Area of Quadrant Explained with Formula and Applications

A circle is characterized as the locus of a considerable number of focuses that are equidistant from the inside. However, a quadrant is one-fourth segment of a circle which is achieved when a circle is partitioned equally into four segments or rather 4 quadrants by two lines that are perpendicular to each other in nature. In this article, let us talk about what a quadrant is, how to compute the area of a quadrant and the area of quadrant formulas with some examples for a better understanding of the concepts of quadrants.  At the end of this section, you will be able to state what a quadrant is and use the area of the quadrant formula to solve your problems.

In this article, students will learn the concept of quadrants, other relevant terms, and examples of accurately identifying and plotting points using quadrants. Students are required to study the concept of quadrants thoroughly in order to solve a wide range of math problems. Knowing how to find the area of quarter circles will serve students well as they advance in math classes and as they take mathematics tests. 

What is Quadrant?

The coordinated frame system has four quarters or segments in it and these quarters or segments are known as the quadrants. These quadrants are all equal in size and area. With respect to a circle, the quarter of a circle is known as a quadrant, which is a segment of 90 degrees. Let us consider four such quadrants attached together. What does it make? It forms a circle. Let’s understand this better with the help of the image given below. You can see that the circle has been divided into four equal parts. Now, these parts are known as quadrants. Each of these quadrants is equal in size and at the midpoint or the center O, they all make a 90-degree right angle. 

How to Calculate the Area of a Quadrant of Circle?

Before we go ahead and learn how to calculate the area of a quadrant of a circle, there are a few things you must know. Stated below are the key factors you must keep in handy before you solve the problem.

In order to find the area of a quadrant of a circle, you first need to know the area of the circle. Here’s a list of things you need to know.

1. The Center of a Circle: All the points of the circle are at an equal distance from the center of the circle. Hence, this is known as the center. 

2. The Radius of a Circle: The distance from the center of the circle to any point on the circle, is known as the radius of the circle. It is denoted by the letter R.

3. The Diameter of a Circle: The diameter is twice the radius of the circle. It is denoted by the letter D.

4. The Circumference of a Circle: The distance around the edge of the circle is called the circumference of the circle.

5. The formula of the circumference of Circle: Circumference = 2πr

6. The Area of a Circle: The total amount of sq units occupied by a circle is known as the area of the circle.

7. The formula of Area of Circle: Area = π x radius x radius or Area of circle = πr2

Now that we know the area of the circle, let’s calculate the area of the quadrant of the circle. 

We know that the circle can be divided into four segments or portions and these portions can be called quadrants. Since there are four quadrants in a circle, you can just divide the area of the circle by 4. 

Therefore, 

Area of a Quadrant = \[\frac{\pi r^{2}}{4}\].

Methods to Calculate the Area of a Quadrant

The area of a quadrant of a circle can be calculated by two methods.

Method 1: By dividing the area of a circle by 4 to proportionate it to the area of the one-fourth part of the circle, we can obtain the area of a quadrant

Area of circle =  \[\pi \times r^{2} \].

One-fourth area of circle = \[\frac{1}{4}\]. 

Area of quadrant = \[\frac{1}{4} \times \pi \times r^{2} \]

Also, Area of quadrant = \[\frac{1}{4} \times \pi \times (\frac{d}{2})^{2} \]

Method 2: By using the area of the sector of a circle, we can obtain the area of a quadrant.

Area of a sector of a circle = \[(\frac{θ}{360º}). \pi. r^{2}\]

where θ is equal to 90° because the quadrant of a circle is a type of sector having a right angle.

Area of a quadrant = area of a sector of a circle of θ as 90°

Area of a quadrant = \[(\frac{90°}{360°}). \pi. r^{2}\].

Area of a quadrant = \[\frac{1}{4}. \pi. r^{2}\]

The Formula for The Perimeter of a Quadrant

Pulling both parts together, the formula for the perimeter (​p​) of a quadrant is:

p = 0.5πr + 2rp = 0.5πr + 2r

This is really easy to use. For example, if you have a quadrant with ​r​ = 10, this is:

p​=( 0.5 x π x 10 ) + ( 2 x 10 ) = 5π + 20 = 15.7 + 20 = 35.7​

Solved Examples

Question 1: The radius of a circle is 6 cm. Find the area of the circle, the perimeter of the circle, and the area of the quadrant of the circle. 

Solution: 

Given, 

The radius = 6 cm 

We know that,

Area of circle = πr2

Circumference = 2πr

Area of a Quadrant = \[\frac{\pi r^{2}}{4}\]

(i) Area of circle

Area of Circle = πr2

Area of Circle = 3.14 * 62

Area of Circle = 3.14 * 6 * 6

Area of Circle = 3.14 * 36

Area of Circle = 113.04 cm2

(ii) Circumference of the Circle

Circumference = 2πr

Circumference = 2 * 3.14 * 6

Circumference = 12 * 3.14

Circumference = 37.68 cm.

(iii) Area of a Quadrant of the Circle

Area of a Quadrant = \[\frac{\pi r^{2}}{4} \]

Area of a Quadrant =  \[\frac{\pi 6^{2}}{4} \]

Area of a Quadrant =  \[\frac{\pi 6 \times 6}{4} \]

Area of a Quadrant = 3.14 * \[\frac{36}{4}\]

Area of a Quadrant = \[\frac{113.04}{4}\]

Area of a Quadrant = 28.26 cm2

Question 2: The radius of a circle is 2m. Find the area of the circle, the perimeter of the circle, and the area of a quadrant of the circle. 

Solution: 

Given, 

The radius of a circle is = 2m 

We know that,

Area of circle = πr2

Circumference = 2πr

Area of a Quadrant = \[\frac{\pi r^{2}}{4}\]

(i) Area of circle

Area of Circle = πr2

Area of Circle = 3.14 * 22

Area of Circle = 3.14 * 4

Area of Circle = 12.56 m2

(ii) Circumference of the Circle

Circumference = 2πr

Circumference = 2 * 3.14 * 2

Circumference = 4 * 3.14

Circumference = 12.56 m.

(iii) Area of a Quadrant of the Circle

Area of a Quadrant = \[\frac{\pi r^{2}}{4}\]

Area of a Quadrant = \[\frac{\pi 2^{2}}{4}\]

Area of a Quadrant = \[\frac{\pi \times 2 \times 2}{4}\]

Area of a Quadrant = \[\frac{3.14 \times 4}{4}\]

Area of a Quadrant = 3.14 m2.