(\frac{1}{3}\)πr²h Volume is given in cubic units, or units³.
If the height of a cone is 7 cm and radius is 3 cm, what is the volume?
Use 3.14 for π. V = \(\frac{1}{3}\)π3²7 V = \(\frac{\pi 63}{3}\) V = π21 V = 65.94 cm³

If you do not know the height but you do know the radius and the length of the side, you can use the Pythagorean Theorem to find the height.
What is b? a² + b² = c² 81 + b² = 225 b² = 144 b = 12 m
V = \(\frac{1}{3}\)πr²h = \(\frac{1}{3}\)π9²12 = \(\frac{972 \pi}{3}\) = 324π = 1,017.36 m³

Find the volume of each cone. Use 3.1 4 for π. Remember that d = 2r. Round answers to the nearest hundredth.

Question 1.
a.

V = ____ in.³
Answer: 200.96 in.³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
height = 12 in.
radius = 4 in.
V = \(\frac{1}{3}\)π4² 12  = \(\frac{\pi 192}{3}\)  = π64  = 200.96 in.³

b.

V = ____ ft.³
Answer: 376.8 ft.³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius = 6 ft
height = 10 ft
V = \(\frac{1}{3}\)π6² 10  = \(\frac{\pi 360}{3}\)  = π120  = 376.8 ft.³

c.

V = ____ cm³
Answer: 39.25 cm³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius = d / 2= 5 / 2 = 2.5 cm
height = 6 cm
V = \(\frac{1}{3}\)π2.5² 6  = \(\frac{\pi 37.5}{3}\)  = π12.5  = 39.25 ft.³

Question 2.
a.

V = ____ m³
Answer: 769.3 m³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius = 7 m
height = 15 m
V = \(\frac{1}{3}\)π7² 15  = \(\frac{\pi 735}{3}\)  = π245 = 769.3 m³

b.

V = ____ in.³
Answer: 50.24 in.³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius = 4 in.
a² + b² = c²
16 + b² = 25
b = 3
height = 3 in.
V = \(\frac{1}{3}\)π4² 3  = \(\frac{\pi 48}{3}\)  = π16 = 50.24 in.³

c.

V = ____ m³
Answer: 314 m³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius = 5 m
a² + b² = c²
25 + b² = 169
b = 12
height = 12 m
V = \(\frac{1}{3}\)π4² 12  = 314 m³

Question 3.
a.

V = ____ ft.³
Answer: 94.2 ft.³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius = d / 2= 6 / 2 = 3 ft
height = 10 ft
V = \(\frac{1}{3}\)π3² 10  = 94.2 ft.³

b.

V = ____ ft.³
Answer: 9231.6 ft.³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius =21 cm
a² + b² = c²
441 + b² = 841
b = 20
height = 20 cm
V = \(\frac{1}{3}\)π21² 20 = 9231.6 ft.³

c.

V = ____ in.³
Answer: 47.1 in.³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius = 3 in.
height = 5 in.
V = \(\frac{1}{3}\)π3² 5  = 47.1 in.³

Find the volume of each cone. Use 3.14 for π. Round answers to the nearest hundredth.

Question 1.
a.

V = ___ cm³
Answer: 167.47 cm³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius = 4 cm
height = 10 cm
V = \(\frac{1}{3}\)π4² 10  = 167.4666666 = 167.47 cm³

b.

V = ___ m³
Answer: 2786.23 m³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius = d / 2 = 22 / 2 = 11 m
height = 22 m
V = \(\frac{1}{3}\)π11² 22  = 2786.22666666 = 2786.23 m³

c.

V = ___ yd.³
Answer: 25.12 yd.³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius = d / 2 = 4 / 2 = 2 yd.
height = 6 yd.
V = \(\frac{1}{3}\)π2² 6  = 25.12 yd.³

Question 2.
a.

V = ___ m³
Answer: 314 m³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius = d / 2 = 10 / 2 = 5 m
height = 12 m
V = \(\frac{1}{3}\)π5² 12  = 314 m³

b.

V = ___ yd.³
Answer: 1526.04 yd.³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius = 9 yd.
height = 18 yd.
V = \(\frac{1}{3}\)π9² 18  = 1526.04 yd.³

c.

V = ___ m³
Answer: 103.62 m³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius = 3 m
height = 11 m
V = \(\frac{1}{3}\)π3² 11  = 103.62 m³

Question 3.
a.

V = ___ ft.³
Answer: 29.31 ft.³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius = d /  2 = 4 / 2 = 2 ft.
height = 7 ft.
V = \(\frac{1}{3}\)π2² 7  = 29.306666 = 29.31 ft.³

b.

V = ___ mi.³
Answer: 29.31 mi.³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius = 2 mi.
height = 7 mi.
V = \(\frac{1}{3}\)π2² 7  = 29.306666 = 29.31 mi.³

c.

V = ___ m³
Answer: 200.96 m³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius = d /2 = 8 / 2 = 4 m
height = 12 m
V = \(\frac{1}{3}\)π4² 12  = 200.96 m³

Question 4.
a.

V = ___ cm³
Answer: 16.49 cm³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius = 1.5 cm
height = 7 m
V = \(\frac{1}{3}\)π1.5² 7  = 16.485 = 16.49 cm³

b.

V = ___ in.³
Answer: 678.24 in.³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius =  6 in.
height =  18 in.
V = \(\frac{1}{3}\)π6² 18  = 678.24 in.³

c.

V = ___ ft.³
Answer: 949.85 ft.³
Volume is the amount of space a three-dimensional figure occupies. The volume of a cone is calculated as \(\frac{1}{3}\)base × height.
This is because a cone occupies \(\frac{1}{3}\) of the volume of a cylinder of the same height. Base is the area of the circle, πr².
radius =  11 ft.
height =  7.5 ft.
V = \(\frac{1}{3}\)π11² 7.5  = 949.85 ft.³