How to find the perimeter of a semicircle

Video Transcript

Determine the perimeter of the figure, using 3.14 to approximate 𝜋.

The perimeter of any shape is the distance around the outside. In this question, we have three sides of a rectangle and a semicircle. The opposite sides of a rectangle are parallel and equal in length. Therefore, the two vertical sides are 45 centimetres. The perimeter is therefore equal to 45 plus 45 plus 35 plus 𝑥, where 𝑥 is the length of the arc of the semicircle.

The circumference of any circle is equal to 𝜋 multiplied by the diameter or 𝜋𝑑. As a semicircle is half of a circle, the length of the arc of the semicircle will be 𝜋𝑑 divided by two. In our semicircle, the diameter is 35 centimetres. This means that 𝑥 is equal to 𝜋 multiplied by 35 divided by two.

We are told to use 3.14 as an approximation for 𝜋. We need to multiply this by 35 and then divide by two. 3.14 multiplied by 35 is 109.9. Dividing this by two gives us 54.95. The length of the arc of the semicircle is 54.95 centimetres. We can now calculate the total perimeter by adding 45, 45, 35, and 54.95. Adding these four numbers gives us 179.95. The perimeter of the figure is 179.95 centimetres.

Explanation:

#"the perimeter (P ) or circumference of a circle is"#

#color(red)(bar(ul(|color(white)(2/2)color(black)(P=2pir=pid)color(white)(2/2)|)))#

#"where r is the radius and d is the diameter"#

#"here it is a semicircle, that is half the circle"#

#rArr1/2xx2pir=pir" is the perimeter"#

#"here " d=10rArrr=10/2=5#

#"the perimeter of this semicircle includes the diameter"#

#rArrP=(pixx5)+10#

#color(white)(rArrP)=(3.142xx5)+10#

#color(white)(rArrP)=15.71+10=25.71" cm"#

The continuous closed line that makes the boundary of a two-dimensional shape is known as the perimeter. It’s easy to work out the perimeter of a shape with straight edges, such as a square, because you simply add the lengths of each edge together. A different formula is required to work out the perimeter of a semi circle (a circle cut in half), because it consists of a curved edge as well as a straight edge. That formula is

P = \frac{1}{2} πd + d

where ​d​ is the diameter of the semi circle.

Apply the number pi (π) to the formula

P = \frac{1}{2} πd + d

Pi is a circle's circumference divided by its diameter and is always the same value, 3.14.

To find the perimeter of a semi circle, you have to know the diameter (the length of its straight edge). For example, if the diameter of your semi circle is 12 centimeters, the formula becomes

P = \frac{1}{2} ×(3.14×12) + 12

Work out

3.14 × 12 = 37.68

Then work out

37.68 ÷ 2 = 18.84

You now have the perimeter of the curved edge of the semi circle. But a semi circle also has a straight edge, which is why you have to add the length of the diameter to calculate the whole perimeter. In this case, work out

18.84 + 12 = 30.84

The perimeter of your semi circle is 30.84 centimeters.

The perimeter of a semicircle is the total length of its boundary. It is easy to calculate the perimeter of shapes that have straight edges, like in the case of a square, where all the sides can be simply added. However, the perimeter of a semicircle has a different formula since a semicircle is the half of a circle, and it consists of a curved and a straight boundary. Let us learn how to calculate the perimeter of semicircle in this lesson.

What is the Perimeter of Semicircle?

The perimeter of a semicircle, which is also known as the circumference of a semicircle, is defined as the total length of its boundary. It is calculated by adding the length of the diameter and half the circumference of the original circle. The unit of the perimeter of the circle is expressed in linear units like "inches", "feet", "meters" or "centimeters" etc.

Circumference of a Semicircle

The circumference of a semicircle is half of the circle's circumference. It is defined as the measurement of the arc that forms a semicircle. It does not include the length of the diameter. As we know that the circumference of the circle formula is 2πR, where R is the radius.

Formula of Circumference of Semicircle

Observe the following semicircle where AB is the diameter of the semicircle marked as 'd' and OM is the radius marked as 'r'. Let the perimeter of the semicircle be "P".

The perimeter of the semicircle is given as P = Half of Circumference of the original circle + Length of Diameter.
Circumference of circle = 2πr
Half of the circumference of the circle = (1/2) × Circumference of circle = (1/2) × 2πr = πr = π(d/2)
Length of diameter = d = 2r

The circumference of the semicircle in terms of the radius is P = C = 2πR/2 = πR
In terms of diameter, the circumference of the semicircle is P = C = π(d/2) + d.

How to Find the Perimeter of Semicircle?

The steps to determine the perimeter of a semicircle are given below:

• Step 1: Find the product of π and the radius of the semicircle.
• Step 2: Find the diameter of the semicircle.
• Step 3: Add the values obtained in the above two steps.
• Step 4: The value thus obtained is the perimeter of the semicircle.

Example: Find the circumference of a semicircle whose radius is 7 units. (Use π = 22/7)

Solution: Given: The radius of the semicircle is 7 units.

