Surface Area and Volume of Spheres

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The surface area and volume of a sphere depend only on its radius r. The surface area is S = 4\pi r^2, and the volume is V = \tfrac{4}{3}\pi r^3. A sphere is the set of all points in space that are the same distance from a center point, like a ball.

Before this lesson: You should know Circumference and Area of Circles, Volume of Pyramids and Cones and Exponents.

Parts of a Sphere

A sphere is all the points in space at a distance r from a center. The radius goes from the center to the surface, and the diameter goes through the center from one side to the other. A great circle is the largest circle you can draw on a sphere. It cuts the sphere into two hemispheres.

Left: a sphere with center O and radius r. A great circle, the largest circle on the sphere, goes around its middle. The sphere has surface area 4 pi r squared and volume four thirds pi r cubed. Right: a hemisphere, half of a sphere, with a flat circular base of radius r. Its volume is two thirds pi r cubed.
A great circle splits a sphere into two hemispheres.

Sphere Formulas

S = 4\pi r^2 \qquad V = \tfrac{4}{3}\pi r^3

The surface area is exactly 4 great circles, since each great circle has area \pi r^2. The volume can be shown with Cavalieri’s principle: a hemisphere holds the same amount as a cylinder minus a cone, both with radius r and height r. That gives \pi r^3 - \tfrac{1}{3}\pi r^3 = \tfrac{2}{3}\pi r^3 for half the sphere.

ShapeSurface areaVolume
Sphere4\pi r^2\tfrac{4}{3}\pi r^3
Hemisphere (with its flat base)2\pi r^2 + \pi r^2 = 3\pi r^2\tfrac{2}{3}\pi r^3

How to Find Surface Area and Volume

  1. Find the radius. If you know the diameter, divide by 2.
  2. For surface area, square the radius and multiply by 4\pi.
  3. For volume, cube the radius and multiply by \tfrac{4}{3}\pi.
  4. Square units for surface area, cubic units for volume. Round only at the end.

Worked Examples

Example 1: Radius given

A sphere has radius 6 cm. Find its surface area and volume.

  1. Write the surface area formula: S = 4\pi(6)^2
  2. Square: S = 4\pi(36)
  3. Multiply: S = 144\pi \approx 452.39
  4. Write the volume formula: V = \tfrac{4}{3}\pi(6)^3
  5. Cube (see exponents): V = \tfrac{4}{3}\pi(216)
  6. Multiply: V = 288\pi \approx 904.78

Answer: S = 144\pi \approx 452.39 cm² and V = 288\pi \approx 904.78 cm³

Example 2: Diameter given

A ball has a diameter of 10 in. Find its surface area and volume.

  1. Find the radius: r = 10 \div 2 = 5
  2. Surface area: S = 4\pi(5)^2 = 4\pi(25)
  3. Multiply: S = 100\pi \approx 314.16
  4. Volume: V = \tfrac{4}{3}\pi(5)^3 = \tfrac{4}{3}\pi(125)
  5. Multiply: V = \frac{500\pi}{3} \approx 523.60

Answer: S \approx 314.16 in² and V \approx 523.60 in³

Example 3: Work backward from the surface area

A sphere has surface area 196\pi m². Find its radius and volume.

  1. Write the formula: 196\pi = 4\pi r^2
  2. Divide both sides by 4\pi: 49 = r^2
  3. Take the positive square root: r = 7
  4. Volume: V = \tfrac{4}{3}\pi(7)^3 = \tfrac{4}{3}\pi(343)
  5. Multiply: V = \frac{1372\pi}{3} \approx 1436.76

Answer: r = 7 m and V \approx 1436.76 m³

Example 4: A hemisphere

A solid hemisphere has radius 3 ft. Find its volume and its total surface area (curved part plus flat base).

  1. Volume is half of a sphere: V = \tfrac{2}{3}\pi(3)^3
  2. Cube: V = \tfrac{2}{3}\pi(27)
  3. Multiply: V = 18\pi \approx 56.55
  4. Curved part is half the sphere’s surface: 2\pi(3)^2 = 18\pi
  5. Flat base is a circle: \pi(3)^2 = 9\pi
  6. Add: S = 18\pi + 9\pi = 27\pi \approx 84.82

Answer: V \approx 56.55 ft³ and S \approx 84.82 ft²

Example 5: Word problem with a water tank

A spherical water tank has a radius of 10 ft. How many cubic feet of water can it hold? One cubic foot holds about 7.48 gallons. About how many gallons is that?

