A Quarter of a Circle - Definition, Formulae & Examples

What Is a Quarter Circle?

The portion of the area formed by two radii of the circle that are perpendicular to each other is called a quarter of a circle. Hence, the one-fourth area/portion of the circumference of a circle is a quarter circle. It is also known as a quadrant or a quadrant of a circle. So, in clear words, we can say that if we divide a circle into four equal parts, each part is a quarter circle or a quadrant. The image of a quarter of a circle is given below.

A Quarter of a Circle

Formulae for Calculating the Area of a Quarter Circle

Now we will derive a formula to calculate the area of a quarter of a circle. Let’s take a circle whose radius is 'r' and diameter 'd'. So we know that d = 2r. Now we can derive the formulae for the area of a quarter circle in terms of radius and diameter.

1. Calculation of the area of a Quarter Circle using Radius

We have studied that the area of a circle is calculated by πr2. Now, we already know that a quarter circle is one-fourth portion of a full circle. Thus, the area of the quarter circle is one-fourth of the area of the circle.

Hence, the area of a quarter circle in terms of radius = πr2 / 4

2. Calculation of the area of a Quarter Circle using Diameter

We know that d = 2r, then r = d/2. Now, we will substitute this in the above formula of calculation of the area of a quarter circle in terms of radius and we will get the area of a quarter circle in terms of diameter.

Hence, the area of a quarter circle in terms of diameter= π(d/2)2/ 4

The area of ¼ of a circle in terms of diameter = πd2 / 16

Area of a Quarter Circle in terms of radius and diameter

Solved Examples

Example 1: The radius of a circular park is 60 metres. A quarter circular portion of this part is used for doing exercise for old people. Find the area of the portion that is used by the old people for exercise. Use π = 3.142.

Solution: The radius of the circular park is given, r = 60 metres.

The area of the portion used by the old people for exercise can be calculated with the help of the formula of the area of a quarter circle.

The portion of the park used by the old people for exercise= πr2 / 4 = (3.142)(60)2 / 4 = 2827.8 square metres.

Answer: The required area used by the old people for exercise is 2827.8 square metres.

Example 2: Rohan ordered a pizza for himself and his three friends. They want to share it equally. The pizza is circular shaped and its diameter is 32 centimetres. Find the amount of pizza that each of them will get. Use π = 3.14.

Solution: The diameter of the pizza is given, d = 32 centimetres.

Since it is mentioned in the problem statement that the pizza is divided into four equal parts, each part is a quarter circle and hence its area can be calculated with the help of the formula of the area of a quarter circle formula.

The radius of the pizza is r = d/2 = 32/2 = 16 centimetres

Area of each portion = πr2 / 4 = (3.14)(16)2 / 4 = 200.96 square centimetres.

Hence the amount of pizza that each of them will get will be 200.96 square centimeters.

Conclusion

In this article, a quarter of a circle is discussed in simple language. The formulae for the calculation of a quarter of a circle are also given. The formulae are provided in terms of radius as well as diameter.