Thus, the perimeter of the semicircle is P = πr + 2r ⇒ P = (22/7 × 7) + (2 × 7)
= (22 + 14) units. Thus, P = 36 units
∴ The perimeter of semicircle is 36 units.

FAQs on Perimeter of Semicircle

What do you Mean by Perimeter of Semicircle?

The perimeter of a semicircle is defined as the total length of its boundary which is calculated by adding the length of the diameter and half the circumference of the original circle. Its unit is expressed in "inches", "feet", "meters" or "centimeters". The perimeter of a semicircle is also known as the circumference of the semicircle.

What is the Formula of Perimeter of Semicircle?

The formula for the perimeter of the semicircle is the sum of the length of the diameter and half the circumference of the original circle. It is written as Perimeter = (πr + 2r), where "r" is the radius of the semicircle and π is a constant with a value of 22/7. In terms of diameter, the formula is expressed as, Perimeter = π(d/2) + d, where 'd' is the diameter of the semicircle.

How to Find the Perimeter of Semicircle?

We can find the perimeter of a semicircle using the following steps:

• Step 1: Find the product of π and r where "r" is the radius of the semicircle.
• Step 2: Find the diameter of the semicircle.
• Step 3: Add the values in Step 1 and Step 2.
• Step 4: The value obtained in Step 3 is the perimeter of the semicircle.

What is the Perimeter of Semicircle of Radius 7 cm?

If the radius of a semicircle is 7 cm, the perimeter can be calculated with the help of the formula: Perimeter = (πr + 2r), where 'r' is the radius and π is equal to 22/7. On substituting the values we get, Perimeter = (πr + 2r) = (22/7 × 7) + (2 × 7) = 22 + 14 = 36 cm.

What is the Difference Between Circumference and Perimeter of Semicircle?

The perimeter of a semicircle and the circumference of a semicircle mean the same. Both of them refer to the total length of the boundary of a semicircle. Hence, circumference is another name for the perimeter of a semicircle.

How to find the Perimeter of Semicircle with Diameter?

If the diameter of a semicircle is given, the perimeter can be calculated with the help of the formula: Perimeter = π(d/2) + d, where 'd' is the diameter of the semicircle and π is equal to 22/7. For example, the diameter of a semicircle is given as 16 units, on substituting the values we get, Perimeter = π(d/2) + d = [π × (16/2)] + 16 = 41.14 units

How to find the Area and Circumference of a Semicircle?

The area of a semicircle can be calculated with the help of the formula, Area = πr2 /2, where 'r' is the radius of the semicircle and π is a constant with a value of 22/7. For example, the radius of a semicircle is 7 units, let us substitute the values in the formula, Area = πr2 /2 = [(22/7) × 72] /2 = 77 square units. The circumference of a semicircle can be calculated with the help of the formula, Circumference = 2πR/2 = πR, where 'R' is the radius of the semicircle and π is a constant with a value of 22/7. For example, the radius of a semicircle is 14 units, let us substitute the values in the formula, Circumference = 2πR/2 = πR = 22/7 × 14 = 44 units.

What is the Circumference of a Semicircle with a Diameter of 8 units?

If the diameter of a semicircle is given, the circumference can be calculated with the help of the formula: Circumference = π(d/2) + d, where 'd' is the diameter of the semicircle and π is equal to 22/7. In this case, the diameter of a semicircle is given as 8 units. On substituting the values we get, Circumference = π(d/2) + d = [π × (8/2)] + 8 = 20.57 units

How do you find the Radius if the Perimeter of Semicircle is Given?

If the perimeter of a semicircle is given, we can find the radius of the semicircle using the same formula, Perimeter = (π + 2)r, where 'r' is the radius and π is equal to 22/7. For example, if the perimeter of a semicircle is 21 units, on substituting the values we get, 21 = (22/7 + 2)r. This gives the value of radius as 4.08 units.

How to Find the Area of Semicircle When the Circumference of Semicircle is Given?

We can find the area of a semicircle when the circumference of the semicircle is given using the following steps:

• Step 1: Find the value of the radius using the formula of the perimeter of the semicircle, P = (πr + 2r), and equate it to the value of the perimeter of the semicircle.
• Step 2: Once, the equation is obtained, solve for "r".
• Step 3: Use the formula, A = (πr2)/2 to determine the value of the area of the semicircle.

What Happens to the Circumference of Semicircle if the Radius of the Semicircle is Doubled?

If the radius of the semicircle is doubled then the circumference will also get doubled. This means if the value of radius gets doubled and becomes (2r), the circumference of semicircle will be P = (π + 2)(2r) = 2(π + 2)r, which is twice the original circumference.

What is the Formula for the Circumference of Semicircle in Terms of the Diameter?

The formula for the circumference of a semicircle is, C = (πr + 2r), where 'r' is the radius. As we know that diameter is twice the value of the radius, thus, d = 2r. The formula for the circumference of a semicircle becomes, C = π(d/2) + d.