Surface Area and Volume of Spheres

Unit 11: Surface Area and Volume · ~15–30 min

A solid with no flat faces

A sphere is the set of all points in space a fixed distance (the radius, r) from a center point. Unlike a prism or a cone, a sphere has no base, no net, and no edges — it's a single continuous curved surface, so its formulas can't be built by "unfolding" faces the way earlier ones were. Instead, both formulas depend only on the radius.

Surface area and volume formulas

A useful sanity check: a sphere's surface area equals exactly four great circles (four times πr²), and its volume is exactly two–thirds the volume of the smallest cylinder that snugly contains it (a cylinder with radius r and height 2r).

• If you're given a diameter instead of a radius, divide by 2 first — both formulas use r, not d.
• Volume grows with the cube of the radius, so even small changes in r produce large changes in volume — doubling r makes the volume eight times as large.

Guided practice

• 1.Find the surface area of a sphere with radius 10 ft, in terms of π.
  Show answer

  S.A. = 4πr² = 4π(10)² = 4π(100) = 400π ft².
• 2.Find the volume of a sphere with radius 3 m, to the nearest tenth.
  Show answer

  V = &frac43;πr³ = &frac43;π(27) = 36π ≈ 113.1 m³.
• 3.A sphere has diameter 14 cm. Find its surface area to the nearest tenth.
  Show answer

  r = 7 cm. S.A. = 4π(7)² = 4π(49) = 196π ≈ 615.8 cm².
• 4.Sphere B has twice the radius of Sphere A. How many times greater is Sphere B's volume?
  Show answer

  Volume depends on r³, so doubling the radius multiplies volume by 2³ = 8. Sphere B's volume is 8 times Sphere A's.
• 5.A hemisphere (half a sphere) has radius 4 in. Find its total surface area, including the flat circular base, in terms of π.
  Show answer

  The curved part is half the sphere's surface area: ½(4πr²) = 2πr² = 2π(16) = 32π. The flat circular base adds πr² = 16π. Total = 32π + 16π = 48π in².