How To Find The Area Of A Quarter Circle Formula And Solved 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.
FAQs on Area Of A Quarter Circle Explained With Formula
1. What is the area of a quarter circle?
The area of a quarter circle is one-fourth of the area of a full circle, given by the formula (πr²)/4.
- Area of a full circle = πr²
- Since a quarter circle is 1/4 of a circle, divide by 4
- So, Area = (πr²)/4
2. What is the formula for the area of a quarter circle?
The formula for the area of a quarter circle is (πr²)/4, where r is the radius.
- π ≈ 3.14 or 22/7
- r = radius of the circle
- Square the radius and divide by 4
3. How do you find the area of a quarter circle step by step?
To find the area of a quarter circle, use the formula (πr²)/4 and follow these steps:
- Step 1: Measure or identify the radius (r).
- Step 2: Square the radius (r²).
- Step 3: Multiply by π.
- Step 4: Divide the result by 4.
4. How do you calculate the area of a quarter circle if the diameter is given?
If the diameter is given, first find the radius and then use (πr²)/4 to calculate the area.
- Radius = Diameter ÷ 2
- Use the formula: (πr²)/4
Area = (π × 5²)/4 = (25π)/4 = 6.25π cm².
5. What is the area of a quarter circle with radius 7 cm?
The area of a quarter circle with radius 7 cm is (49π)/4 cm².
- Formula = (πr²)/4
- r = 7 cm
- Area = (π × 7²)/4 = (π × 49)/4
6. Why do we divide by 4 to find the area of a quarter circle?
We divide by 4 because a quarter circle represents one-fourth (1/4) of a full circle.
- A full circle measures 360°.
- A quarter circle measures 90°.
- Since 90° is 1/4 of 360°, we divide the full area πr² by 4.
7. What is the difference between a quarter circle and a semicircle?
A quarter circle is one-fourth of a circle, while a semicircle is half of a circle.
- Area of quarter circle = (πr²)/4
- Area of semicircle = (πr²)/2
- Quarter circle angle = 90°
- Semicircle angle = 180°
8. What are the units of the area of a quarter circle?
The area of a quarter circle is measured in square units such as cm², m², or in².
- If radius is in centimeters, area is in cm².
- If radius is in meters, area is in m².
9. Can you give a real-life example of the area of a quarter circle?
A real-life example of the area of a quarter circle is calculating the space of a curved garden corner or a pizza slice shaped like 90°.
- If a garden corner has radius 4 m:
- Area = (π × 4²)/4 = (16π)/4 = 4π m²
10. What are common mistakes when finding the area of a quarter circle?
A common mistake when finding the area of a quarter circle is forgetting to divide by 4 after using πr².
- Using diameter instead of radius without halving it.
- Forgetting to square the radius.
- Not including square units in the final answer.
