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Surface area and volume of spheres

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Surface Area and Volume of a Sphere

A sphere's surface area is 4 pi r squared and its volume is four-thirds pi r cubed, both depending only on the radius. Unlike a cone or cylinder, a sphere has no flat net since its curved surface cannot unroll flat. Learn both formulas with a worked example.

What a sphere is

A sphere is a perfectly round 3-D shape — every point on its surface is the same distance (the radius, r) from its center, like a ball. Unlike a cone or a cylinder, a sphere has no flat faces or edges, and its curved surface cannot be unrolled into a flat net.

Surface area and volume of a sphere A sphere with radius r. Its surface area is 4 pi r squared and its volume is four thirds pi r cubed. Unlike a cone or cylinder, a sphere has no flat net; its curved surface cannot be unrolled flat. r SA = 4πr² V = (4/3)πr³ A sphere has no flat net — its curved surface cannot unroll flat.
A sphere with radius r; its surface area and volume both depend only on r.

A sphere's surface area is SA = 4πr² — exactly four times the area of a great circle (a flat circle cut through the sphere's center), which connects back to circumference of a circle.

A sphere's volume is V = (4/3)πr³. Both formulas use only the radius — there is no height or slant height to measure, since every direction from the center reaches the surface at the same distance.

Take a sphere with radius 3.

  • Surface area: SA = 4π(3)² = 4π(9) = 36π ≈ 113.1 square units.
  • Volume: V = (4/3)π(3)³ = (4/3)π(27) = 36π ≈ 113.1 cubic units.

(These two numbers matching is a coincidence of r=3 — surface area and volume measure different things and generally are not equal.)

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