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. 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.
What do you mean by the perimeter of a semicircle?
Since, 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 πR+2R, which can also be written as R(π+2) 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 \[\frac{\pi R^{2}} {2}\]
where R= Radius of the semicircle.
= Constant named Pi, approximately equal to 3.142.
The unit of area of a semicircle is m2 or cm2
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: 2πR
where R= Radius of the semicircle.
= Constant named Pi, approximately equal to 3.142.
The unit of the 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
Solved Questions
Question 1
A semicircle has a diameter of 100 metres. 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(π−2)
We need R to calculate the perimeter of semicircle,
Radius = \[\frac{diameter}{2} = \frac{100}{2} \] = 50 metre
π = 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
π=3.142
Perimeter=?
The formula to calculate the perimeter of a semicircle is, R(π−2)
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(π−2)
We need R to calculate the perimeter,
Radius = \[\frac{diameter}{2} = \frac{8}{2} \] = 4 cm
π = 3.142
Therefore, perimeter = 4 ( 3.142 + 2 ) = 5.142 cm
Using the area of a semicircle formula = \[\frac{\pi R^{2}}{2}\]
Therefore, area of the semicircle is \[\frac{3.142 \times 4 \times 4}{2}\] = 25.14 cm2
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 exactly half of the cake.
Solution
Let’s list down the given information,
Radius= 5cm
Area=?
Area of semicircle formula is = \[\frac{\pi R^{2}}{2}\]
Therefore, area of the semicircle is \[\frac{3.142 \times 5 \times 5}{2}\] = 39.275 cm2
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 cm
= 3.142
Circumference of a circle = 2πR
Therefore, the circumference of the circle = 3.142 \[\times\] 3.5 \[\times\] 2= 21.944 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=?
The formula to calculate the perimeter of a semicircle is, R(π+2)
We need R to calculate the perimeter,
Radius = \[\frac{diameter}{2} = \frac{10}{2} \] = 5 cm
= 3.142
Therefore, perimeter = 5 ( 3.142 + 2 ) = 25.71 cm
Question 7
Find the area of a semicircle whose radius is 49 cm.
Solution
We are given the radius of the semicircle, i.e., r = 49 cm
Now, the area of a semicircle = \[\frac{\pi R^{2}}{2}\]
Thus, the area of the semicircle would be \[\frac{1}{2} \times \frac{22}{7} \times 49 \times 49 \] = 539 cm2
Question 8
Find out the perimeter of a semicircle if its diameter is 7 cm.
Solution
We are given the diameter of the semicircle, i.e., d = 7 cm.
Now, the perimeter of a semicircle = R ( - 2)
However, the perimeter of a semicircle with the use of its diameter = \[(\frac{1}{2}(\pi d)) + d \]
Therefore, the perimeter would be \[(\frac{1}{2} \times \frac{22}{7} \times 7) + 7 \] = 11 + 7 = 18 cm.
Fun Facts
In geometry, when a plane region is bounded by three semicircles with three apexes in such a way that the corner of each semicircle is connected on the side of the baseline (on the same side as their diameters), it forms what is known as an arbelos. And did you know? There is an actual sculpture based on the mechanism/concept of an arbelos in Kaatsheuvel, in the Netherlands.
According to Thales’ Theorem, any triangle that is inscribed within a semicircle with a vertex located at each of the endpoints of the shape and a third vertex somewhere else on the semicircle is a right-angled triangle.
A semicircle can also be used as a lemma in any general method. This is done by transforming any polygonal shape into a copy of itself but the area should be of any other given polygonal shape.
Parbelos, a figure similar to an arbelos, doesn’t use semicircles. Instead, it makes use of parabola segments.
FAQs on Semicircle
1. What are the properties of a semi-circle?
The half-circle we get from cutting a complete circle into two halves along the diameter line, is known as a semicircle. Some of the main properties of a semicircle are as follows:
It is a one-dimensional locus of points that ends up forming half a circle.
The full arc of a semicircle is always 180°.
It tends to have only a single line of symmetry; reflection symmetry.
The angle at its circumference is 90°.
It has a straight side and a curved side along with two vertices where both these sides meet at the two ends of the straight side.
It is also often referred to as a half-disk.
All the perpendicular lines intersecting the semicircle tend to be concurrent at the center of the circle (which contains the given semicircle).
2. What is an inscribed angle? What is the angle inscribed in a semicircle?
An inscribed angle is usually formed in a circle with the help of two chords that tend to have a common endpoint on that circle. The measure of this type of an angle is always half the measure of the intercepted arc. And according to the Inscribed Angle Theorem, an angle inscribed within a semicircle tends to be 90°, i.e., it’s a right angle. This is due to the fact that the intercepted arc tends to measure 180°. So, naturally, any angle that is corresponding to it and is inscribed within, would measure half of it, which makes it a right angle.
3. What do you call a half of a semicircle?
The most common half of a semicircle (or the one-fourth of a circle) is referred to as a quadrant. It refers to the four quarters present in the coordinate plane system and each of these four sections are known as quadrants. The area of a quadrant is equal to one-fourth of the area of the given circle. Its formula is as follows:
Area of a quadrant = \[\frac{1}{4} \times \] area of a circle = \[\frac{\pi R^{2}}{4}\].
The area of a quadrant can also be calculated in another way out of the diameter of the circle.
The units of area of a quadrant include m2, inch2, and cm2.
4. How do you find the diameter of a semicircle?
A semicircle is the half of a circle and the diameter is considered to be its base. The diameter of a semicircle is twice its radius which thus makes the radius of a semicircle half its diameter.
5. The area of a semicircle is 567.05 cm2. Find 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 = \[\frac{\pi R^{2}}{2}\]
\[\frac{(3.142 \times R \times R)}{2} \] = 567.057 cm2 , R2 = 361.182
Therefore, radius(R) = 19 cm.
6. 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 = \[\frac{Diameter}{ 2} = \frac{12}{2} \]= 6 cm
π = 3.142
Therefore, perimeter = 6(3.142+2) = 30.85 cm
Area of a semicircle is = \[\frac{\pi R^{2}}{2}\]
Therefore, area of the half circle is \[\frac{(3.142 \times 6 \times 6)}{2} \] = 30.85 cm2.
Circumference= π × R
Therefore, the circumference of the circle = 3.142 × 6 × 2 = 37.704 cm
