Home » Math Theory » Geometry » Surface Area of a Sphere

# Surface Area of a Sphere

## Introduction

In the world of mathematics, various shapes and figures have their own unique set of properties and formulas. One of these is a sphere, a three-dimensional figure that appears frequently in our everyday life, such as in the form of balls, globes, and bubbles. In this article, we will provide the concept of a sphere’s surface area and explore its practical applications.

Understanding the concept of the surface area of a sphere is typically suitable for students in the 8th grade or higher. The idea is generally introduced after students fully understand area and volume in two-dimensional shapes and simple three-dimensional shapes, like cylinders and prisms.

## Math Domain

The surface area of a sphere falls under the domain of Geometry. More specifically, it pertains to the measurement and properties of three-dimensional shapes.

## Applicable Common Core Standards

The topic is most closely aligned with the following Common Core State Standards for Mathematics:

CCSS.MATH.CONTENT.8.G.C.9: “Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.”

Although this standard explicitly mentions volumes, understanding the surface area is an integral part of understanding the volumes of these figures.

## Definition of the Topic

A sphere is a 3D geometric shape that is perfectly symmetrical about its center. It is the collection of all points in space that are equidistant from a given point, known as the center.

A sphere’s surface area is the total area that the sphere’s surface covers. It is measured in square units. A mathematical constant π (pi) has an approximate value of 3.14159.

## Key Concepts

The surface area of a sphere does not depend on its orientation in space.

A sphere’s surface area is always positive.

SA = 4πr2 is the formula for calculating a sphere’s surface area.

## Discussion with Illustrative Examples

A sphere is a 3D figure with a curved surface where every point is equidistant from the center. A sphere’s surface area measures the overall area the object’s surface takes up.

Computing the Surface Area of a Sphere

Let us consider this example:

In preparing for the celebration of the Songkran Festival, the locals are making water balls. Two sizes are available: large and small. The radius length of the smaller water ball is 2.5 inches, while the larger one is twice the radius length of the smaller water ball. Compute the surface area of the two water balls.

Solution

Now, let us have this example when the radius is the measurement we need to find.

Water guns are common when people celebrate Songkran Festival. A water gun is given on the left. Compute the radius of its spherical water storage.

Solution

Since we know that SA=50.24 sq. in., let us equate it to the surface area formula 4πr² and solve for the radius.

4πr²=50.24  (Apply the formula)

(4)(3.14)r²=50.24

12.56 r²=50.24

$\frac{12.56 r^2}{12.56}$=$\frac{50.24}{12.56}$ (Divide both sides by 12.56)

r²=4

$\sqrt{r^2}$=$\sqrt{4}$ (Take the square root of both sides)

r=2 (Simplify)

The radius length of spherical water gun storage is 2 inches.

## Examples with Solution

Example 1

Determine the surface area of a sphere with a radius of 5 units.

Solution

Substitute r = 5 into the formula:

SA = 4π(5)² = 4π(25) = 100π square units.

Example 2

Suppose we have a sphere with a radius of 3 units, find its surface area.

Solution

Using the formula, SA = 4πr², we substitute r = 3 into the formula:

A = 4π(3)² = 4π(9) = 36π square units.

Example 3

If a sphere’s surface area is 200π square units, what is the radius of the sphere?

Solution

Here, we know the surface area and need to find the radius.

Given A = 200π, we set this equal to the formula and solve for r:

4πr²=200π

Divide both sides by 4π:

r²=$\frac{200π}{4π}$ = 50

Take the square root of both sides:

r =$\sqrt{50}$ ≈ 7.07 units

Therefore, the sphere’s radius is approximately 7.07 units.

## Real-life Application with Solution

A basketball has a diameter of approximately 24 cm. What is the surface area of the basketball?

Solution

First, note that the radius is half of the diameter. So, the radius of the basketball is 24 cm ÷ 2 = 12 cm.

Substitute r = 12 into the formula:

SA = 4π(12)² = 4π(144) = 576π square cm.

So, the surface area of the basketball is approximately 576π square cm.

This application is essential, for instance, when designing a basketball. Knowing the surface area can help determine the amount of material needed to cover the ball.

## Practice Test

1. Determine the surface area of a sphere with a radius of 7 units.

2. What is its radius if a sphere’s surface area is 144π square units?

3. Find the sphere’s surface area with a diameter of 10 units.

4. A spherical balloon has a radius of 5 cm. Find the surface area of the balloon.

5. A sphere has a surface area of 300π square units. What is its diameter?

6. If a sphere has a diameter of 15 units, what is its surface area?

1. 196π square units

2. 6 units

3. 100π square units

4. 100π square cm.

5. Approximately 17.32 units.

6. 225π square units.

### What is the distinction between a sphere’s surface area and volume?

A sphere’s surface area is the total area that the surface of the sphere covers, while its volume is the amount of space inside the sphere. Surface area and volume are calculated using different formulas.

### Why is the formula for the surface area of a sphere A = 4πr²?

The formula comes from calculus, specifically the process of integration. The objective is to break the sphere into infinitesimally small disks, find the surface area of each disk, and then add them up.

### Can a sphere’s surface area be negative?

No, the surface area of a sphere cannot be negative. The radius of a sphere is always positive, and since the formula for the surface area involves squaring the radius, the result is always positive.

### What does the π in the formula for the surface area of a sphere represent?

The π is a mathematical constant known as Pi. π  is the ratio of the circumference or perimeter of any circle to its diameter. Its approximate value is 3.14159.