Area of a Circle

If you know the radius of a circle, you can easily find its area. The area of a circle with radius \large{r} is calculated by multiplying the constant \color{blue}\large\pi to the square of the radius, \large{{\color{red}r}^2}. By the way, the irrational number \large\pi is the ratio of the circumference of a circle to its diameter which is approximately equal to 3.1416 when rounded to four decimal places.

formula to find area of the circle which is pi times radius square

Examples of Finding the Area of Circle

1) What is the area of a circle with radius 4 inches?

As you might have guessed, this is a very straightforward problem. The measure of the radius is given to us which is \color{red}4 inches. The only thing we have to do is substitute the value of the radius into the formula then simplify.

Notice that in our answer above, we just kept the area in terms of \pi. We did not multiply the number 16 to the approximate numerical value of \pi, i.e. \pi = 3.14.

If you leave your answer as 16\,\pi , you got it right since it is the exact value of the area. However, if you wish your answer as an approximation then you can multiply it through with the estimated value of \pi which in this case we used \color{blue}3.14, rounded to two decimal places.

Therefore, the area of the circle is about 50.24 square inches.

However, if you want a more precise value of the area, you can use the calculator’s internal value of \pi. Usually, we can retrieve that by pressing a secondary key. In TI-84, you press the 2nd key followed by the caret key ^.

Now, rounding the answer to two decimal places, we have 50.27 square inches.


2) Find the area of a circle with a diameter of 10 centimeters.

Here’s the deal. You are assumed to give your answer in terms of \pi unless you are explicitly told to write the area as an approximation. If your teacher wants it to be an approximation, this is where you replace \pi with some numerical value.

Now, let’s sketch a circle with a diameter 10 cm.

To calculate the area of a circle, we will need the radius. However, we can easily figure out what the radius is if we know the diameter. The radius of a circle is half the diameter. Thus, {\Large{{10} \over 2}} = 5.

Plug in the value of the radius into the formula then simplify.

Therefore, the area of the circle is 25\pi square centimeters.


3) What is the area of a circle if the diameter has endpoints \left( { - 7,2} \right) and \left( {5,2} \right)?

Let’s plot the endpoints of the diameter. Draw a line segment connecting them and make sure it passes through the center.

Now, we want to find the diameter of the circle then divide it by 2 to get the radius. In this situation, the diameter can be found easily by inspection. Since the endpoints of the diameter lie on the horizontal line y=2, the length of the diameter is the absolute value of the difference of the x-coordinates.

\left| { - 7 -5 } \right| = \left| { - 12} \right| = 12

Because the diameter is 12 units, its radius must be 6 units. The only thing we have to do is substitute the value of the radius into the formula.

Therefore, the area of the circle is 36\,\pi square units.


This lesson is in progress. Thank you for your patience!