Volume of Cylinders, Cones and Spheres: practice solutions, Grade 8 – download the PDF

1 A cylinder ★★★

a. \(V = \pi r^2 h = \pi \times 5^2 \times 10 = 250\pi\). The exact volume is \(250\pi \text{ cm}^3\).

b. \(250 \times 3.14 = 785\). The volume is about \(785 \text{ cm}^3\).

2 A cone ★★★

\(V = \dfrac{1}{3}\pi r^2 h = \dfrac{1}{3} \times \pi \times 9 \times 7 = \dfrac{63\pi}{3} = 21\pi\).

The volume is \(21\pi \text{ in}^3\).

3 A sphere ★★★

\(V = \dfrac{4}{3}\pi r^3 = \dfrac{4}{3} \times \pi \times 27 = 36\pi\).

The volume is \(36\pi \text{ ft}^3\).

4 True or false? ★★★

a. False. The cone has one third of the volume of the cylinder, because three cones fill the cylinder.

b. False. Volume is measured in cubic units such as \(\text{cm}^3\); square units are used for area.

5 Diameter given ★★★

The radius is \(8 \div 2 = 4\) m.

\(V = \pi \times 4^2 \times 5 = 80\pi \approx 80 \times 3.14 = 251.2\).

The volume is about \(251.2 \text{ m}^3\).

6 Cone from cylinder ★★★

The cone holds one third of the cylinder: \(90 \div 3 = 30\).

The cone has a volume of \(30 \text{ cm}^3\).

7 A glass ball ★★★

\(V = \dfrac{4}{3}\pi \times 6^3 = \dfrac{4}{3} \times 216\pi = 288\pi\).

\(288 \times 3.14 = 904.32\). The volume is about \(904.3 \text{ cm}^3\).

8 A can of soup ★★★

a. The radius is 3.5 cm. \(V = \pi \times 3.5^2 \times 10 = 122.5\pi \approx 384.65\). The volume is about \(384.65 \text{ cm}^3\).

b. \(384.65 \text{ cm}^3 \approx 384.65 \text{ mL}\), which is less than 400 mL. So the can cannot hold 400 mL of soup.

9 A water tank ★★★

\(V = \pi \times 2^2 \times 6 = 24\pi \approx 75.36 \text{ ft}^3\).

\(75.36 \times 7.48 \approx 563.69\). The tank holds about 564 gallons.

10 A sand pile ★★★

The radius is 3 m. \(V = \dfrac{1}{3}\pi \times 3^2 \times 2 = 6\pi \approx 18.84 \text{ m}^3\).

\(18.84 \div 4 = 4.71\). The truck needs 5 trips, since a fifth trip is needed for the last part.

11 Find the height ★★★

\(\dfrac{1}{3}\pi \times 4^2 \times h = 48\pi\), so \(\dfrac{16}{3}h = 48\).

\(h = 48 \times \dfrac{3}{16} = 9\). The height is 9 cm.

12 Find the radius ★★★

\(\pi r^2 \times 8 = 200\pi\), so \(8r^2 = 200\) and \(r^2 = 25\).

Since \(r\) is a length, \(r = 5\). The radius is 5 cm.

13 A glass globe ★★★

The radius is 5 cm. \(V = \dfrac{4}{3}\pi \times 5^3 = \dfrac{500\pi}{3}\).

\(\dfrac{500 \times 3.14}{3} = \dfrac{1570}{3} \approx 523.3\). The volume is about \(523.3 \text{ cm}^3\).

14 A bowl ★★★

\(V = \dfrac{2}{3}\pi \times 6^3 = \dfrac{2}{3} \times 216\pi = 144\pi\).

\(144 \times 3.14 = 452.16\). The bowl holds about \(452.16 \text{ in}^3\).

15 An ice cream cone ★★★

Cone: \(\dfrac{1}{3}\pi \times 3^2 \times 10 = 30\pi\). Hemisphere: \(\dfrac{2}{3}\pi \times 3^3 = 18\pi\).

Total: \(30\pi + 18\pi = 48\pi\). With \(\pi \approx 3.14\): \(48 \times 3.14 = 150.72\).

The total volume is \(48\pi \text{ cm}^3 \approx 150.72 \text{ cm}^3\).

16 A metal pipe ★★★

Outer cylinder: \(\pi \times 5^2 \times 20 = 500\pi\). Hole: \(\pi \times 3^2 \times 20 = 180\pi\).

Metal: \(500\pi - 180\pi = 320\pi \approx 1{,}004.8\).

The pipe contains about \(1{,}004.8 \text{ cm}^3\) of metal.

17 A ball in a tube ★★★

a. Tube: \(\pi \times 3^2 \times 6 = 54\pi\). Ball: \(\dfrac{4}{3}\pi \times 3^3 = 36\pi\).

b. \(\dfrac{36\pi}{54\pi} = \dfrac{2}{3}\). The ball fills two thirds of the tube.

c. Empty space: \(54\pi - 36\pi = 18\pi \approx 56.52\). About \(56.52 \text{ cm}^3\) is empty.

18 Changing dimensions ★★★

Original: \(\pi \times 3^2 \times 5 = 45\pi\).

a. \(r = 6\): \(\pi \times 36 \times 5 = 180\pi\), which is 4 times the original.

b. \(h = 10\): \(\pi \times 9 \times 10 = 90\pi\), which is 2 times the original.

c. \(r = 6, h = 10\): \(\pi \times 36 \times 10 = 360\pi\), which is 8 times the original.

19 Pouring a cone ★★★

Water: \(\dfrac{1}{3}\pi \times 4^2 \times 12 = 64\pi \text{ cm}^3\).

In the glass: \(\pi \times 4^2 \times h = 16\pi h\). So \(16\pi h = 64\pi\) and \(h = 4\).

The water rises 4 cm. (Same base and the cone is three times as tall as this height, so this agrees with the one-third rule.)

20 Radius of a sphere ★★★

\(\dfrac{4}{3}\pi r^3 = 972\pi\), so \(\dfrac{4}{3}r^3 = 972\) and \(r^3 = 972 \times \dfrac{3}{4} = 729\).

Since \(9^3 = 729\), \(r = 9\). The radius is 9 m.

21 A marble in a fish tank ★★★

The water level rises by a layer whose volume equals the ball’s volume.

Ball: \(\dfrac{4}{3}\pi \times 3^3 = 36\pi\). Layer: \(\pi \times 15^2 \times h = 225\pi h\).

\(225\pi h = 36\pi\), so \(h = \dfrac{36}{225} = 0.16\). The water level rises by 0.16 cm.

22 A grain silo ★★★

Cylinder: \(\pi \times 4^2 \times 10 = 160\pi\). Cone roof: \(\dfrac{1}{3}\pi \times 4^2 \times 3 = 16\pi\).

Total: \(176\pi\). With \(\pi \approx 3.14\): \(176 \times 3.14 = 552.64\).

The silo has a volume of \(176\pi \text{ m}^3 \approx 552.64 \text{ m}^3\).