- 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

The following diagram gives the formula for the area of a circle. Scroll down the page for more examples and solutions on how to use the formula and also a proof or the formula.

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.

Example:

Find the area of a circle with radius 4cm.

Examples:

1. Find the area of a circle with radius 3cm.

2. Find the area of a circle with diameter 20cm.

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

Example:There are two circles such that the radius of the larger circle is three times the radius of the smaller circle.

(a) How many times the circumference of the larger circle is the circumference of the smaller circle?

(b) What is the ratio of the area of the larger circle to the area of the smaller circle?

Example 1: Janell wants to replace the glass in her mirrors. She can buy glass for $0.89 per square inch. If the price includes tax, how much would she pay, to the nearest penny?

Example 2: The rectangle has a length of 21 inches and each circle is congruent. What is the area of one circle?

Example 3: A tire from Karen's car is shown below. What is the closest distance traveled, in feet, after 3 full rotations of the tire?

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.

Examples: The area of a circle is 12.56 yd

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.

Example:

The area of the circle is 12.56 cm

The shape is made from 4 semicircles of diameter 8m and a square of side 8m. The radius of each circle is therefore 4m, and you have the equivalent of 2 whole circles. Then add on the area of the square.

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

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