Area of a Circle – Definition, Formula, Derivation, Examples, FAQs

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What Is the Area of a Circle?

The area of a circle is the region occupied by the circle within its boundary (circumference) in a 2D plane. 

If the radius of a circle is r, the area is defined as A = πr2, where is a mathematical constant with approximate value of 3.14 or $\frac{22}{7}$. 

Area of a circle is measured in square units

Radius, circumference, area

Parts of a Circle

A circle is a two-dimensional closed geometric shape that is a collection of all points which are at a fixed distance from a fixed point (center). This fixed distance is called the radius of the circle.

Let’s learn about different parts of a circle. It will help us in understanding the different formulas used to calculate the area of a circle.

PartDefinition
CenterThe fixed point at the middle that is equidistant from all the points from every point on the boundary of the circle.
RadiusThe fixed distance from the center to any point on the circumference (boundary) of the circle. All radii (plural) of a circle are equal.
DiameterA straight line that connects any two points on the circumference while passing through the center of the circle. 
CircumferenceThe length of the boundary of a circle.
ChordA line segment that connects any two points on the circumference of a circle.
TangentA straight line that passes by touching the circumference at one point.
ArcA part of the circle’s circumference.
SegmentA region enclosed by an arc and a chord.
Sector A region enclosed by 2 radii and an arc.
Parts of a Circle

Area of Circle Formulas

DescriptionFormula
Area of a circle using radiusA = πr2
Area of a circle using diameterA $= \frac{\pi}{4} d^{2}$
Area of a circle using circumferenceA $= \frac{C^{2}}{4\pi}$

How to Find the Area of a Circle

Let’s discuss methods to find the area of a circle based on the information available.

  • How to Find the Area of a Circle with Radius

The formula for the area of a circle (A) using its radius r is as follows:

A = πr2

Example: A circle has a radius of 5 inches. Find the area of the circle. 

A = πr2

A = 3.14 × (5 inches)2

A = 3.14 × 25 inch2

A = 78.5 inch2

  • How to Find the Area of a Circle with Diameter

We know that A = πr2 and r $= \frac{d}{2}$

The formula for the area of a circle (A) using its diameter (d) as follows:

A $= \frac{\pi}{4} d^{2}$

Example: A circle has a diameter of 10 inches. Find the area of the circle.

A $= \frac{\pi}{4} d^{2}$

A $= \frac{\pi}{4} \times $(10 inches)2

A $= (\frac{3.14159}{4}) \times 100$ inch2

A $= 78.54$ inch2

  • How to Find the Area of a Circle with Circumference

Circumference = C = 2πr 

Thus, r $= \frac{C}{2\pi}$

Substitute this value in the formula A = πr2.

A $= \pi (\frac{C}{2\pi})^{2}$

A $= \frac{C^{2}}{4\pi}$

Example: A circle has a circumference of 31.42 units. Find the area of the circle.

A $= \frac{C^{2}}{4\pi}$

A $= \frac{(31.42)^{2}}{4 \times 3.14}$

A = 78.6 inch2

Derivation of Area of a Circle Formula

You can visualize and derive the area of a circle by the following two methods:

  1. Derivation of the area of a circle formula using the area of a rectangle

The figure shows a circle divided into 16 equal sectors of equal areas. If the sectors are cut out from the circle and placed as a parallelogram-shaped figure, the area of the circle will be equal to the area of this figure.

Visualizing the area of a circle using a rectangle

The pink-colored sectors will cover half of the circle and thus have half the circumference. The blue-colored sectors will cover the other half.

Radius = r

Circumference = 2πr

Circumference of Pink Sectors $= \frac{1}{2} \times 2\pi r = \pi r$ 

Circumference of Blue Sectors = πr

We can see that the parallelogram-like shape will appear like a rectangle if the number of sectors is increased. 

Deriving the formula for the area of a circle using area of a rectangle

Length of the rectangle = πr

Breadth = r

Area of rectangle = l × b

Putting the values from the figure into the formula

A = πr × r

A = πr2

Area of a circle =  πr2

  1. Derivation of the area of a circle formula using the area of a triangle

The figure below shows concentric circles throughout the radius. Imagine that the circle is cut along a radius and opened up to form a shape that resembles a triangle. 

The circumference of a circle will become the base of the triangle, and the radius will become its height.

Circumference = 2πr

Radius = r = Height of the Triangle

Deriving the Area of a circle using area of a triangle

So, the area of the triangle (A) will be equal to the area of the circle. We have

Since, Area of a triangle $= \frac{1}{2}$ × base × height

So, putting the values from the figure into the formula

A $= \frac{1}{2} × (2\pi r) × r$

A = πr2

Real-world Examples of Applications of Area of a Circle

Let us take a couple of real-world applications of the area of a circle.

Example: Tracy has a table with a circular tabletop having a diameter of 3 feet. Would a tablecloth with an area 5 square feet be sufficient for covering the tabletop?

A circular table top with diameter 3 feet

The radius of the tabletop will be half of the diameter.

$r =  \frac{d}{2} = \frac{3}{2} = 1.5$ feet

Now, the area of the circular tabletop can be calculated using the formula:

A = πr2

A = π(1.5)2

A = 7.065 square feet

Thus, the tablecloth having an area 5 square feet won’t be enough to cover the tabletop.

