In this lesson, we will learn

- the formula for the area of a circle
- how to find the area of a circle given radius or diameter
- how to solve word problems using the area of a circle
- when given the area, how to find the radius or diameter
- when given the area, how to find the circumference
- how to prove the formula for the area of a circle

Related Topics: More Geometry Lessons

A circle is a closed curve formed by a set of points on a plane that are the same distance from its center. The area of a circle is the region enclosed by the circle. The area of a circle is equals to pi (*π*) multiplied by its radius squared.

Pi (*π*) is the ratio of the circumference of a circle to its diameter. Pi is always the same number for any circle. The value of *π* (pi) is approximately
3.14159265358979323846...
but usually rounding to 3.142 should be sufficient

The area of a circle is given by the formula:

A = πr^{2}(see a mnemonic for this formula)

where *A* is the area and *r* is the radius.

Since the formula is only given in terms of radius, remember to change from diameter to radius when necessary. The radius is equals to half the diameter.

Example 1:

Find the area the circle with a diameter of 10 inches.

Solution:

Step 1: Write down the formula: | A = πr^{2} |

Step 2: Change diameter to radius: | |

Step 3: Plug in the value: | A = π5^{2} = 25π |

Answer: The area of the circle is 25*π* ≈ 78.55 square inches.

Example 2:

Find the area the circle with a radius of 10 inches.

Solution:

Step 1: Write down the formula: | A = πr^{2} |

Step 2: Plug in the value: | A = π10^{2} = 100π |

Answer: The area of the circle is 100*π* ≈ 314.2 square inches.

See also Area of a Sector

Worksheet to calculate the area of circle

Worksheet to calculate circumference and area of circle when given diameter or radius.

The following video shows how to use the formula to calculate the area of the circle given the radius.

The following video shows how to use the formula to calculate the area of the circle given the radius or the diameter

The following videos show how to solve word problems using the area of circles.

How many square meters of grass are covered if a 20-meter diameter center circle of a soccer field is painted red?

When a nine-inch diameter cream pie is thrown, how many square inches of target will it cover?

How many square inches of paper remain from a 12-inch square after a 12-inch diameter circle is cut out of the square?

The biggest circle you can draw on this paper would have a diameter of 8.5 inches. What is the area of the biggest circle you can make from this page?

From the formula *A = πr*^{2}, we see that we can find the radius of a circle by dividing its area by *π* and then get the positive square-root. The diameter is then twice the radius.

This video shows how to find the radius or diameter of a circle when given the area.

This video shows how to find the diameter of a circle when given the area.

A gallon of paint covers 400 square feet. What is the biggest diameter circle of grass that can be covered as a target for a skydiver with a gallon of paint?

This video shows how to find the diameter of a circle when given the area.

To break the record of 500 square feet of circular puddle from a water balloon, a circle would need to be bigger than what diameter?

Worksheet to calculate problems that involve the radius, diameter, circumference and area of circle.

Worksheet 1, Worksheet 2 on word problems that involve circles.

To find the circumference of a circle when given the area, we first use the area to find the radius. Then, we use the radius to find the circumference of the circle.

The following video shows how to find the circumference of a cricle given the area.

The following video shows some examples to calculate areas of circles and also composite shapes with circles or segments of circles.

This video shows a graphical proof of the formula of a circle.

It involves dividing the circle into many sectors and rearranging the sectors to form a rectangle. The base of the rectangle is shown to be *πr* and the height of the rectangle is *r*. The area of the rectangle is then the product of *πr* and *r*. The area of the circle which is equal to area of the rectangle is then *πr*^{2}.

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