A quadrant is one-fourth of a circle. Since a full circle measures 360 degrees, a quadrant covers 90 degrees. When we divide a circle into four equal parts, each part is called a quadrant. All four quadrants have the same size and shape. When combined, they make up the full circle again.

The perimeter of a quadrant is the total length around its boundary. This includes one-fourth of the circle’s outer edge (the arc) and the two straight sides (radii) connecting the arc to the center. So, the formula to find the perimeter of a quadrant is:

Perimeter = (1/4 × Circumference of the Circle) + 2 × Radius

In geometry, the coordinate plane is divided into four parts by the x-axis (horizontal) and y-axis (vertical). These parts are also called quadrants. Each quadrant is an infinite region in the plane.

When we talk about a quadrant of a circle, we mean one of the four equal parts formed by dividing the circle. These quadrants are not just geometric shapes but are also important in graphs and coordinate systems.

In the illustration above, the area highlighted as ABO is one of the circle’s four quadrants, and the angle AOB forms a right angle in the region’s middle.

What is the Perimeter of the Quadrant of a Circle?

The perimeter of a quadrant of a circle is the total length around its boundary. Since a quadrant is one-fourth of a full circle, its curved part is one-fourth of the circle’s circumference. Along with this curved part, a quadrant also has two straight sides, both equal to the radius of the circle. So, to find the perimeter of a quadrant, we add one-fourth of the circumference to twice the radius. This gives us the full boundary length of the quadrant. It’s a simple way to measure the edge of one-quarter of a circle.

Here, we will look at how to calculate a quadrant’s perimeter. Exactly one-fourth of any circle is a quadrant. At its centre, it has a 90-degree angle.

Let r represent the circle’s radius.

Circumference of circle = \( 2\pi r \)

Arc length of quadrant = \( \frac{circumference }{4}\) = \(\frac{\pi r}{2}\)

Perimeter of quadrant = arc + 2 radii

\( = \frac{\pi r}{2} + 2r\)

\( = r\left ( \frac{\pi }{2} + 2 \right )\)

So, Perimeter \( \left ( P \right ) = r\left ( \frac{\pi }{2} + 2 \right )\)

How to find the Perimeter of a Quadrant of a Circle?

Let us understand how we can find the perimeter of a quadrant of a circle through an example.

Imagine cutting a circular cake with a 10 cm radius into four equal pieces. Determine the perimeter of a single-circle quadrant.

Solution: As we known \( \pi = \frac{22}{7}\)

r is given 10 cm

We now simply apply the formula perimeter of the quadrant of a circle

\( = \frac{\pi r}{2} + 2r\)

\( = 10\left ( \frac{3.14 }{2} + 2 \right )\)

71.4 cm

Perimeter of a single-circle quadrant is 71.4 cm

Perimeter of a Semi Circle

The perimeter of a semicircle refers to the total length of the curved boundary that forms half of a circle. To calculate the perimeter of a semicircle, we need to consider the curved section, which is the semicircular arc, and the straight line segment, which is the diameter of the circle. The formula for the perimeter of a semicircle is derived by summing the lengths of the semicircular arc and the diameter. 

The length of the semicircular arc is half the circumference of the full circle, given by [latex]\frac{1}{2} \pi d[/latex], where [latex]d[/latex] represents the diameter. The length of the diameter is simply the distance across the circle passing through its center. Therefore, the perimeter of a semicircle is given by [latex]\frac{1}{2} \pi d + d[/latex].

Perimeter of Quadrant of an Ellipse

There are 4 equal sections within the ellipse, one in each quadrant.

Perimeter of ellipse = \(2\pi \left [ \sqrt{\frac{a^{2}+b^{2}}{2}} \right ]\)

Arc length of ellipse = \(2\pi \frac{\frac{\sqrt{a^{2}+b^{2}}}{2}}{4}\)

Perimeter of quadrant = arc +ab = \(2\pi \frac{\frac{\sqrt{a^{2}+b^{2}}}{2}}{4}+ab\)

where a = minor axis and b = major axis

Example 1: Calculate the area and perimeter of a quadrant of a circle of radius 21 cm.

Solution: The perimeter of a quadrant is also known as the length of the arc, and the area of a quadrant is a fraction of the total area of the circle. Let's proceed with the calculations:

Perimeter (Length of the Arc) of a Quadrant:

We will use the formula [latex]P = \frac{1}{4} \times 2\pi r + 2r[/latex], where [latex]r = 21[/latex] cm.

Substituting the values into the formula:

[latex]P = \frac{1}{4} \times 2\pi (21 , \text{cm}) + 2(21 , \text{cm})[/latex]

[latex]P = \frac{1}{4} \times 2 \times 3.14 \times 21 + 2 \times 21[/latex]

[latex]P = 33 + 42[/latex]

[latex]P = 75[/latex]

Therefore, the perimeter of the quadrant is 75 centimeters.

Area of a Quadrant:

The area of a quadrant is equal to one-fourth of the area of the entire circle. The formula to calculate the area of a quadrant is:

[latex]A = \frac{1}{4} \times \pi r^2[/latex]

Substituting the given radius of 21 cm, we have:

[latex]A = \frac{1}{4} \times \pi \times 21^2[/latex]

[latex]A = \frac{1}{4} \pi (441 , \text{cm}^2)[/latex]

[latex]A = \frac{1}{4} \times 3.14 \times 441[/latex]

[latex]A = 346.185 , \text{cm}^2[/latex]

The perimeter of a quadrant of a circle of radius r is:

The perimeter of a quadrant is the total length of the boundary of a quadrant. Thus, it is equal to the sum of the length of the curved section of the quadrant and the straight line segments. In this case, the curved section is one-fourth of the circumference of the full circle, which is [latex]\frac{1}{4} \times 2\pi r[/latex], and the straight line segment is the radius of the circle, which is 2r.

Therefore, the perimeter of a quadrant of a circle with radius r is given by [latex]P = \frac{1}{4} \times 2\pi r + 2r[/latex].

Example 2: Determine the perimeter of the quadrant with a radius of 6.2 cm.

Solution: Perimeter \( \left ( P \right ) = r\left ( \frac{\pi }{2} + 2 \right )\)

\( \pi \) = 3.14 , r = 6.2 cm

\( = 6.2 \left ( \frac{3.14}{2} + 2 \right )\)

= 22. 13 cm

As a result, a circle with a radius of 6.2 cm has a quadrant whose perimeter is 22.13 cm.

Example 3: Calculate the 7 cm-radius quadrant’s perimeter.

Solution: r= 7 cm and \( \pi = \frac{22}{7}\)

\( \left ( P \right ) = r\left ( \frac{\pi }{2} + 2 \right )\)

\( = \left ( 7 \right )\left [ \frac{22}{14}+ 2 \right ]\)

\( = 7\left [ \left ( \frac{11}{7} \right ) + 2 \right ]\)

\( = 7\left ( \frac{11+14}{7} \right )\)

= 25 cm

The 7 cm-radius quadrant’s perimeter is 25 cm

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