Such a calculation is useful in day-to-day life scenarios to know the precise measurements of circular objects, thus minimizing the wastage of money and resources.

Facts about Area of a Circle

  • The word “circle” itself comes from the Latin word “circulus,” meaning “a small ring.”
  • Archimedes is known for his method of exhaustion, which involved inscribing and circumscribing polygons around a circle to approximate its area.

Conclusion

In this article, we learned how to calculate the area of a circle. We discussed formulas used to find the area of a circle based on the information known to us. We also derived the formula for the area of a circle using the area of a rectangle and area of a triangle. Now, let’s solve a few examples and practice MCQs.

Solved Examples on Area of a Circle

Example 1: If the diameter of a circular garden is 50 feet, find the area of the garden.

Solution:

Diameter of the circular garden (d) = 50 feet

The radius of the circular garden (r) $= \frac{d}{2} = 25$ feet

The area of the circular garden (A) = πr2

A = π(25)2

A = 625π square feet

Therefore, the area of the circular garden is 625π square feet.

Example 2: Calculate the area of a circle with a radius of 10 feet in terms of π.

Solution: 

Radius (r) = 10 feet

The area of the circle (A) = πr2

A = π(10)2

A = 100π

Therefore, the area of the circular swimming pool is 100π square feet.

Example 3: What is the area of a circular window with a diameter of 3 feet?

Solution:

Diameter (d) = 3 feet

Radius (r) $= \frac{d}{2}$ 

r = 1.5 feet

So, A = πr2

A $= \pi \times (1.5)^{2}$

A = 2.25π

Therefore, the area of the circular window is 2.25π square feet.

Example 4: What is its area if a circular table measures 12π feet around its edge?

Solution:

Circumference of the circular table (C) = 12π

The area of the circular table when the circumference is given $= A = (\frac{C}{2π})^{2}$

A $= (\frac{12\pi}{2\pi})^{2}$

A = 62

A = 36 feet2

So, the area of the circular table is 36 feet2. We have another way to find out the area of the circular table when its circumference is provided.

A $= \frac{C^{2}}{4\pi}$

A $= \frac{12\pi^{2}}{4\pi}$

A $= \frac{144\pi}{4\pi}$

A = 36 square feet

Example 5: If a pizza restaurant charges $0.10 per square inch of pizza, what would be the cost of a pizza with a 14-inch diameter? (Round to the nearest whole number.)

Solution:

Diameter of the pizza = 14 inches

Radius of the pizza $= \frac{14}{2} = 7$ inches

We need to calculate the area first. For the area of the pizza, we can use the formula (A) = πr2

A = π × 72 = 49π square inches

Cost per square inch =  $0.10

Cost of a pizza with 49π square inches area $= 49\pi \times \$0.10 = \$15.386 = \$15$ (Rounded to the nearest whole number.)

Therefore, the cost of the pizza is $\$15$.

Practice Problems on Area of a Circle

Area of a Circle - Definition, Formula, Derivation, Examples, FAQs

Attend this quiz & Test your knowledge.

1

What is the area of a circle whose diameter is 8 units?

$8\pi \;unit^{2}$
$16\pi\;unit^{2}$
$64\pi\;unit^{2}$
$32\pi \;unit^{2}$
CorrectIncorrect
Correct answer is: $16\pi\;unit^{2}$
d = 2 units
r = 4 units.
A $= \pi r^{2} = \pi (4)^{2} = 16\pi$ square units
2

What is the radius of a circle with an area of 36π square inches?

6 inches
12 inches
18 inches
36 inches
CorrectIncorrect
Correct answer is: 6 inches
A $= \pi r^{2}$
$36\pi = \pi r^{2}$
$r = \sqrt{36} = 6$ inches.
3

What is the area of a circle with a circumference of 18π units?

$81\pi\; unit^{2}$
$9\pi\; unit^{2}$
$81\; unit^{2}$
$324\pi\; unit^{2}$
CorrectIncorrect
Correct answer is: $81\; unit^{2}$
$C = 2\pi r$
Thus, $r = \frac{C}{2\pi} = \frac{18\pi}{2\pi} = 9$ units
Now, we can put the value of r into the area formula to calculate the area of the circle.
A $= \pi r^{2}$
A $= \pi(9)^{2} = 81\pi\; unit^{2}$
4

What is the area of a unit circle?

2π square units
$\pi^{2}$ square units
π square units
$\frac{\pi}{2}$ square units
CorrectIncorrect
Correct answer is: π square units
Radius of unit circle = 1 unit
A $= \pi r^{2} = \pi (1)(1) = \pi$ square units

Frequently Asked Questions about Area of a Circle

Area of a circle $= \pi r^{2}$, where r is the radius of the circle.

The area of a circle can never have a negative value. The area of a circle is the amount of space it encloses given by A = πr2 . Both π and r are positive, thus area is always a positive value.

The radius of a circle is always a positive value and not a negative one. The radius of a circle is the distance from the center to any point on its boundary. It is always a positive value as per its definition.

The area of a circle is directly proportional to the square of radius. The circumference of a circle is directly proportional to the radius.

The surface area is a term used in reference with three-dimensional objects. It refers to the total area of all its faces or surfaces. A circle is a plane two-dimensional shape, so it does not have a “surface area”; instead, it just has an area. So, the surface area of a circle is nothing but its area.