Semicircle

What is a Semicircle?

  • A semicircle is a half-circle that is formed by cutting a whole circle into two halves along a diameter line. 

  • A line segment known as the diameter of a circle cuts the circle into exactly two equal semicircles.

  • The semicircle has only one line of symmetry which is the reflection symmetry.

  • The semicircle is also referred to as a half-disk.

  • Since the semicircle is half of the circle (360 degrees), the arc of the semicircle always measures 180 degrees.         

 The dotted lines represent the circumference.        

What Do You Mean By the Perimeter of a Semi-Circle?

  • ince, you know that the semicircle is half of a circle. You might think that the perimeter of a semicircle is half the perimeter of the circle. But that’s not true! 

  • The perimeter of a semicircle is \[\pi R + 2R\], which can also be written as \[R\left( {\pi  + 2} \right)\] by factoring out R.

where, R= Radius of the semicircle.

π = Constant named Pi, approximately equal to 3.142

The unit of perimeter of a semicircle is cm, metre.

What Do You Mean By the Area of a Semicircle?

  • The area of a half circle generally refers to the space inside the semicircle or the area or region enclosed by it.

  • Here, the area of a semicircle is half the area of a circle.

  • The area of a semicircle or the area of half circle  is  \[\frac{{\pi {{\text{R}}^2}}}{2}\] .

where, R= Radius of the semicircle.

Π = Constant named Pi, approximately equal to 3.142.

The unit of area of a semicircle is \[{m^2}\] or\[c{m^2}\].

What Do You Mean By the Circumference of a Semicircle?

  • The perimeter and the circumference of a semicircle are the same.

  • The circumference of a semicircle is: \[2\pi R\].

where, R= Radius of the semicircle.

Π = Constant named Pi, approximately equal to 3.142.

The unit of circumference of a semicircle is m or cm.        

What Do You Mean By the Angle Inscribed in a Semicircle?

  • The angle inscribed within a circle is always equal to 90 degrees. 

  • This inscribed angle is formed by drawing a line from each end of the diameter to any point on the semicircle. 

  • No matter where the line touches the semicircle the angle is always equal to 90 degrees. 


Formulas to Remember

Half circle area formula

\[\frac{{\pi {{\text{R}}^2}}}{2}\]

Perimeter of semicircle formula

\[R\left( {\pi  - 2} \right)\]

Circumference of  semicircle formula

\[2\pi R\]


QUESTIONS

Question 1 

A semicircle has a diameter of 100 meters. Find the perimeter of the semicircle using the semicircle formula.

Solution

Let’s list down the given information,

Diameter=100 m

Perimeter=?

Formula to calculate the perimeter of a semicircle is, \[R\left( {\pi  - 2} \right)\]

We need R to calculate the perimeter of semicircle,

\[Radius = \frac{{Diameter}}{2} = \frac{{100}}{2} = 50{\text{ }}meter\]

\[\Pi  = 3.142\]

Therefore, perimeter = 50(3.142+2) = 257.1 cm


Question 2

Riya’s school basketball court has 2 semicircles both at each end. The semicircles have 6 -foot radii. What is the perimeter of one semicircle of the court?

Solution

Let’s list down the given information,

Radius=6 foot

\[\Pi  = 3.142\].

Perimeter=?

Formula to calculate the perimeter of a semicircle is, \[R\left( {\pi  - 2} \right)\]

Therefore, the perimeter of the semicircle at one end of the court is, 6(3.142+2) = 30.72 cm


Question 3

The circle given below in Figure 2.1 has a diameter of 8 cm. Find the following:

  • Perimeter of the semicircle

  • Area of the semicircle

Solution

Let’s list down the given information,

Diameter= 8cm

Perimeter=?

Area=?

Using  the perimeter of  semicircle formula that is, \[R\left( {\pi  - 2} \right)\]

We need R to calculate the perimeter,

\[Radius = \frac{{Diameter}}{2} = \frac{8}{2} = 8{\text{ }}cm\]

\[\Pi  = 3.142\]

Therefore, perimeter = 4(3.142+2) = 5.142 cm

Using the area of a semicircle formula = \[\frac{{\pi {{\text{R}}^2}}}{2}\]

Therefore, area of the semicircle is \[\frac{{3.142 \times 4 \times 4}}{2}\] = 25.14 \[c{m^2}\].


Question 4

In the Figure 2.2 given below the radius of a circular cake made by Hannah is 5 cm. Find the area of the exactly half of the cake.

Solution

Let’s list down the given information,

Radius= 5cm

Area=?

Area of semicircle formula is = \[\frac{{\pi {{\text{R}}^2}}}{2}\]

Therefore, area of the semicircle is \[\frac{{3.142 \times 5 \times 5}}{2}\]= 39.275\[c{m^2}\]


Question 5

Find the circumference of the semi-circle whose diameter is 7 cm.

Solution

Let’s list down the given information,

Diameter = 7cm

Circumference =?

\[Radius = \frac{{Diameter}}{2} = \frac{7}{2} = 3.5{\text{ }}cm\]

\[\Pi  = 3.142\]

Circumference of a circle = \[2\pi  \times R\]

Therefore, the circumference of the circle = \[3.142\; \times 3.5 \times 2 = {\text{ }}21.944{\text{ }}cm\]


Question 6

George has a garden outside his house which is in the shape of a circle with a diameter of 10 yards. George wants to fence exactly half of the garden. Find the perimeter of the part he wants to fence.

Solution 

Let’s list down the given information,

Diameter= 10 cm

Perimeter=?

Formula to calculate the perimeter of a semicircle is, \[R\left( {\pi  + 2} \right)\]

We need R to calculate the perimeter,

\[Radius = \frac{{Diameter}}{2} = \frac{{10}}{2} = 5{\text{ }}cm\] 

\[\Pi  = 3.142\] 

Therefore, perimeter = 5(3.142+2) = 25.71 cm

FAQ (Frequently Asked Questions)

1. Find the area, perimeter and circumference of the semicircle from the figure 2.3 given below.

Let’s list down the given information,

Diameter= 12cm

Perimeter=?

Area=?

Circumference=?

Formula to calculate the perimeter of a semicircle is, R(π+2)

We need R to calculate the perimeter,

Radius = Diameter / 2 = 12 / 2 = 6 cm

π = 3.142

Therefore, perimeter = 6(3.142+2) = 30.85 cm

Area of a semicircle is = (πR2)/2

Therefore, area of the half circle is (3.142 × 6 × 6)/2 = 30.85 cm2.

Circumference= π × R

Therefore, the circumference of the circle = 3.142 × 6 × 2 = 37.704 cm

2. The area of a semicircle is 567.05 cm2Find the radius of the semicircle?

Let’s list down the given information,

Area =567.057 cm2.

Radius=?

π = 3.142

Area of a semicircle is = (πR2)/2

(3.142 x R x R)/2 = 567.057 cm2 , R2 = 361.182

Therefore, radius(R) = 19 cm.

3. What is half of a semicircle called?

The quadrant is the most common half-semicircle. 

4. What is the perimeter of a semicircle?

The perimeter of a semicircle is R(π+2).