How to Find the Area of a Quarter Circle?

When a circle is divided into \(4\) equal parts, \(4\) quadrants are formed, and each of these quarters is known as a quadrant. In this guide, you will learn more about the quarter circle and how to find its area.

How to Find the Area of a Quarter Circle?

A quarter-circle is one-fourth part of the whole circle. So, the area of a quarter circle is the fourth part of the whole area of a circle.

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Step by step guide to finding the area of a quarter circle

A circle is a collection of points that are at a constant distance from a fixed point. This fixed point and fixed distance are called “center” and “radius”, respectively. A quarter-circle is one-fourth. So the area of a quarter circle is exactly one-fourth of the area of a full circle.

Area of a quarter circle using radius

We know that the area of a circle is \(πr^2\). A quarter-circle is one-fourth portion of a full circle, so its area is one-fourth of the area of the circle. Thusthe area of a quarter circle in terms of radius \(\color{blue}{= \frac{πr^2}{4}}\)

Area of a quarter circle using the diameter

Since we have \(d = 2r\), \(r= \frac{d}{2}\). By substituting it in the above formula, we can get the area of a quarter circle in diameter.

The area of a quarter circle \(\color{blue}{= \frac{π(\frac{d}{2})^2}{ 4 }= \frac{πd^2}{16}}\)

How to find the area of a quarter circle?

Here are the steps to find the area of a quarter circle:

  • If the radius \((r)\) is given, immediately replace it with the formula \(\frac{πr^2}{4}\).
  • If the diameter \((d)\) is given, solve \(d = 2r\) for \(r\) and use the formula \(\frac{πr^2}{4}\) (or) immediately replace the value of d in the formula \(\frac{πd^2}{16}\).
  • When the circumference \((C)\) is given, solve \(C = 2πr\) for \(r\) and replace it in the formula \(\frac{πr^2}{4}\).
  • When area \((A)\) is given, solve \(A = πr^2\) for \(r\) and replace it with the formula \(\frac{πr^2}{4}\) (or) simply find \(\frac{A}{4}\).

Finding Area of a Quarter Circle – Example 1:

A circle has a diameter of \(32\space cm\), find the area of a quarter circle. \((\pi =3.14)\)

Solution:

Diameter of circle \(= 32\space cm\)

Area of a quarter circle \(= \frac{πd^2}{16}\)

\(=\frac{3.14 × 32^2}{16}\)

\(=\frac{3.14 × 32×32}{16}\)

\(=\frac{3215.36}{16}\)

\(=200.96\space cm^2\)

Exercises for Finding Area of a Quarter Circle

  1. Find the area of the quadrant with a radius \(24\space cm\).
  2. Find the area of the quadrant with a diameter \(16 \space in\)
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{452.16\space cm^2}\)
  2. \(\color{blue}{50.24 \space in^2}\)

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