How to Calculate the Area of a Quadrant with Solved Examples
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.
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.
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.
The Diameter of a Circle: The diameter is twice the radius of the circle. It is denoted by the letter D.
The Circumference of a Circle: The distance around the edge of the circle is called the circumference of the circle.
The formula of the circumference of Circle: Circumference = 2πr
The Area of a Circle: The total amount of sq units occupied by a circle is known as the area of the circle.
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.
FAQs on Area of Quadrant: Complete Guide for Students
1. What is the area of a quadrant of a circle and what is its formula?
A quadrant is one-fourth of a circle, formed by two perpendicular radii and the connecting arc. Since a full circle has an area of πr², the area of a quadrant is simply one-fourth of that. The formula to calculate the area of a quadrant is:
Area of Quadrant = (1/4) * πr²
Here, 'r' represents the radius of the circle.
2. How do you calculate the perimeter of a quadrant?
The perimeter of a quadrant is the total length of its boundary. Unlike the area, you must account for both the straight sides and the curved arc. The boundary consists of:
- Two straight sides, which are the radii (r) of the circle.
- One curved side, which is the arc (one-fourth of the circle's circumference).
The formula is: Perimeter = 2r + (1/4) * (2πr) = 2r + (πr/2).
3. How can you find the area of a quadrant if you only know the circle's circumference?
If you are given the circumference (C) instead of the radius (r), you can still find the area of the quadrant in two steps:
- First, find the radius: Use the circumference formula, C = 2πr, to solve for r. This gives you r = C / (2π).
- Then, calculate the area: Substitute this value of 'r' into the area of the quadrant formula, A = (1/4)πr².
For example, if the circumference is 44 cm, the radius is 44 / (2 * 22/7) = 7 cm. The area is (1/4) * (22/7) * 7² = 38.5 cm².
4. What is the difference between the area of a quadrant and the area of a sector?
This is a common point of confusion. A sector is a portion of a circle enclosed by two radii and an arc, corresponding to a central angle (θ). A quadrant is a specific, special type of sector where the central angle is always 90 degrees. Therefore, every quadrant is a sector, but not every sector is a quadrant. The formula for a sector's area is (θ/360) * πr², which becomes (90/360) * πr² = (1/4)πr² for a quadrant.
5. How are the “sides” of a circle's quadrant defined?
A quadrant of a circle does not have 'sides' in the same way a polygon like a square does. Its boundary is composed of both straight and curved lines:
- Two Straight Boundaries: These are the two radii of the circle that meet at the circle's center, forming a 90-degree angle.
- One Curved Boundary: This is the arc of the circle, which represents exactly one-fourth of the total circumference.
6. How does doubling the radius of a circle affect the area of its quadrant?
The area of a quadrant is directly proportional to the square of its radius (r²). This means any change in the radius has an exponential effect on the area.
If you double the radius (from r to 2r), the new area will be (1/4)π(2r)² = (1/4)π(4r²) = 4 * [(1/4)πr²].
So, doubling the radius increases the area of the quadrant by a factor of four.
7. Can you provide a real-world example of calculating the area of a quadrant?
Imagine a circular garden sprinkler that covers a radius of 10 metres but is set to water only a 90-degree section of the lawn. This watered section is a quadrant. To find its area:
- Formula: Area = (1/4) * π * r²
- Substitute values: Area = (1/4) * 3.14 * (10 m)²
- Calculate: Area = (1/4) * 3.14 * 100 = 78.5 m².
The sprinkler waters an area of 78.5 square metres.
