Circles can be witnessed from micro to macro-objects around us. For example, our Earth is a flat circle; the moon is a gigantic circle; your dog might be playing with a circular ball, and much more. A circle is a two-dimensional shape whose locus is equidistant from the circle’s center. The circle’s boundary is known as its perimeter or the circumference, while the area bounded by its boundaries is known as the area. Do you know about the parts of a circle? And how is the area related to them?
Understanding the different parts of the circle
The midpoint inside any circle is its center. The line drawn from one end of the circle touching the periphery to the other edge across the center of the circle is known as the circle’s diameter. Half of this diameter is known as radius. What do we say if the diameter does not pass through the center? Then, the diameter will no longer be the diameter. Instead, it will be called a chord. A chord is a line drawn from one periphery to the other periphery of the circle without crossing the center. From this, one can conclude that the longer chord present in the circle will be the circle’s diameter.
A circle can possess an infinite number of chords, diameters, and radii. Have you ever eaten a slice of pizza? What did you observe? That slice is a part of the circular pizza, separated across the center. Therefore, a part of the circle cut through the center, making a specific angle with the center known as a sector. How is the circle area related to these parts of the circle? What is the formula for the area of a circle?
Area of circle formula: The universal form
The formula for the area of a circle is given by 𝜋r2, where r is the radius of the circle. The value of 𝜋, pronounced pi, is 22/7 or 3.14. Pi is the ratio of the circumference to the diameter of a circle. It is one of the widely used mathematical constants. The area of the circle is measured in terms of square units. Whatever be the unit of the radius, diameter, or circumference, the same will be for the area of the circle but in terms of squares. It is denoted as units².
The circular table will have an area, the circular plate will have an area, the ball will have an area, and like this, many more objects will have a circular area. Area defines the space required to keep the object in a particular place. So if we need to place a circular disc in the cupboard, we need to find the area, i.e., how much space will be needed. After area, does the circle have volume? Since a circle is a two-dimensional figure, it does not possess volume. Then what about its surface area?
The surface area of a circle formula: 2D version of a circle
Since the surface area is the property of a three-dimensional figure, the two-dimension circle does not have a surface area like that of the sphere. However, to find the area of the sphere that is needed to be placed somewhere can be found by the surface area. And this surface area is the same as the area of a circle. Hence, any circle’s surface area will be the same as the area of the circle, i.e., 𝜋r2.
Area of circle formula in terms of diameter
The diameter is double the radius, i.e., we can write diameter = 2 x radius. Therefore, the radius will be d/2. From the area of the circle 𝜋r2, putting the value of r in terms of d, we get,
A = 𝜋 (d/2)2 = 𝜋/4 d2. This is the area of a circle in terms of diameter.
Area of circle formula in terms of circumference
The circumference of a circle is the boundary of the circle, denoted by 2𝜋r, where r is the radius of the circle. Therefore, C = 2𝜋r. Taking out the value of r from here and putting in the area of a circle formula, we get,
A = 𝜋 (C/2𝜋)2 = C2/ 4𝜋. This is the area of the circle formula in terms of the circumference of the circle.
Area of a sector of a circle formula
After knowing the area of a circle when radius, diameter, and circumference are given, it’s time to find the area of the circle when the sector is given. A sector is a portion of a circle, sometimes referred to as a wedge. When two radii are drawn from the center of a circle to the edge of the circle, the region bounded by these two radii and the circumference is known as the sector. Just like a slice of pizza? Yes, a pizza slice is an example of a sector if the slice is cut through the center of a circle, not across the circumference.
How to measure the central angle subtended by the circle? Well, use a protractor. Place the protractor between the two radii and measure the central angle. In many questions, the angle is given in the question. So, no one has to bother measuring the angle.
The area of the sector when the angle and radius of the sector are given is denoted by,
Asector = 𝜃/360° 𝜋r², where 𝜃 will be in degrees.
Asector = ½ r² 𝜃, where 𝜃 is in radians.
The relationship between radians and degrees is given as radian = degree x 𝜋/180
After finding the area of the sector, it can be used to find the area of the circle.
Area of circle = Area of sector x 360 / Central angle, here the central angle is in degrees.
We got to know about all the ways to find the area of a circle, but where does the universal formula come from?
Deriving the area of a circle formula
Why is the formula to find the area of a circle 𝜋r²? To derive the area of a circle formula, divide a circle into various triangles, such that these triangles can be joined in the form of a rectangle as shown in the figure. The more the number of sections, the more clear the rectangle shape will be.
We know, that the area of a rectangle is the product of its length and breadth. From the given diagram, the length of the rectangle is half of the circumference of the circle, which is denoted by 𝜋r. The breadth of the rectangle is the radius.
Therefore, the area of the rectangle is length x breadth = 𝜋r x r = 𝜋r², which is the area of the circle formula.
Learn more about the area of the circle formula by solving examples
Example 1: What is the area of a circle if the radius is 60 m?
Solution: According to the area of a circle formula,
A = 𝜋 r², where r is the radius
Putting the values in the formula, we get, A = 𝜋 60².= 11304 m².
Example 2: Find the area of a circle whose largest chord measures 32 cm.
Solution: We know that the largest chord in a circle is its diameter. Hence, using the area of a circle formula,
A = 𝜋/4 d², where d is the diameter of the circle.
Putting the values in the formula, we get, A = 𝜋/4 32²= 803.84 cm².
Example 3: Find the area of a circle whose sector’s area is 6𝜋 units and the angle subtended at the center is 45 degrees.
Solution: From the area of the sector formula, we know,
Acircle= Asector(360/C), where C is the central angle in degrees.
Therefore, the area of the circle will be,
A = 6𝜋 (360/45) = 48𝜋 = 150.72 units.
Example 4: Find the net area for the given circle.
Solution: In the given figure, two circles can be seen. The bigger one has a diameter of 11 cm and the smaller one has a diameter of 3.5 cm. According to the area of a circle formula, A = 𝜋/4 d², where d is the diameter of the circle.
Putting the values in the formula, we get,
The area of the bigger circle = AB = 𝜋/4 11² = 94.985 cm²
The area of the smaller circle = AC = 𝜋/4 3.52 = 9.616 cm²
To find the net area of the circle, subtracting the smaller circle’s area from the bigger circle, we get, 94.985 – 9.616 =85.369 cm².