How to Calculate Perimeter and Area of a Circle with Examples
A circle is a two-dimensional shape made up of points that are stretched out from a given point (centre) on the plane by a fixed or constant distance (radius). The fixed point is referred to as the circle's origin or centre, and the fixed distance between each point and the origin is referred to as the radius. Calculating the perimeter and area of a circle is one of the earliest ways to learn how to work with pi. Find out how to calculate these measurements in this article that breaks down the formulas used for calculating these values.
A circle
What is the Perimeter of a Circle?
The boundary or overall arc length of a circle's perimeter is known as its perimeter. A circle's circumference is the term used to describe its edge. Therefore, since the perimeter of a circle provides information on its circumference, the formula is also referred to as the circumference of a circle.
The Perimeter of Circle Formula
The circle's radius acts as one of the three variables in the perimeter of a circle formula, along with two constants.
Perimeter of a circle $=2 \pi r=\pi d$ units
Here,
$r=$ radius of the circle
$d=$ diameter of the circle
Perimeter of Circle Formula
How to Find the Area of the Circle?
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 circle area formula is $A=\pi r^2$.
To calculate the area of the circle, we must know its radius or its diameter. It is calculated by multiplying $\pi$ by the radius squared: $A=\pi r^2$
Here, r is the radius of the circle.
We can also calculate the area of a circle based on its diameter.
We already know that the radius can be expressed as the diameter divided by two:
$D=2 r$
$r=\dfrac{D}{2}$
If we substitute this expression for the radius in the formula for the area of the circle, we obtain the formula as a function of the diameter:
$A=\pi\left(\dfrac{D}{2}\right)^2$
$A=\pi\left(\dfrac{D^2}{4}\right)$
Where $D$ is the diameter of the circle.
Circle Area Formula
The image below shows the area of the circle formula, with $r$ as the radius and $d$ as the diameter.
Circle Area Formula
Solved Examples
Q 1. What is the area of a circular park with a diameter of 8 feet?
Ans: Given that the diameter of the circular park is 8 feet.
Thus, the radius of the park is half the diameter. That is, the radius is 4 feet. Use the formula for the area of a circle, $A=\pi r^2$, where $r$ is the circle's radius. Substitute 4 for $r$ in the formula.
$A=\pi(4)^2$
$A=\pi(16)$
$A=3.14(16)$
$A=50.24 f t^2$
Therefore, the park's area is $50.24 \mathrm{ft}^2$.
Q 2. What is the perimeter of a circle pizza if the area is $25 \mathrm{~cm}^2$?
Ans: The circle area formula is:
$A=\pi r^2$
We substitute the area for the value of $25 \mathrm{~cm}^2$ that gives us the statement:
$25=\pi r^2$
Then we isolate $r$ and operate to obtain its value:
$r =\sqrt{\dfrac{25}{\pi}}$
$r =\sqrt{7.95}$
$r =2.82 \mathrm{~cm}$
Put the value in the perimeter of the circle formula:
$P=2 \pi r$
We substitute the value of the radius in the circumference of a circle formula:
$P=2 \pi(2.82)$
$P=2(3.14)(2.82)$
$P=17.71 \mathrm{~cm}$
The circle's perimeter is $17.71 \mathrm{~cm}$.
Practice Questions
Q 1: What is the perimeter of a circle knowing its diameter is 2m?
Ans: 6.28 m
Q 2: Find the area of a circle with a radius of 20 cm.
Ans: $1256.63 \mathrm{~cm}^2$
Summary
We discussed here in the article the perimeter and area of a circle. The space a circle takes up on a two-dimensional plane is known as the area of the circle. Alternately, the area of the circle is the area included inside the circumference or perimeter of the circle. $A=\pi r^2$, where $r$ is the circle's radius, is the formula for calculating a circle's surface area.
The length of the circle's edge determines its circumference. This implies that a circle's circumference and perimeter are equal. The circle's circumference will equal the length of the rope that precisely encircles its edge.
FAQs on Perimeter and Area of Circle Explained
1. What are the fundamental formulas for calculating the perimeter and area of a circle?
The two primary formulas related to a circle's measurements are:
- Perimeter (Circumference): The formula is C = 2πr, where 'C' is the circumference, 'r' is the radius, and π (pi) is a mathematical constant approximately equal to 3.14159. The perimeter is the total length of the circle's boundary.
- Area: The formula is A = πr², where 'A' is the area and 'r' is the radius. The area represents the total space enclosed within the circle's boundary.
2. What is the difference between the perimeter and the area of a circle?
The key difference lies in what they measure. The perimeter, also known as the circumference, is the one-dimensional distance around the circle—like a fence around a circular park. It is measured in linear units (e.g., cm, m). In contrast, the area is the two-dimensional space inside the circle—like the grass covering the park. It is measured in square units (e.g., cm², m²).
3. How can you calculate the area of a circle if you only know its diameter?
You can calculate the area directly from the diameter. First, remember that the radius (r) is half of the diameter (d), so r = d/2. Substitute this into the area formula (A = πr²):
- A = π * (d/2)²
- A = π * (d²/4)
So, the formula to find the area directly from the diameter is A = (πd²)/4. Simply square the diameter, multiply by π, and then divide by 4.
4. What role does Pi (π) play in the perimeter and area of a circle?
Pi (π) is a fundamental constant that represents the ratio of a circle's circumference to its diameter. Its value is approximately 3.14159. It is an irrational number, meaning its decimal representation never ends or repeats. Pi is essential because it provides a universal constant to relate the linear dimension (radius or diameter) of any circle to its perimeter and area, ensuring the formulas C = 2πr and A = πr² work for circles of all sizes.
5. If you double the radius of a circle, how does it affect its perimeter and area?
The effects are different for perimeter and area:
- Perimeter: Since the perimeter formula is C = 2πr, doubling the radius (from r to 2r) will result in a new circumference C' = 2π(2r) = 2 * (2πr). Therefore, doubling the radius doubles the perimeter.
- Area: The area formula is A = πr². Doubling the radius gives a new area A' = π(2r)² = π(4r²) = 4 * (πr²). Therefore, doubling the radius quadruples the area.
6. How do you find the area and perimeter of a semicircle?
A semicircle is half of a circle, so its calculations include both the curved part and the straight diameter.
- Area of a Semicircle: It is simply half the area of a full circle: A = (πr²)/2.
- Perimeter of a Semicircle: This is a common point of error. The perimeter is the length of the arc (half the circumference) plus the length of the straight diameter. So, the formula is P = πr + 2r, or P = r(π + 2).
7. What are some real-world examples where calculating a circle's perimeter and area is important?
Calculating the perimeter and area of circles is crucial in many fields:
- Construction and Engineering: To find the amount of fencing needed for a circular garden (perimeter) or the amount of concrete for a circular base (area).
- Manufacturing: To determine the material needed to create circular objects like gears, pipes, or pizza pans.
- Sports: To mark the boundary of a circular athletic field (perimeter) or a centre circle in basketball (area).
- Science: To calculate the cross-sectional area of wires or pipes for fluid dynamics and electrical calculations.
8. Can two different circles have the same perimeter but a different area?
No, this is impossible for circles. The perimeter (circumference) of a circle is uniquely determined by its radius (C = 2πr), and its area is also uniquely determined by the same radius (A = πr²). If two circles have the same perimeter, they must have the same radius. Consequently, they must also have the same area. A unique perimeter corresponds to a unique area for any given circle.
