How to Calculate the Area of a Circle

Last week, we have discussed how to calculate the circumference of a circle. In this post, we learn how to calculate the area of a circle. The area of a circle which we will denote by A is equal to the product of \pi and the square of its radius r. Putting it in equation, we have

A = \pi r^2.

In the examinations, the value of \pi is specified. They usually use 3.14, 3.1416 or \frac{22}{7}.

If you can recall, the radius is the segment from the center to the point on the circle as shown below. The radius is half the diameter. The diameter is the longest segment that you can draw from one point on the circle to another. It always passes through the center.

Note: We also use the term radius to refer to the length of the radius and diameter as the length of the diameter.  


Now that we have reviewed the basic terminologies, let us have some examples on how to calculate the area of a circle.

Example 1

What is the area of a circle with radius 8 centimeters. Use \pi = 3.14.


A = \pi r^2

A = (3.14)(8^2)

A = (3.14)(64)

A = 200.96

So, the area of the circle is 200.96 square centimeters (sometimes abbreviated as sq. cm.)

Be Careful! Length is measured in units and area is measured in square units. For example, the radius given is in inches (length), the answer for area is in square inches. So, since the Civil Service Exam is multiple choice, the examiner could place units and square units in the choices.

Example 2

Find the area of a circle with diameter 14 centimeters. Use \pi = \frac{22}{7}.


Notice that the given is the diameter, so we find the radius. Since the diameter is twice its radius, we divide 14 centimeters by 2 giving us 7 centimeters as the radius. Now, let’s calculate the area.

A = \pi r^2

A = (\frac{22}{7}) (7^2)

A = 22(7)

A = 154 square centimeters.

Example 3

Find the radius of a circle with area 6.28 square meters. Use \pi = 3.14.


In this problem, area is given. We are looking for the radius. We still use the original formula and make algebraic manipulations later, so we don’t have to memorize a lot of formulas.

A = \pi r^2

We substitute the value of area and \pi.

6.28 = 3.14 r^2

We are looking for r, so we isolate r to the right side (recall how to solve equations).

\displaystyle \frac{6.28}{3.14} = \frac{3.14r^2}{3.14}

2 = r^2

Since, we have a square, we get the square root of both sides. That is

\sqrt{2} = \sqrt{r^2}

\sqrt{2} = r

So, radius is square root of 2 meters or about 1.41 meters.

In this calculation, 2 is not a perfect square. Since you are not allowed to use calculator, they probably won’t let you calculate for the square root of number. So, in this case,  the final answer is that the radius of the circle is square root of 2 meters (meters, not square meters).

That’s all for now. In the next post, we will be working on problems involving area of a circle.

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