  1. Write the volume formula: V = \tfrac{4}{3}\pi(10)^3
  2. Cube: V = \tfrac{4}{3}\pi(1000)
  3. Multiply: V = \frac{4000\pi}{3} \approx 4188.79
  4. Convert to gallons: 4188.79 \times 7.48 \approx 31{,}332.2

Answer: about 4,188.79 ft³, or about 31,332 gallons

Common Mistakes

MistakeExample of the errorFix
Squaring instead of cubing for volumeV = \tfrac{4}{3}\pi(6)^2Volume uses r^3: \tfrac{4}{3}\pi(216).
Using the diameterS = 4\pi(10)^2 for a ball 10 in acrossUse r = 5.
Mixing up the formulasS = \tfrac{4}{3}\pi r^2S = 4\pi r^2 and V = \tfrac{4}{3}\pi r^3.
Forgetting the flat base of a solid hemisphereS = 2\pi r^2 onlyA solid hemisphere also has a circle: add \pi r^2.
Rounding \pi earlyUsing 3.14 and then rounding againKeep \pi and round once at the end.

Practice Problems

Give answers with \pi exactly, then rounded to the nearest hundredth.

  1. Find the surface area and volume of a sphere with radius 3.
  2. Find the surface area and volume of a sphere with radius 9.
  3. Find the surface area and volume of a sphere with diameter 14.
  4. A sphere has surface area 64\pi. Find its volume.
  5. A sphere has volume 36\pi. Find its radius.
  6. Find the volume and total surface area of a solid hemisphere with radius 6.
  7. A great circle of a sphere has area 25\pi. Find the sphere’s surface area.
  8. An ice cream cone is a cone with radius 3 cm and height 10 cm, topped with a hemisphere of ice cream with radius 3 cm. Find the total volume.
  9. A ball has a radius of 12 cm. How much leather covers it?
  10. If you double the radius of a sphere, what happens to its surface area and its volume?
Show answers
  1. Surface area: S = 4\pi(3)^2 = 36\pi \approx 113.10
    Volume: V = \tfrac{4}{3}\pi(3)^3 = \tfrac{4}{3}\pi(27) = 36\pi \approx 113.10
  2. Surface area: S = 4\pi(9)^2 = 4\pi(81) = 324\pi \approx 1017.88
    Volume: V = \tfrac{4}{3}\pi(9)^3 = \tfrac{4}{3}\pi(729) = 972\pi \approx 3053.63
  3. Find the radius: r = 14 \div 2 = 7
    Surface area: S = 4\pi(49) = 196\pi \approx 615.75
    Volume: V = \tfrac{4}{3}\pi(343) = \frac{1372\pi}{3} \approx 1436.76
  4. Write the formula: 64\pi = 4\pi r^2
    Divide both sides by 4\pi: 16 = r^2
    Take the square root: r = 4
    Volume: V = \tfrac{4}{3}\pi(64) = \frac{256\pi}{3} \approx 268.08
  5. Write the formula: 36\pi = \tfrac{4}{3}\pi r^3
    Multiply both sides by 3: 108\pi = 4\pi r^3
    Divide both sides by 4\pi: 27 = r^3
    Take the cube root (see cube roots): r = 3
  6. Volume: V = \tfrac{2}{3}\pi(6)^3 = \tfrac{2}{3}\pi(216) = 144\pi \approx 452.39
    Curved part: 2\pi(6)^2 = 72\pi
    Base: \pi(6)^2 = 36\pi
    Add: S = 108\pi \approx 339.29
  7. Write the great circle area: \pi r^2 = 25\pi
    Divide both sides by \pi: r^2 = 25
    Surface area is 4 great circles: S = 4(25\pi) = 100\pi \approx 314.16
  8. Cone (see volume of pyramids and cones): \tfrac{1}{3}\pi(3)^2(10) = \tfrac{1}{3}\pi(90) = 30\pi
    Hemisphere: \tfrac{2}{3}\pi(3)^3 = \tfrac{2}{3}\pi(27) = 18\pi
    Add: 30\pi + 18\pi = 48\pi \approx 150.80 cm³
  9. Surface area: S = 4\pi(12)^2 = 4\pi(144) = 576\pi \approx 1809.56 cm²
  10. Surface area depends on r^2, so it is multiplied by 2^2 = 4. Volume depends on r^3, so it is multiplied by 2^3 = 8. You will see this idea again in similar solids.

FAQ

What is the formula for the volume of a sphere?

V = \tfrac{4}{3}\pi r^3, where r is the radius.

What is the formula for the surface area of a sphere?

S = 4\pi r^2. It equals the area of four great circles.

How do you find the volume of a hemisphere?

Take half of the sphere’s volume: V = \tfrac{2}{3}\pi r^3.

How do you find the radius from the volume?

Multiply the volume by 3, divide by 4\pi, then take the cube root.

Summary

  • A sphere is all points the same distance r from a center.
  • Surface area: S = 4\pi r^2. Volume: V = \tfrac{4}{3}\pi r^3.
  • A hemisphere has volume \tfrac{2}{3}\pi r^3 and total surface area 3\pi r^2.
  • Use the radius, cube it for volume, and round at the end.

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Next lesson: Similar Solids

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