## Quarter Circle – Introduction

In our everyday lives, we come across many shapes in the form of various objects, like a square table, a rectangular tray, a football in the shape of a circle, oval eggs, etc.

Have you ever observed the shape of a slice of pizza when divided into 4 equal parts? It looks like a triangle, but one of the sides is actually curvy. This shape is called a quarter circle.

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## What Is a Quarter Circle?

A quarter of anything means one-fourth of that object or thing. Thus, by a quarter of a circle, we mean one-fourth of the circle. We can also get a quarter circle by dividing a semicircle into two halves.

The following image shows division of a circle in quarters.

In mathematical terms, the area created by two perpendicular radii and one-fourth area of the circumference of a whole circle is called a quarter circle. It is also often referred to as a **quadrant**.

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## Area of a Quarter Circle

One-fourth area of a whole circle is known as the area of a quadrant or a quarter of a circle.

Quarter circle formula for area can be given as:

Area of a quarter circle $=$ Area of circle $/ 4 = \frac{π\text{r}^2}{4}$

**Note:**

The value of $π = 22/7$ or, $3.14$

## Perimeter of a Quarter Circle

The perimeter of a quarter circle consists of two radii and the circumference of a quarter circle (which is the one fourth part of the circumference of a circle). Hence, the formula for calculating it would be as follows:

Perimeter of a Quarter Circle $= \text{r} + \text{r} + (\frac{2π\text{r}}{4}) = 2\text{r} + (\frac{π\text{r}}{2})$

## Solved Examples

**Example 1: **If the radius of a circle is 6 cm, what will be the area of a quarter of its circle?

**Solution:**

Area of a quadrant $= π\frac{\text{r}^2}{4}$

$= \frac{3.14 \times 6 \times 6}{4}$

$= \frac{113.04} {4}$

$= 28.26 \text{cm²}$

**Example 2:** If the diameter of a circle is 32 cm, what will be the area of a quarter circle?

**Solution:**

Diameter of the circle $= 32$ cm

So, radius of the circle $= \text{diameter}/2 = 32/2 = 16 \text{cm}$

Area of quadrant $= \frac{π\text{r}}{24} = \frac{3.14 \times 16 \times 16} {4} = 200.96 \text{cm²}$

**Example 3:** A circular park has a radius of 35 yards. One-quarter of the park has a playground for toddlers. Find the perimeter of the playground.

**Solution:**

The total radius of the park $(\text{r}) = 35$ yards

The perimeter of the quarter of the park $= 2\text{r} +$ $(\frac{π\text{r}}{2})$

$= 2 \times 35 + ( \frac{3.14 \times 35} {2})$

$= 70 + 54.95$

$= 124.95$ yards

## Practice Problems

## Quarter Circle

### Choose the correct formula for calculating the area of a quarter circle.

The formula to calculate the area of a quarter of a circle is $\frac{\text{πr}^2}{4}$

### Sam ordered a 16-inch pizza for himself and his 3 friends. They want to share the pizza equally among themselves. Find the quantity of pizza each one got. $(\text{Use} π = 3.14)$

Radius of the pizza $(\text{r}) = d / 2$

$= 16 / 2 = 8$ inches

Area of each quarter portion of pizza $= \frac{\text{πr}^2}{4} = \frac{3.14 \times 8 \times 8}{4}$

$= 50.24$ square inches

### What is the perimeter of a quarter circle whose radius is 14 units?

Given radius of the circle $= 14$ units

The perimeter of a quarter circle $= 2r + (\frac{\text{πr}}{2})$

$= 2 \times 14 + (\frac{3.14 \times 14}{2}$)

$= 28 + 21.98$

$= 49.98$ units

## Frequently Asked Questions

**What is another term used for a quarter circle?**

A quarter-circle is also known as a quadrant and is formed by dividing a whole circle into four equal parts.

**Is the radius of a quarter circle different from the radius of the whole circle?**

No, the radius of a quadrant will always be equal to the radius of the whole circle.

**How can we calculate the area of a quarter circle by using the radius?**

If the circle has a radius *r*, then area of the quadrant $= π\text{r}² / 4$