Area of quadrant 

A circle is a two-dimensional shape whose circumference consists of multiple points which are equidistant from a fixed point in the inside of the shape. A circle when cut into four equal parts gives 4 quadrants. Therefore, a quadrant is the one-forth part of a circle. It can also be defined as a sector of a circle formed by two radii intersecting each other at right angles.

Area of a quadrant

The area of a quadrant of a circle can be derived from the area of a circle. For that, we must know certain points about the area of a circle.

1. The centre O of the circle is equidistant from all the points on the circle. Therefore it is the mid-point of the circle.
2. The radius of a circle is the line segment joining the centre of a circle to any point on the boundary of the circle. It is expressed as ‘r’.
3. The diameter of a circle is the line segment joining one point on the boundary of a circle to another point present on its boundary while passing through the centre of the circle. It is also the longest chord of the circle. It is double the length of the radius and is denoted by ‘d’.

   diameter = 2 . radius

   d = 2 . r

   which implies

   r = d/2

4. Perimeter of a circle is defined as the measure of the overall boundary of the circle. The perimeter of a circle is known as the circumference of a circle. It is mathematically formulated as

   circumference/perimeter = 2 . π. r

   circumference/perimeter = 2 . π . d/2

5. Area of circle is the space occupied by the circle in a plane. Mathematically,

   area = π . r²

   area = π . (d/2)²

Calculating the area of a quadrant

The area of a quadrant of a circle can be calculated by two methods.

1. Method 1
   As we know, a quadrant is one-forth part of a circle. So to find the area of a quadrant, we can divide the area of circle by 4 to proportionate it to the area of the one-forth part of the circle.

   Area of circle = π . r²

   One-forth area of circle = ¼ . area of circle

   ¼ . Area of circle = ¼ . π . r²

   Area of quadrant = ¼ . π . r²

   Also, Area of quadrant = ¼ . π . (d/2)²

2. Method 2
   As we know, quadrant of a circle is also a sector of the circle. By using the area of sector of a circle, we can obtain the area of a quadrant.

   Area of a sector of a circle = (θ/360º) . π . r²

   where θ is equal to 90° because quadrant of a circle is a type of sector having a right angle.

   Area of a quadrant = area of sector of a circle of θ as 90°

   Area of a quadrant = (90°/360°) . π . r²

   Area of a quadrant = 1/4 . π . r²