If a circle is divided into divided into four equal parts, then each part is known as a quadrant of a circle. The four circle quadrants each cover an angle of 90 degrees which adds up to a total of 360 degrees.

The quadrants are labeled the first quadrant, second quadrant, third quadrant, and fourth quadrant in an anti-clockwise manner as shown in the image below.

We now give the formulae for calculating the perimeter and area of a quadrant. Since quadrants are a quarter (one-fourth) of the circle, it is expected that the area and circular boundary of a quadrant be (1/4)^{th} of that of a circle.

**Formulae for a circular quadrant:**

Note that when we calculate the perimeter, that is the entire boundary of the quadrant we add 2 times the radius to the circular boundary.

This is because the quadrant consists of a circular part and 2 straight lines whose length is equal to the radius.

**Example 1:**

Suppose that a circular cake having a radius of 10cm is cut into four equal parts. Calculate the area and perimeter of a single quadrant of the circle.

* Solution*: We know that

*π*= 22/7 and we are given r = 10cm. We now simply apply the formulae stated above.

Area of a Quadrant = *π*r^{2}/4 = 22/7 * 100/4 = 78.57 cm^{2}.

Perimeter of a Quadrant= *π*r/2+ 2r = 22/7 * 10/2 + 20 = 15.71 + 20 = 35.71cm.

**Example 2:**

Given a quadrant having a radius of 35 cm, calculate its area and the length of the circular arc/circular boundary of the quadrant.

* Solution*: We know that r = 35cm.

Area of a Quadrant = *π*r^{2}/4 = 22/7 * 35^{2}/4 = 962.5 cm^{2}.

Length of circular arc = *π*r/2 = 22/7 * 25/2 = 55cm.

**The Quadrant Circle in Trigonometry:**

The unit circle having a radius equal to 1 plays an important role in the definition of the trigonometric ratios of sine and cosine. When we draw the radius at a particular angle *θ* it intersects the circle at some point P.

The y-coordinate of that point is defined to be sin(*θ*) and the x-coordinate is defined to be cos(*θ*).

Depending on the quadrant in which the angle lies we can predict whether the sine or cosine of that angle will be positive or negative.

For example, if *θ* lies in the second quadrant then since the x-coordinate here is negative and the y-coordinate positive we conclude that cos(*θ*) takes a negative value and sin(*θ*) takes a positive value.