Before knowing what is the area of a quarter circle, let us recall what is a circle and a quarter circle. A circle is a locus (collection) of points that are at a fixed distance from a fixed point. This fixed point and the fixed distance are called the "center" and "radius" respectively. A quarter-circle is one-fourth portion of a circle. So the area of a quarter circle is exactly one-fourth of the area of the full circle.

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Let us learn the formula for the area of a circle along with its proof, a few solved examples, and practice questions.

1.What is a Quarter Circle (Quadrant)?
2.Area of a Quarter Circle Formulas
3.How to Find the Area of a Quarter Circle?
4. FAQs onArea of a Quarter Circle

What Is a Quarter Circle (Quadrant)?


The area (or) portion that is formed by two radii that are perpendicular to each other and one-fourthportion of the circumference of a circle is known as a quarter circle. This is also known as a quadrant of a circle. i.e., if we divide a circle into 4 equal parts, each part is a quarter circle (or) a quadrant.


Area of a Quarter Circle Formulas


Consider a circle of radius 'r' and diameter 'd'. We know that d = 2r. Let us derive the formulas for the area of a quarter circle in terms of radius and diameter.

Area of a Quarter Circle Using Radius

We know that the area of a circle isπr2. As we learned already in the previous section, a quarter circle is one-fourth portion of a full circle and thus its area is one-fourth of the area of the circle.

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

Area of a Quarter Circle Using Diameter

Since d = 2r, we have r = d/2. Substituting this in the above formula, we can get the area of a quarter circle in terms of diameter.

The area of a quarter circle =π(d/2)2/ 4 =πd2/ 16

Thus,the area of a quarter circle in terms of diameter =πd2 / 16

Note:Here,π is a mathematical constant whose value is considered to be 22 / 7 (or)3.141592...

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How to Find the Area of a Quarter Circle?


Consider a circle of radius 'r'. Here are the steps to find the area of thequarter circle.

If the radius (r) is given then straight away substitute it in the formulaπr2/ 4.If the diameter (d) is given then either solve d = 2r for 'r'and use the formulaπr2/ 4 (or) straight away substitute the value of d in the formulaπd2 / 16.If the circumference (C) is given then solve C = 2πr for 'r' and substitute it in the formulaπr2/ 4.If area(A) is given then either solve A =πr2for 'r'and substitute it in the formulaπr2/ 4 (or) simply find A / 4.
Now that we have understood the formula ad method to find area of a quarter circle, let us have a look at a few solved examples for better understanding.

Solved Examples on Area of a Quarter Circle


Example 1:The radius of a circular park is 40 yards. A quarter circular portion of this part is allotted for playing equipment. Find the area of the portion that is allotted for the playing equipment. Useπ = 3.142.

Solution:

The radius of the circular park is, r = 40 yards.

The area of the portion allotted for playing equipment can be found by using the area of a quarter circle formula.

The portion of the park allotted for playing equipment =πr2/ 4 =(3.142)(40)2/ 4 =1256.8 square yards.

Answer: The required area of playing equipment =1256.8 square yards.

Example 2:James ordered a pizza for him and his 3 friends. They want to share it equally. The pizza is circular shaped and its diameter is 16 inches. Using the area of a quarter circle formula, find the amount of pizza that each of them got. Useπ = 3.14.

Solution:

The diameter of the given pizza is, d = 16inches.

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Since the pizza is divided into 4 equal parts, each part is a quarter circle and hence its area can befound by using the area of a quarter circle formula.