# Surface Area of a Sphere

Related Topics: More Geometry Lessons

In these lessons we will learn
• the formula of the surface area of a sphere.
• how to calculate the surface area of a sphere.
• how to calculate the surface area of a hemisphere.
• how to solve problems about the surface area of spheres.
• how to prove the formula of the surface area of a sphere.

### Surface Area of a Sphere

A sphere is a solid in which all the points on the round surface are equidistant from a fixed point, known as the center of the sphere. The distance from the center to the surface is the radius.

Surface area of a sphere is given by the formula

Surface Area of sphere = 4πr2

where r is the radius of the sphere.

Example:

Calculate the surface area of a sphere with radius 3.2 cm

Solution:

Surface area of sphere
= 4π r2
= 4π (3.2)2
= 4 × 3.14 × 3.2 × 3.2
= 128.6 cm2

Worksheet to calculate the surface area of spheres.

How to find the surface area of a sphere?
Example:
Find the surface area of a sphere with r = 4 ft How to calculate the surface area of a sphere given the radius or diameter?
Example:
Find the surface area of a sphere with diameter = 22 cm

### Surface Area of a hemisphere

A hemisphere is half a sphere, with one flat circular face and one bowl-shaped face.

The surface area of a hemisphere is equal to the area of the curve surface plus the area of the circular base.

Surface area of hemisphere

= × surface area of sphere + area of base

= × 4π r2 + π r2 = 3π r2

Example:

Calculate the surface area of a hemisphere with radius 4 cm

Solution:

Surface area of hemisphere

= × surface area of sphere + area of base

= × 4π r2 + π r2

= 3π r2

= 3π (4)2

= 150.86 cm2

### Problems about surface area of spheres

Problem: What is the radius of the sphere given the surface area?
Example: What is the radius of a sphere with a surface area of 900π? Problem:
The surface area of a sphere is 5024 square meters. What is the volume of the sphere? Use 3.14 for pi. Round your answer to the nearest cubic meter. Problem:
The radius of a sphere is tripled.
a) Describe the effect on the volume.
b) Describe the effect on the surface area.

### Proof of the formula of the surface area of a sphere

This video demonstrates that the surface area of a sphere equals the area of 4 circles. (It is not a formal proof) The formula for the surface of a sphere is derived by summing up small ring elements of area along its perimeter. (uses calculus) These two videos explain the Archimedes method of deriving the surface area of a sphere.
Part1: Part 2:

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