Volume of Spheres

Show your work for each problem.

1.
Which formula gives the volume of a sphere with radius $r$ ?
1. a) $V = \frac{4}{3}\pi r^3$
2. b) $V = \frac{1}{3}\pi r^2 h$
3. c) $V = \pi r^2 h$
4. d) $V = 4\pi r^2$
Answer: $V = \frac{4}{3}\pi r^3$
The volume of a sphere is $V = \frac{4}{3}\pi r^3$ . $4\pi r^2$ has only $r^2$ (it is not a volume), $\pi r^2 h$ is the cylinder and $\frac{1}{3}\pi r^2 h$ is the cone.
2.
A sphere has a radius of $3$ cm. Its volume is $V = \square\,\pi$ cm³. What number goes in the box?
Answer: 36
$r^3 = 27$ and $\frac{4}{3} \times 27 = 36$ , so $V = 36\pi$ cm³.
3.
A ball has a diameter of $10$ cm. Which value of $r$ do you put into $V = \frac{4}{3}\pi r^3$ ?
1. a) $2.5$ cm
2. b) $5$ cm
3. c) $20$ cm
4. d) $10$ cm
Answer: $5$ cm
$r = \frac{d}{2} = \frac{10}{2} = 5$ cm. Using $10$ would make the volume $8$ times too big.
4.
Find the volume of a sphere with radius $6$ m. Leave the answer in terms of $\pi$ .
Answer: $288\pi$ m³
1. What is $r^3 = 6^3$ ? 216
2. What is $\frac{4}{3} \times 216$ ? 288
5.
A sphere has a volume of $500$ cm³. It is cut exactly in half through its center. What is the volume of one hemisphere, in cm³?
Answer: 250
The hemisphere has the same radius, but half the volume: $500 \div 2 = 250$ cm³.
6.
A ball has a diameter of $6$ cm. Find its volume using $\pi \approx 3.14$ . Round to the nearest hundredth.
Answer: 113.04
1. What is the radius in cm? 3
2. $V = \square\,\pi$ : what is $\frac{4}{3} \times r^3$ ? 36
3. What is the volume in cm³ with $\pi \approx 3.14$ ? 113.04
7.
The radius of a sphere is doubled. What happens to its volume?
1. a)It doubles
2. b)It becomes 8 times as big
3. c)It becomes 4 times as big
4. d)It becomes 6 times as big
Answer: It becomes 8 times as big
$(2r)^3 = 8r^3$ , so the volume is multiplied by $2^3 = 8$ . Example: $r = 1$ gives $\frac{4}{3}\pi$ , $r = 2$ gives $\frac{32}{3}\pi$ .
8.
A marble has a radius of $1.5$ cm. Its volume is $V = \square\,\pi$ cm³. What number goes in the box? (Write it as a decimal.)
Answer: 4.5
$r^3 = 3.375$ and $\frac{4}{3} \times 3.375 = 4.5$ , so $V = 4.5\pi$ cm³ (about $14.13$ cm³ with $3.14$ ).
9.
A salad bowl is a hemisphere with an inside radius of $9$ cm. How much can it hold? Use $\pi \approx 3.14$ and round to the nearest hundredth.
Answer: 1526.04
1. What is $r^3 = 9^3$ ? 729
2. Hemisphere: $V = \square\,\pi$ . What is $\frac{2}{3} \times 729$ ? 486
3. What is the volume in cm³ with $\pi \approx 3.14$ ? 1526.04
10.
A sphere has a volume of $288\pi$ cm³. What is its radius in cm?
Answer: 6
$\frac{4}{3} r^3 = 288$ , so $r^3 = 288 \times \frac{3}{4} = 216$ and $r = \sqrt[3]{216} = 6$ cm.
11.
A sphere fits exactly inside a cylinder (same radius, and the cylinder's height equals the sphere's diameter). What fraction of the cylinder's volume does the sphere fill?
1. a) $\frac{1}{3}$
2. b) $\frac{2}{3}$
3. c) $\frac{3}{4}$
4. d) $\frac{1}{2}$
Answer: $\frac{2}{3}$
Cylinder: $\pi r^2 \times 2r = 2\pi r^3$ . Sphere: $\frac{4}{3}\pi r^3$ . Ratio: $\frac{4}{3} \div 2 = \frac{2}{3}$ . This is Archimedes' famous result.
12.
A basketball has a diameter of $24$ cm. Find its volume using $\pi \approx 3.14$ . Round to the nearest whole cm³.
Answer: 7235
1. What is the radius in cm? 12
2. What is $12^3$ ? 1728
3. $V = \square\,\pi$ : what is $\frac{4}{3} \times 1728$ ? 2304
4. What is the volume in cm³, rounded to a whole number? 7235
13.
A round water tank is a sphere with a diameter of $2$ m. Find its volume using $\pi \approx 3.14$ . Round to the nearest hundredth of a m³.
Answer: 4.19
1. What is the radius in m? 1
2. What is $r^3$ ? 1
3. What is $\frac{4}{3} \times 3.14 \times 1$ , rounded to the nearest hundredth? 4.19
14.
A sphere has a volume of $972\pi$ cm³. Find its radius.
Answer: $r = 9$ cm
1. Divide both sides of $\frac{4}{3}\pi r^3 = 972\pi$ by $\pi$ : $\frac{4}{3} r^3 = \square$ 972
2. Multiply by $\frac{3}{4}$ : what is $r^3$ ? 729
3. Take the cube root: what is $r$ in cm? 9
15.
A ball with radius $6$ cm is packed in a cylindrical box with radius $6$ cm and height $12$ cm. How much empty space is left in the box? Leave the answer in terms of $\pi$ .
Answer: $144\pi$ cm³
1. Cylinder: $V = \pi r^2 h = \square\,\pi$ . What is $6^2 \times 12$ ? 432
2. Ball: $V = \square\,\pi$ . What is $\frac{4}{3} \times 6^3$ ? 288
3. Empty space $= \square\,\pi$ cm³. What is $432 - 288$ ? 144
16.
Which holds more: a sphere with radius $3$ cm or a cube with edges of $5$ cm? (Use $\pi \approx 3.14$ .)
1. a)The sphere
2. b)It cannot be decided
3. c)They hold the same amount
4. d)The cube
Answer: The cube
Sphere: $36\pi \approx 113.04$ cm³. Cube: $5^3 = 125$ cm³. The cube holds more, even though the sphere is $6$ cm wide.
17.
Eight small chocolate balls, each with radius $1$ cm, are melted and made into one big ball. What is the radius of the big ball?
Answer: $R = 2$ cm
1. One small ball: $V = \frac{4}{3}\pi \times \square$ . What is $1^3$ ? 1
2. The big ball has $\frac{4}{3}\pi R^3 = 8 \times \frac{4}{3}\pi$ . What is $R^3$ ? 8
3. What is $R$ in cm? 2
18.
A dome shaped like a hemisphere has a volume of $144\pi$ m³. What is its radius in m?
Answer: 6
$\frac{2}{3} r^3 = 144$ , so $r^3 = 144 \times \frac{3}{2} = 216$ and $r = \sqrt[3]{216} = 6$ m.