ShapesProperties
- One curved face, no edges and no corners.
- Every flat slice through a sphere is a circle. A slice through the centre gives the largest one, called a great circle.
- Of all solids with a given surface area, the sphere holds the most.
Why the formulas work
Archimedes showed that a sphere takes up exactly two thirds of the cylinder that just contains it. That cylinder has radius r and height 2r, so its volume is 2πr³, and two thirds of that is (4/3)πr³. He also showed that the surface of the sphere equals the curved surface of the same cylinder: 2πr × 2r = 4πr², exactly four times the area of a great circle.
Worked example
Take a sphere with radius 7 cm.
- Volume. Volume = (4 ⁄ 3) πr3. Here: (4 ⁄ 3) × π × 7 × 7 × 7 = 1,436.76 cm³ (with π = 22/7: 1,437.33 cm³).
- Surface area. Surface area = 4πr2. Here: 4 × π × 7 × 7 = 615.75 cm² (with π = 22/7: 616 cm²).
Finding a missing measurement
From the surface area, r = √(S ÷ 4π): 616 cm² gives r² = 49, so r = 7 cm, taking π = 22/7. From the volume, r is the cube root of 3V ÷ 4π: 38,808 cm³ gives r³ = 9261, so r = 21 cm.
Mistakes to avoid
- Using r² in the volume or r³ in the surface area.
- Using the diameter in place of the radius.
- Doubling the radius makes the surface four times and the volume eight times as large.
Where you see it
Cricket balls, footballs, marbles, laddoos, ball bearings, soap bubbles and, very nearly, the Earth and the Moon are spheres.
Try these
Work each one out, then tap to check.
A sphere has radius 10.5 cm. Find its volume and surface area. Take π = 22/7.
Volume = 4,851 cm³; Surface area = 1,386 cm².
A sphere has radius 3 cm. Find its volume and surface area. Take π = 22/7.
Volume = 113.14 cm³; Surface area = 113.14 cm².