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 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.
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 one-fourth 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 πr2. 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...
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 πr2 / 4.
- If the diameter (d) is given then either solve d = 2r for 'r' and use the formula πr2 / 4 (or) straight away substitute the value of d in the formula πd2 / 16.
- If the circumference (C) is given then solve C = 2πr for 'r' and substitute it in the formula πr2 / 4.
- If area(A) is given then either solve A = πr2 for 'r' and substitute it in the formula πr2 / 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
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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 = πr2 / 4 = (3.142)(40)2 / 4 = 1256.8 square yards.
Answer: The required area of playing equipment = 1256.8 square yards.
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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 = πr2 / 4 = (3.14)(8)2 / 4 = 50.24 square inches.
Method 2
Area of each portion = πd2 / 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 one-fourth of the area of a full circle of radius 'r'. i.e., the area of the quarter circle = πr2 / 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 πr2 / 4.
- If d is given use the formula πd2 / 16.
- If C is given, then solve C = 2πr for 'r' and use the formula πr2 / 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 πr2 / 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 πr2 / 4 (or)
- Straight away substitute the value of d in the formula πd2 / 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 one-fourth 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 = πr2 / 4.
What Is the Formula for Perimeter of a Quarter Circle?
A quarter-circle is made up of two radii and one-fourth 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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