About Area & Circumference of a Cirlce
- Calculating Area and Circumference of a Circle
- What is a Circle?
- Examples on finding a Radius and a Diameter of a Circle
- Circumference of a Circle
- Examples on finding Circumferences of a Circle
- Perimeter of a Circle
- Example on Calculating the Perimeter of a Semi-Circle
- Area of a Circle
- Examples on finding Areas of Circles
- Frequently Asked Questions
Area and Circumference of a Circle
We see circular shapes everywhere. When we go to eat at a pizzeria, the pizza we love to eat is circular in shape. Birthdays and cakes go hand in hand and most cakes too have a circular shape. Do you know how a cake mold is made? It depends on the area of the cake that is to be made. Let us know in the sections below how to find the area.
What is a Circle?
A circle is defined as the set of all points in a plane that are the same distance from the center. The radius is the distance between the circle’s center and any point on it. The diameter is the distance all around the circle through the center.
- A circle’s diameter ‘d’ is equal to twice its radius ‘r’. A circle’s radius ‘r’ is equal to half of its diameter ‘d’.
Diameter: \( d=2r \)
Radius: \( r=\frac{d}{2} \)
Examples on finding a Radius and a Diameter of a Circle
Example 1. The diameter of a circle is 21.670 feet. Find the radius?
Solution:
As we know, the radius of a circle, \( r=\frac{d}{2} \)
\( =\frac{21.670}{2} \) (Substituted d as 21.670 from the given data)
\( =10.835\) (Divided “d” value with 2)
Therefore, the radius of the given circle is 10.835 feet.
Example 2. The radius of a circle is 4.2 meters. Find the diameter?
Solution:
As we know, the diameter of a circle, \( d=2r \)
\( =2(4.2) \) (Substituted r as 4.2 from the given data)
\( =8.4\) (Multiplied “r” value with 2)
Therefore, the diameter of the given circle is 8.4 meters.
Example 3. The diameter of a circle is 16 centimeters. Find the radius?
Solution:
As we know, the radius of a circle, \( r=\frac{d}{2} \)
\( =\frac{16}{2} \) (Substituted d as 16 from the given data)
\( =8 \) (Divided “d” value with 2)
Therefore, the radius of the given circle is 8 centimeters.
Example 4. The radius of the circle is 9 yards. Find the diameter?
Solution:
As we know, the diameter of a circle, \( d=2r \)
\( =2(9) \) (Substituted r as 4.2 from the given data)
\( =18\) (Multiplied “r” value with 2)
Therefore, the diameter of the given circle is 18 yards.
Circumference of a Circle
The circumference of a circle is the distance all around the circle. For every circle, the circumference to diameter ratio is the same and its value is represented by the Greek letter ‘’, called ‘pi’. The values 3.14 and \( \frac{22}{7} \) are two approximations for the value of \( \pi \).
- It’s easier to use \( \frac{22}{7} \) as an estimate of ‘’ when the radius or diameter is a multiple of 7.
- A circle’s circumference ‘C’ is equal to ‘’times the diameter ‘d’ or multiplied by twice the radius ‘r’ and \( \pi \).
\( C=\pi d \) or \( C=2\pi r \)
.
Examples on finding Circumference of a Circle
Example 1. How do you calculate the circumference of a flying disc with a radius of 6 inches? Use 3.14 for \( \pi \).
Solution:
As we know, the formula for the circumference of a circle is:
\( C=2\pi r \)
\( C=2\times 3.14\times 6 \) (Substitute 3.14 for and 6 for “r”)
\( C=37.68 \) (Multiply)
The circumference is about 37.68 inches.
Example 2. What is the circumference of a watch face with a diameter of 30 millimeters? Use \( \frac{22}{7} \) for \( \pi \).
Solution:
As we know, the formula for circumference of a circle is
\( C=\pi d \)
\( C=\frac{22}{7}\times 30 \) (Substitute \( \frac{22}{7} \) for and 30 for “d”)
\( C=94.28 \) (Multiply)
The circumference is about 94.28 inches.
Perimeter of a Circle
We are familiar with the term ‘perimeter’ but in the case of circles, we use different terminology. The perimeter of a circle is called its circumference. A semicircle’s perimeter differs from a circle’s circumference. The perimeter is simply the outline of a straight-sided shape, whereas the circumference is the length of the circle’s outline.
