Take a round-shaped paper and fold it in half. Have you noticed the shape you got? It is called a semicircle. It is thus half the circumference of the full circle. Semicircle is one of the most common shapes, we find in our real life. A protractor, a speedometer, a taco are all examples of semicircles.
Definition
‘Semi’ means half, thus semicircle is a half-circle. It is formed when a line passing through the center of the circle touches the two ends forming an intercepted arc.
Thus a semicircle consists of the diameter of the circle and its connecting arc. Semicircle being half a circle, its arc always measures (360°/2 = 180°) and thus is referred to as a half-disc. It has only one line symmetry, the reflection symmetry.
Semicircle
Formulas
Area
The area of a semicircle is the space enclosed by the semicircle. Since a semicircle is exactly half a circle, its formula can be obtained by dividing the area of a circle by 2.
As we know,
Area of a Circle = πr2
Thus,
Area of a semicircle = πr2/2, here π = 3.141, r = radius
The area of a semicircle is expressed in square units (m2, cm2, etc)
Area of a Semicircle Formula
Let us solve an example to clear your concept.
How to Find the Area of a Semicircle
Find the area of a semicircle with a radius of 6 cm.
Solution:
As we know, Area (A) = πr2/2, here π = 3.141, r = 6 cm = 3.141 × (6)2/2 = 56.538 cm2
How to Find the Radius of a Semicircle
Using the above formula for calculating the area, we also find the radius of the semicircle if the area of the semicircle is known.
Let us solve a problem to clear your concept.
The area of a semicircle is 555.55 cm2. Find the radius of the semicircle.
Solution:
As we know, Area (A) = πr2/2, here area = 555.55 cm2, π = 3.141 555.55 = 3.141 × r2/2 r2 = 555.55 × 2/3.141 r2 = 353.74 r = 18.80 cm
Perimeter
The perimeter of a semicircle is the sum of half the circle’s circumference, plus the diameter of the semicircle. It is also called the circumference of a semicircle. Thus, it is not half the perimeter of a circle.
As the perimeter of a circle is given by 2πr or πd
So, the perimeter of the semicircle = ½ (2πr) + d => πr + d => πr + 2r, here π = 3.141, r = radius
or, ½ πd + d, here π = 3.141, d = diameter
The perimeter of a semicircle is expressed in units of m, cm, etc.
Perimeter of a Semicircle Formula
Let us solve an example to clear your concept.
How to Find the Perimeter of a Semicircle
A semicircle has a diameter of 60 m. Find the perimeter of the semicircle.
Solution:
As we know, Perimeter (P) = ½ πd + d, here π = 3.141,d = 60 m = (½ × 3.141 × 60 + 60) m = 154.23 m
The basketball court of your school has 2 semicircles at each end. The semicircles have 12 cm radii. What is the perimeter of one semicircle of the court?
Solution:
As we know, Perimeter (P) = πr + 2r, here π = 3.141, r = 12 cm = (3.141 × 12) + (2 × 12) cm = 61.69 cm
Find the circumference of the semicircle whose diameter is 22 cm.
Solution:
As we know, Circumference is same as perimeter of a semicircle. Thus, Perimeter (P) = ½ πd + d, here π = 3.141,d = 22 cm = (½ × 3.141 × 22 + 22) cm = 56.55 cm
A circle has a diameter of 16 m. Find the area and perimeter of a semicircle.
Solution:
As we know, Area (A) = πr2/2, here π = 3.141, r = d/2 = 16/2 = 8 m = 3.141 × (8)2/2 = 100.51 cm2 Again, Perimeter (P) = ½ πd + d, here π = 3.141,d = 16 cm = (½ × 3.141 × 16 + 16) cm = 41.12 cm
Michael has a circular-shaped garden outside his house with a radius of 6 yd. Now, he wants to fence exactly half of the garden. Calculate the perimeter he wants to fence.
Solution:
As we know, Perimeter (P) = πr + 2r, here π = 3.141, r = 6 yd = (3.141 × 6 + 2 × 6) yd = 30.846 yd Hence, Michael will need to fence a perimeter of 30.846 yd
Calculate the arc length of a semicircle with a diameter of 6 cm and a perimeter of 30 cm.
Solution:
As we know, The arc length of a semicircle can be obtained by subtracting the diameter of the semicircle from its perimeter. Thus, Arc Length (L) = Perimeter of the semicircle – Diameter of the semicircle = (30 – 6) cm = 24 cm
Angle Inscribed in a Semicircle
An inscribed angle of a semicircle is an angle formed by drawing a line from each endpoint of the diameter to a point on the semicircle arc.
Angle Inscribed in a Semicircle
Thus, an inscribed angle has a measure that is half the measure of the arc that subtends it. Since a semicircle is half of a circle the angle subtended by the arc that forms the semicircle measures 180°. Therefore, all inscribed angle of a semicircle is 180°/2 = 90° Thus, an angle inscribed in a semicircle is a right angle.
