In these lessons we will learn

### Surface Area of a Sphere

**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

### 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:

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

- 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.

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 = 4π

r^{2}

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

= 4*π r*^{2}

= 4*π* (3.2)^{2
}= 4 × 3.14 × 3.2 × 3.2

= 128.6 cm^{2}

Worksheet to calculate the surface area of spheres.

Example:

Find the surface area of a sphere with r = 4 ft

Example:

Find the surface area of a sphere with diameter = 22 cm

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

= × 4*π **r*^{2} + *π **r*^{2 }= 3*π* *r*^{2}

**Example: **

Calculate the surface area of a hemisphere with radius 4 cm

**Solution:**

Surface area of hemisphere

= × surface area of sphere + area of base

= × 4*π **r*^{2} + *π **r*^{2}

= 3*π* *r*^{2}

= 3*π* (4)^{2}

= 150.86 cm^{2}

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.

Part1: Part 2:

Rotate to landscape screen format on a mobile phone or small tablet to use the **Mathway** widget, a free math problem solver that **answers your questions with step-by-step explanations**.

You can use the free Mathway calculator and problem solver below to practice Algebra or other math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations.

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