Example on Calculating the Perimeter of a Circle
Example 1. The length of the straight side is 3 meters. The circumference of a circle with a diameter of 3 meters is one-half the distance around the curved part. What is the perimeter?
Solution:
As per the given question, divide the circumference by 2.
\( \frac{C}{2}=\frac{\pi d}{2} \)
\( =\frac{3.14\times 3}{2} \) (Substitute 3.14 for \( \pi \) and 3 for “d”)
\( =\frac{9.42}{2} \) (Multiplied \( \pi \) with “d”)
\( =4.71 \) (Divided)
So, the perimeter is about 3 + 4.71 = 7.71 meters.
Example 2. Find the perimeter of the semicircular region.
Solution:
As per the given question, divide the circumference by 2.
\( \frac{C}{2}=\frac{\pi d}{2} \)
\( =\frac{3.14\times 2}{2} \) (Substitute 3.14 for \( \pi \) and 2 for “d”)
\( =\frac{6.28}{2} \) (Multiplied \( \pi \) with “d”)
\( =3.14 \) (Divided)
So, the perimeter is about 2 + 3.14 = 5.14 feet.
Example 3. Find the perimeter of the semicircular region.
Solution:
As per the given question, divide the circumference by 2.
\( \frac{C}{2}=\frac{\pi d}{2} \)
\( =\frac{3.14\times 7}{2} \) (Substitute 3.14 for \( \pi \) and 7 for “d”)
\( =\frac{21.98}{2} \) (Multiplied \( \pi \) with “d”)
\( =10.99 \) (Divided)
So, the perimeter is about 7 + 10.99 = 17.99 centimeters.
Area of a Circle
The product of ‘\( \pi \)’ and the square of the radius ‘r’ determines the area ‘A’ of the circle. \( A=\pi r^2\) is the algebraic formula for area.
Examples on finding Areas of Circles
Example 1. Find the area of the circle. Use \( \frac{22}{7} \) for \( \pi \).
Solution:
Estimate \( 3\times 8^2=3\times 64=192 \)
The area of the circle, \( A=\pi r^2 \)
\( =\frac{22}{7}\times 8^2 \) (Substitute \( \frac{22}{7} \) for \( \pi \) and 8 for ‘r’)
\( =\frac{22}{7}\times 64 \) (Evaluate \( 8^2 \) and divide out the common factor)
\( =201.14 \) (Multiply)
The area is about 201 square centimeters. It’s reasonable compared to the estimated value of \(192 \sim 201.14 \).
Example 2. Find the area of the circle. Use 3.14 for \( \pi \).
Solution:
Estimate \( 3\times 14^2=3\times 196=588 \)
The radius is \( \frac{22}{8}=14 \) inches.
The area of the circle, \( A=\pi r^2 \)
\( =3.14\times 14^2 \) (Substitute 3.14 for \( \pi \) and 14 for ‘r’)
\( =3.14\times 196 \) (Evaluate \( 14^2 \) and divide out the common factor)
\( =615.44 \) (Multiply)
The area is about 615 square centimeters. It’s reasonable compared with the estimated value \(588 \sim 615.44 \).
Example 3. Find the area of a semicircle? From the given figure:
Solution:
The area of the semicircle is one-half the area of a circle with a diameter of 40 feet. The radius of the circle is \( \frac{40}{2}=20 \) feet.
\( \frac{A}{2}=\frac{\pi r^2}{2} \) (Divided the area by 2)
\( =\frac{3.14\times 20\times 20}{2} \) (Substitute 3.14 for \( \pi \) and 20 for ‘r’)
\( =\frac{3.14\times 400}{2} \) (Evaluate 20 square)
\( =628 \) (Simplified)
So, the area of the semicircle is about 628 square feet.
Yes, for circles, the circumference and perimeter both are the same. But for semicircles, the perimeter is different from the circumference.
The name ‘pi’ was not first used to denote the number until the 18th century, roughly two millennia after Archimedes calculated the significance of the number 3.14. In other words, the ancient Greeks who discovered the idea did not choose the Greek letter that was used to represent it.