The area of a circle is the space occupied by the circle in a two-dimensional plane. Alternatively, the space occupied within the boundary/circumference of a circle is called the area of the circle. The formula for the area of a circle is A = πr2, where r is the radius of the circle. The unit of area is the square unit, for example, m2, cm2, in2, etc. Area of Circle = πr2 or πd2/4 in square units, where (Pi) π = 22/7 or 3.14. Pi (π) is the ratio of circumference to diameter of any circle. It is a special mathematical constant.

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The area of a circle formula is useful for measuring the region occupied by a circular field or a plot. Suppose, if you have a circular table, then the area formula will help us to know how much cloth is needed to cover it completely. The area formula will also help us to know the boundary length i.e., the circumference of the circle. Does a circle have volume? No, a circle doesn't have a volume. A circle is a two-dimensional shape, it does not have volume. A circle only has an area and perimeter/circumference. Let us learn in detail about the area of a circle, surface area, and its circumference with examples.

 1 Circle and Parts of a Circle 2 What Is the Area of Circle? 3 Area of Circle Formulas 4 Derivation of Area of a Circle Formula 5 Surface Area of Circle Formula 6 Real-World Example on Area of Circle 7 FAQs on Area of Circle

## Circle and Parts of a Circle

A circle is a collection of points that are at a fixed distance from the center of the circle. A circle is a closed geometric shape. We see circles in everyday life such as a wheel, pizzas, a circular ground, etc. The measure of the space or region enclosed inside the circle is known as the area of the circle. Radius: The distance from the center to a point on the boundary is called the radius of a circle. It is represented by the letter 'r' or 'R'. Radius plays an important role in the formula for the area and circumference of a circle, which we will learn later.

Diameter: A line that passes through the center and its endpoints lie on the circle is called the diameter of a circle. It is represented by the letter 'd' or 'D'.

Diameter formula: The diameter formula of a circle is twice its radius. Diameter = 2 × Radius

d = 2r or D = 2R

If the diameter of a circle is known, its radius can be calculated as:

r = d/2 or R = D/2

Circumference: The circumference of the circle is equal to the length of its boundary. This means that the perimeter of a circle is equal to its circumference. The length of the rope that wraps around the circle's boundary perfectly will be equal to its circumference. The below-given figure helps you visualize the same. The circumference can be measured by using the given formula: where 'r' is the radius of the circle and π is the mathematical constant whose value is approximated to 3.14 or 22/7. The circumference of a circle can be used to find the area of that circle.

For a circle with radius ‘r’ and circumference ‘C’:

π = Circumference/Diameterπ = C/2r = C/dC = 2πr

Let us understand the different parts of a circle using the following real-life example.

Consider a circular-shaped park as shown in the figure below. We can identify the various parts of a circle with the help of the figure and table given below. In a CircleIn our parkNamed by the letter
CentreFountainF
CircumferenceBoundary
ChordPlay area entrancePQ
RadiusDistance from the fountain to the Entrance gateFA
DiameterStraight Line Distance between Entrance Gate and Exit Gate through the fountainAFB
Minor segmentThe smaller area of the park, which is shown as the Play area
Major segmentThe bigger area of the park, other than the Play area
Interior part of the circleThe green area of the whole park
Exterior part of the circleThe area outside the boundary of the park
ArcAny curved part on the circumference.

The area of a circle is the amount of space enclosed within the boundary of a circle. The region within the boundary of the circle is the area occupied by the circle. It may also be referred to as the total number of square units inside that circle.

The area of a circle can be calculated in intermediate steps from the diameter, and the circumference of a circle. From the diameter and the circumference, we can find the radius and then find the area of a circle. But these formulae provide the shortest method to find the area of a circle. Suppose a circle has a radius 'r' then the area of circle = πr2 or πd2/4 in square units, where π = 22/7 or 3.14, and d is the diameter.

Area of a circle, A = πr2 square units

Circumference / Perimeter = 2πr units

Area of a circle can be calculated by using the formulas:

Area = π × r2, where 'r' is the radius.Area = (π/4) × d2, where 'd' is the diameter.Area = C2/4π, where 'C' is the circumference.

### Examples using Area of Circle Formula

Let us consider the following illustrations based on the area of circle formula.

Example1: If the length of the radius of a circle is 4 units. Calculate its area.

Solution:Radius(r) = 4 units(given)Using the formula for the circle's area,Area of a Circle = πr2Put the values,A = π42A =π × 16A = 16π ≈ 50.27

Answer: The area of the circle is 50.27 squared units.

