Surface Area of a Sphere

Statement

The surface area of a sphere is the total area of its curved surface. A sphere is perfectly round, with every point on the surface the same distance (radius \(r\)) from the centre. The formula for the surface area is:

\[ S = 4\pi r^2 \]

This tells us that the surface area depends only on the square of the radius.

Why it’s true

• The formula can be derived using calculus or by comparing the sphere with a cylinder. Archimedes discovered that the surface area of a sphere equals the curved surface area of a cylinder with the same height and diameter.
• The factor \(4\pi\) arises naturally from geometry: a circle’s area is \(\pi r^2\), and the surface of a sphere is essentially “four times” this area.
• It is a universal property: no matter the size of the sphere, its surface area is always proportional to \(r^2\).

Recipe (how to use it)

1. Measure or identify the radius \(r\).
2. Square the radius (\(r^2\)).
3. Multiply by \(4\pi\).
4. Leave the answer in terms of \(\pi\) (exact) or approximate with a decimal.
5. Include correct units (cm², m², etc.).

Spotting it

You use this formula whenever you have a 3D ball shape (football, globe, orange, etc.) and need the total curved area.

Common pairings

• Volume of a sphere: \(V = \tfrac{4}{3}\pi r^3\).
• Area of a circle: \(A = \pi r^2\).

Mini examples

1. Given: \(r=3\). Answer: \(S=4\pi \times 9 = 36\pi \approx 113.10\).
2. Given: \(r=7\). Answer: \(S=4\pi \times 49 = 196\pi \approx 615.75\).

Pitfalls

• Using diameter instead of radius — remember \(r=\tfrac{d}{2}\).
• Forgetting to square the radius.
• Mixing surface area with volume.

Exam strategy

• Underline whether the question wants “exact” or “to 1 decimal place”.
• If given diameter, halve it first to find radius.
• Always add units at the end.

Summary

The surface area of a sphere is given by \(S=4\pi r^2\). It depends only on the square of the radius, making it one of the simplest and most elegant surface area formulas. Always take care to use radius, not diameter, and to square correctly.