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Area of a Quarter Circle
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 quartercircle is onefourth portion of a circle. So the area of a quarter circle is exactly onefourth of the area of the full circle.
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 on Area 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 onefourth portion 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 is πr^{2}. As we learned already in the previous section, a quarter circle is onefourth portion of a full circle and thus its area is onefourth of the area of the circle.
Thus, the area of a quarter circle in terms of radius = πr^{2} / 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 = πd^{2} / 16
Thus, the area of a quarter circle in terms of diameter = πd^{2} / 16
Note: Here, π is a mathematical constant whose value is considered to be 22 / 7 (or) 3.141592...
How to Find the Area of a Quarter Circle?
Consider a circle of radius 'r'. Here are the steps to find the area of the quarter circle.
 If the radius (r) is given then straight away substitute it in the formula πr^{2} / 4.
 If the diameter (d) is given then either solve d = 2r for 'r' and use the formula πr^{2} / 4 (or) straight away substitute the value of d in the formula πd^{2} / 16.
 If the circumference (C) is given then solve C = 2πr for 'r' and substitute it in the formula πr^{2} / 4.
 If area(A) is given then either solve A = πr^{2} for 'r' and substitute it in the formula πr^{2} / 4 (or) simply find A / 4.
Now that we have understood the formula and method to find the area of a quarter circle, let us have a look at a few solved examples for better understanding.
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. Use π = 3.142.
Solution:
The radius of the circular park, 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 = πr^{2} / 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. Use π = 3.14.
Solution:
The diameter of the given pizza is, d = 16 inches.
Since the pizza is divided into 4 equal parts, each part is a quarter circle and hence its area can be found by using the area of a quarter circle formula.
Method 1
The radius of the pizza is, r = d/2 = 16/2 = 8 inches.
Area of each portion = πr^{2} / 4 = (3.14)(8)^{2} / 4 = 50.24 square inches.
Method 2
Area of each portion = πd^{2} / 16 = (3.14)(16)^{2} / 16 = 50.24 square inches.
Answer: The area of each portion of pizza = 50.24 square inches.
FAQs on Area of a Quarter Circle
What Is a Quarter of a Circle Called?
When a circle is divided into 4 equal parts, 4 quarters are formed and each of these quarters is known as a "quadrant".
What Is the Area of a Quarter Circle?
The area of a quarter circle is onefourth of the area of a full circle of radius 'r'. i.e., the area of the quarter circle = πr^{2} / 4.
How To Calculate the Area of a Quarter Circle?
If r, d, C, and A are the radius, diameter, circumference, and area of a circle, one of these pieces of information is sufficient to find the area of a quarter circle as explained below.
 If r is given use the formula πr^{2} / 4.
 If d is given use the formula πd^{2} / 16.
 If C is given, then solve C = 2πr for 'r' and use the formula πr^{2} / 4.
 If A is given, then find A / 4.
What Is the Area of a Quarter Circle in Terms of Radius?
Consider a circle of radius 'r'. Then the area of a quarter circle in terms of r is πr^{2} / 4.
How To Find the Area of a Quarter Circle Using the Diameter?
If r and d are the radius and the diameter of a circle, then we know that d = 2r. If the value of 'd' is given, then we can find the area of a quarter circle in one of the following ways:
 Find 'r' using r = d/2 and then use the formula πr^{2} / 4 (or)
 Straight away substitute the value of d in the formula πd^{2} / 16.
What Is the Area of Quadrant of a Circle?
The area of a quadrant of a circle is nothing but the area of a quarter circle and hence it is onefourth of the area of a full circle. i.e., if 'r' is the radius of a full circle, then the area of quadrant of a circle = πr^{2} / 4.
What Is the Formula for Perimeter of a Quarter Circle?
A quartercircle is made up of two radii and onefourth portion of the circumference of a circle. So the perimeter of a quarter circle of radius 'r' is, r + r + (2πr)/4 = 2r + πr/2.
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