Example 2: The length of the largest chord of a circle is 12 units. Find the area of the circle.

Solution:Diameter(d) = 12 units(given)Using the formula for the circle's area,Area of a Circle = (π/4)×d2Put the values,A = (π/4) × 122A = (π/4) × 144A = 36π ≈ 113.1

Answer: The area of the circle is 113.1 square units.

## Area of a Circle Using Diameter

The area of the circle formula in terms of the diameter is: Area of a Circle = πd2/4. Here 'd' is the diameter of the circle. The diameter of the circle is twice the radius of the circle. d = 2r. Generally from the diameter, we need to first find the radius of the circle and then find the area of the circle. With this formula, we can directly find the area of the circle, from the measure of the diameter of the circle. ## Area of a Circle Using Circumference

The area of a circle formula in terms of the circumference is given by the formula $$\dfrac{(Circumference)^2}{4\pi}$$. There are two simple steps to find the area of a circle from the given circumference of a circle. The circumference of a circle is first used to find the radius of the circle. This radius is further helpful to find the area of a circle. But in this formulae, we will be able to directly find the area of a circle from the circumference of the circle. ## Area of a Circle-Calculation

The area of the circle can be conveniently calculated either from the radius, diameter, or circumference of the circle. The constant used in the calculation of the area of a circle is pi, and it has a fractional numeric value of 22/7 or a decimal value of 3.14. Any of the values of pi can be used based on the requirement and the need of the equations. The below table shows the list of formulae if we know the radius, the diameter, or the circumference of a circle.

 Area of a circle when the radius is known. πr2 Area of a circle when the diameter is known. πd2/4 Area of a circle when the circumference is known. C2/4π

Why is the area of the circle is πr2? To understand this, let's first understand how the formula for the area of a circle is derived. Observe the above figure carefully, if we split up the circle into smaller sections and arrange them systematically it forms a shape of a parallelogram. When the circle is divided into even smaller sectors, it gradually becomes the shape of a rectangle. The more the number of sections it has more it tends to have a shape of a rectangle as shown above.

The area of a rectangle is = length × breadth

When we compare the length of a rectangle and the circumference of a circle we can see that the length is = ½ the circumference of a circle

Area of circle = Area of rectangle formed = ½ (2πr) × r

Therefore, the area of the circle is πr2, where r, is the radius of the circle and the value of π is 22/7 or 3.14.

The surface area of a circle is the same as the area of a circle. In fact, when we say the area of a circle, we mean nothing but its total surface area. Surface area is the area occupied by the surface of a 3-D shape. The surface of a sphere will be spherical in shape but a circle is a simple plane 2-dimensional shape.

If the length of the radius or diameter or even the circumference of the circle is given, then we can find out the surface area. It is represented in square units. The surface area of circle formula = πr2 where 'r' is the radius of the circle and the value of π is approximately 3.14 or 22/7.

Ron and his friends ordered a pizza on Friday night. Each slice was 15 cm in length.

Calculate the area of the pizza that was ordered by Ron. You can assume that the length of the pizza slice is equal to the pizza’s radius.

Solution:

A pizza is circular in shape. So we can use the area of a circle formula to calculate the area of the pizza.

Area of Circle formula = πr2 = 3.14 × 15 × 15 = 706.5

Area of the Pizza = 706.5 sq. cm.

Example 4: A wire is in the shape of an equilateral triangle. Each side of the triangle measures 7 in. The wire is bent into the shape of a circle. Find the area of the circle that is formed.

Solution:

Perimeter of the Equilateral Triangle: Perimeter of the triangle = 3 × side = 3 × 7 = 21 inches.

Since the perimeter of the equilateral triangle = Circumference of the circle formed.

Thus, the perimeter of the triangle is 21 inches.

Circumference of a Circle = 2πr = 2 × 22/7 × r = 21. r = (21 × 7)/(44) = 3.34.

Therefore, the Radius of the circle is 3.34 cm. Area of a circle = πr2 = 22/7 ×(3.34)2 = 35.042 square inches.

Therefore, the area of a circle is 35.042 square inches.

Example 5: The time shown in a circular clock is 3:00 pm. The length of the minute hand is 21 units. Find the distance traveled by the tip of the minute hand when the time is 3:30 pm.

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Solution:

When the minute hand is at 3:30 pm, it covers half of the circle. So, the distance traveled by the minute hand is actually half of the circumference. Distance $$= \pi$$ (where r is the length of the minute hand). Hence the distance covered = 22/7 × 21 = 22 × 3 = 66 units. Therefore, the distance traveled is 66 units.