Area of a Quarter Circle: Definition, Formula, Examples

What Is the Area of a Quarter Circle?

One-fourth of the area of a whole circle is called the area of a quarter of a circle. Area of a quarter circle is also known as the area of a quadrant. 

The following image shows division of a circle in four quarters.

Have you ever noticed how a pizza slice looks after we cut a whole pizza into four equal pieces? This one-fourth part of a circle is called a quarter circle.

A quarter of anything means one-fourth of that object or thing. Thus, by a quarter of a circle, we mean one-fourth of the circle. Four of these shapes will make up a full circle. We can also get a quarter circle by dividing a semicircle into two halves.

Area of a Quarter Circle Definition

In mathematical terms, the area created by two perpendicular radii and one-fourth area of the circumference of a whole circle is called the area of a quarter circle. It is also often referred to as a quadrant. These quadrants are all equal in size and area.

If we divide the area of a particular circle by 4, we can get the area of a quarter circle. It’s that easy!

Consider a circle with a radius “r” and a diameter “d.” 

We know that the area of a circle is calculated using the formula $\pi r^2$ .

Here,$\pi = 3.14 = \frac{22}{7}$ . 

Area of Quarter Circle: Formula Using Radius

The radius of a quarter of a circle is the same as the radius of the given circle.So let’s say if a circle has a length of radius “r,” then the area of circle $= \pi r^2$. 

A quarter circle is $\frac{1}{4}$ of the area of a whole circle.

Now, by dividing the area of the whole circle by 4, we can get the formula for a quarter circle.

Thus, the area of a quarter circle $= \frac{\pi r^2}{4}$ or $\frac{1}{4} \pi r^2$.

Area of a Quarter Circle: Formula Using Diameter

For any circle, its diameter is double of its radius.

Thus, $d = 2r$

We may calculate the area of a quarter circle in terms of diameter by substituting this value into the formula above.

Then, the area of circle $= (\frac{d}{2})^2$

                                         $= \frac{\pi d^2}{4}$

Area of a quarter circle $= \frac{1}{4} \times \frac{\pi d^2}{4}$

                                           $= \frac{\pi d^2}{16}$

 Thus, the area of a quarter circle in terms of diameter $= \frac{\pi d^2}{16}$

How To Find the Area of a Quarter Circle

Step 1: Note down the given radius or the diameter of the circle.

Step 2: If the radius “r” is given, substitute the value of “r” in the formula $\frac{\pi r^2}{4}$.

Step 3: If the diameter “d” is given, substitute the value of “d” in the formula $\frac{\pi d^2}{16}$.

Step 4: Area is measured in square units. Don’t forget to assign the appropriate unit to the calculated value of the area of a quarter circle.                

Facts about Area of a Quarter Circle

1. A circle is a two-dimensional shape and so is the quarter circle.

2. A quarter circle is formed when a circle is divided into four equal portions.

3. A quarter circle can also be formed by dividing a semicircle into two equal parts. 

4. A quarter circle is also known as a quadrant.

5. The circumference of a quarter circle is one-fourth of that of a circle with the same radius.

Conclusion

The area of a quarter circle is one fourth of the area of a circle. In this article, we learned in detail about the area of the quarter circle, formulas. Let’s solve a few examples and practice problems for better understanding.

Solved Examples on Area of a Quarter Circle

1. Linda ordered a 12-inch pizza for herself and her 3 friends. They want to share the pizza equally among themselves. Find the quantity of pizza each one got. (Use $ \pi = 3.14)

Solution:

Given data: The diameter of the pizza $= 12$ inches.

$ \pi = 3.14$

We know that the area of a quarter circle in terms of diameter $= \frac{\pi d^2}{16}$

   So,  $A = \frac{3.14 \times (12)^2}{16}$  

               $= \frac{3.14 \times 144}{16}$

              $= \frac{452.16}{16}$

                $= 28.26$ square inches.

The quantity of pizza each one got is 28.26 $in^2$.

2. Find the area of the quarter circle with the help of the image given below.

Solution: 

Given data:

The radius of the circle $= 7$ inches

We know that the area of a quarter circle in terms of its radius $= \frac{\pi r^2}{4}$

           $A =\frac{\pi r^2}{4}$

              $= \frac{3.14 \times 7^2}{4}$

              $= \frac{3.14 \times 49}{4}$

              $= \frac{153.86}{4}$

          $A  = 38.46$ square inches

The area of the quarter circle $= 38.46\; in^2$

3. The radius of a quarter circle shaped table is 3 ft. What is the area of the table? ( Use $ \pi = 3.14$)

Solution:

Given data:

The radius of the table $= 3$ ft

We know that the area of a quarter circle in terms of its radius $= \frac{\pi r^2}{4}$

             $A = \frac{\pi r^2}{4}$

               $= \frac{3.14 \times 3^2}{4}$

               $= \frac{3.14 \times 9}{4}$

             $= \frac{28.26}{4}$

             $=  7.065$ square ft.

The area of the quarter circle shaped table $= 7.065\; ft^2$

4. A circular park has a diameter of 50 yards. Only a quarter of the park is allotted for an open gym. Find the area allotted for the gym.

Solution:

Given data: Diameter of the park $= 50$ yards

We know that the area of a quarter circle in terms of diameter $= \frac{\pi d^2}{16}$

  So,  $A = \frac{3.14 \times(50)^2}{16}$  

               $= \frac{3.14 \times 2500}{16}$

              $= \frac{7850}{16}$

                $= 490.62$ square yards

The area of the park allotted for the gym is 490.62 square yards.

Practice Problems on Area of a Quarter Circle

Frequently Asked Questions on Area of a Quarter Circle