1. Sphere
A sphere is a perfectly round 3D object where every point on the surface is the same distance from the centre. This distance is called the radius \(r\).
Common examples include balls, marbles, bubbles, and planets. Unlike prisms or cones, a sphere has no flat faces, no edges, and no vertices.
2. Surface Area of a Sphere
The surface area of a sphere refers to the area of its entire curved surface. Since a sphere has no flat faces, its total surface area is simply its curved surface area.
The formula for the total surface area of a sphere is:
\( \text{TSA} = 4\pi r^2 \)
2.1. Why the Formula Makes Sense (Simple Idea)
A sphere is made entirely of curved surface. You can imagine approximating it using many tiny squares or by slicing it into thin circular strips. The total area adds up to four times the area of a circle of radius \(r\). That's why the formula becomes:
\( 4 \times \pi r^2 \)
2.2. Example
Find the surface area of a sphere with radius \(r = 7\,\text{cm}\).
\( \text{TSA} = 4\pi r^2 = 4 \times \pi \times 7^2 = 196\pi \)
Using \(\pi = 3.14\):
\( 196 \times 3.14 = 615.44\,\text{cm}^2 \)
3. Real-Life Applications
The surface area of a sphere is useful in many real-world contexts, such as:
- Designing sports balls (cricket ball, football, basketball)
- Painting spherical tanks or domes
- Measuring heat radiation from spherical objects (like stars)
- Calculating the material needed to make spherical ornaments
Whenever the whole sphere is exposed or needs to be covered, you use the formula \(4\pi r^2\).