Surface Area and Volume • Topic 5 of 7

Spheres

A sphere is a perfectly round solid; every point on its surface is the same distance (the radius) from the centre. Its volume is (4/3)πr³ and its surface area is 4πr². The volume uses the cube of the radius, so it grows fast as r increases. For radius 3, the volume is (4/3)π(27) = 36π and the surface area is 4π(9) = 36π. Cube the radius for volume, square it for surface area, and keep answers in terms of π. Choose carefully between the two formulas — volume (cubic units, r³) versus surface area (square units, r²) — since the SAT tests both.

A sphere with radius from the centreSphererV = (4/3)πr³    S = 4πr²

✅ Solved examples

1. A sphere has radius 3. Find its volume in terms of π.
(4/3)π(27) = 36π.
2. A sphere has radius 3. Find its surface area.
4π(9) = 36π.
3. A sphere has radius 6. Find its volume.
(4/3)π(216) = 288π.
4. A sphere has radius 5. Find its surface area.
4π(25) = 100π.

✏️ Practice — try these, take hints as needed

1. A sphere has radius 3. Find its surface area in terms of π.
4πr².
4π(9).
—
36π.
2. A sphere has radius 2. Find its surface area.
4π(4).
—
—
16π.
3. A sphere has radius 3. Find its volume.
(4/3)π(27).
—
—
36π.
4. A sphere has radius 6. Find its surface area.
4π(36).
—
—
144π.
5. A sphere has radius 10. Find its surface area.
4π(100).
—
—
400π.

📝 Topic test — 8 questions

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