Arpita Srivastava Content Writer
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A semicircle can be considered as a single-dimensional locus of points that forms half a circle. The protractor, which is commonly known as ‘D’, is the shape of a semicircle.
- The angle of the arc of the semicircle is 180o or π radians.
- It has 1 line of symmetry along its vertical axis.
- The shape resembles that of a circular arc, forming an angle of 180 degrees.
- The term semicircle is often used to represent either a closed curve or a circular disk.
- It is a two-dimensional figure that is half of the circle.
- This is used to construct the arithmetic and geometric means of two lengths using a compass.
- Half watermelon and an igloo are all real-life examples of semicircles.
- The formula for semicircle can be mathematically represented as:
Area of a Semicircle = πR2/ 2 square units
Perimeter of a Semicircle = (πR + 2R)
- where R is the radius
Key Terms: Semicircle, Circle, Radian, Half circle, Diameter, Radius, Perimeter, Area, Angle, Diameter, Radius, Area of Semicircle, Perimeter of Semicircle, Semicircle Formula
What is Semicircle?
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A semicircle is a figure that is formed when a line passing through the centre touches the two ends of the circle. It is half a circle, i.e. a closed shape where the diameter and half arc of the original circle are present.
- The half arc is also known as a semicircular arc.
- The two semicircles, if joined back, should give us the original circle.
- Two semicircles cut from the same circle are identical to each other.
- You can also refer to them as mirror images of each other.
- When you pass a diameter that equally divides the circle into two halves, then you get 2 semicircles, i.e. half circle.
- The half of the semicircle will form an arc of 180 degrees.
Example of What is Semicircle?Example: Taco, tunnel and Japanese fan are real-life examples of semicircle. |
Also Read:
| Related Concepts | ||
|---|---|---|
| Tangent Circle Formula | Circle Definition | Tangent to a Circle |
| Chord of Circle | Central Angle of a Circle | Great Circle Formula |
Shape of Semicircle
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A semicircle will form a closed two-dimensional shape. It cannot be considered equivalent to a polygon as it has curved edges. The shape consists of one curved shape, which is equivalent to its circumference.
- Since a semicircle is formed by dividing the circle into half as a result, its area will also be half.
- The lines intersecting perpendicularly are concurrent with the point of the centre.
- It tends to follow reflection symmetry.
- The diameter of the circle divides the figure into two equal halves.
- Each of the two halves created will form a semicircle.
Properties of Semicircle
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The properties of semicircle are as follows:
- The straight edge of a semicircle is called the diameter.
- It has two sides where one side is curved, and the other side is straight.
- The two-dimensional image has two corners.
- Angle formed within the figure is equal to 90 degrees.
- Centroid of the semicircle will lie exactly in the middle along the vertical radius.
Perimeter of Semicircle
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The perimeter of the semicircle is equivalent to the sum of the diameter and half of the circumference of the circle. It should be kept in mind that the perimeter is not equal to half of the perimeter of a circle.
- There is confusion when calculating the perimeter of a semicircle, thinking that it is simply dividing the perimeter of the circle by two i.e
2 π r ÷ 2 = π r
r is radius and π is a constant whole value is \(\frac {22} {7}\)
- This is a wrong formula as this only counts the arc of the semicircle
- The mistake we are making here is that the semicircle is not only the arc.
- It is also when the two endpoints of the arc are joined by the diameter (which is double the radius i.e. 2r)
- So, the correct formula for perimeter of semicircle is
π r + 2r = r(π+2)
- where r is radius and π is a constant whole value is \(\frac {22} {7}\)
Example of Perimeter of SemicircleExample: Next to Diana's house is a garden in the shape of a circle with a diameter of 14 yards. Diana wants to use only half the garden for a party, what is the perimeter of the part she wants to use? Ans: We know that the diameter = 14 yards, we need to find the radius. Radius = Diameter/2 = 14/2 = 7 yards So, the perimeter of a semicircle = R(π + 2) where π is 3.14 approx. Perimeter = 7 (3.14 + 2) Perimeter = 7 × 5.14 Perimeter = 35.98 yards |
Area of Semicircle
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Area of semicircle is the space enclosed by the arc and diameter of the circle. It is half of the semicircle. It refers to the inner space or region of the circle.
- In the case of area, it is much simpler and it is the exact half of the area of the circle π r2 divided by 2 i.e (π r2)/2.
- It include diameter of the segment from one end to the other end of the arc.
- This will result in the formation of unique shape in geometry.
- So area of semicircle can be represented as:
Area of Semicircle = 1/2 (π r2)
Example of Area of SemicircleExample: The radius of a circular pie is 14 units. Find the area of half of the pie. Ans: As we know that the radius = 14 units. The area of a semicircle = πR2/ 2 square units. So, by substituting the value of the radius, Area = ((22/7) × 14 × 14)/2 Area = (22 × 14) Area = 308 square units |
Area of the semicircle
Angle Inscribed within a Semicircle
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The angle inscribed within a semicircle is 90° always. This angle is formed by joining two lines from opposite ends of the diameter at an arbitrary point on the semicircle.
- This angle is 90° every time without exception no matter what is the length of the diameter or the lines or where they touch the arc.
- As we know, a semicircle is half of a circle, so the angle formed which makes a circle into a semicircle is 180 degrees.
Semicircle Formula
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The important semicircle formula are as follows:
| Category | Data |
|---|---|
| Area | (πr2)/2 |
| Perimeter | (½)πd + d; when diameter is given |
| πr + 2r | |
| Central angle | 180 degrees |
| Angle in a semicircle | 90 degrees |
Things to Remember
- Semicircle is a part of NCERT Class 10 Mathematics Geometry.
- It is formed by cutting the circle with the arc on one end and the diameter on the other end.
- Perimeter and circumference mean the same and are used alternatively.
- It is the locus of points that form a circle.
- When a train passes through a tunnel, then we can say the tunnel is in the shape of a semicircle.
Also Read:
Sample Questions
Ques. Calculate the perimeter of a circle with a diameter of 14 cm. (2 marks)
Ans. The radius of the circle is 14/2 = 7 cm.
The formula for the perimeter of a circle is r(π+2) where r=7cm.
Therefore, The perimeter will be 7(\(\frac {22} {7}\)+ 2) = 7(\(\frac {22 + 14} {7}\)) = 36 cm.
Ques. Calculate the area of a circle with a radius of 21 cm. (2 marks)
Ans. The radius of the circle is 21 cm.
The formula for the area of a circle is ( π r2 ) where r=21 cm.
Therefore, The area of the semicircle will be half of it.
It will be \(\frac {22} {7}\) X 21 X 21 X \(\frac {1} {2}\) =11 X 3 X 21 = 693 cm2
Ques. Calculate the circumference of a circle with a radius of 6 cm. (2 marks)
Ans. The radius of the circle is 6 cm.
The formula for the circumference of a circle is 6(π+2) where r=6 cm.
Therefore, The circumference will be 6(\(\frac {22} {7}\)+ 2)=6(\(\frac {22 + 14} {7}\))=30.86 cm.
Ques. Calculate the area of a semicircle with a diameter of 26 cm. (2 marks)
Ans. The radius of the semicircle is 26/2 = 13 cm.
The formula for the area of a circle is π r2 where r =13 cm.
Therefore,
The area of the semicircle will be \(\frac {22} {7}\) X 13 X 13 X \(\frac {1} {2}\) =11 X 13 X 13 X 17 = 265.57 cm2
Ques. Calculate the radius of a semicircle with a circumference of 360 cm. (2 marks)
Ans. The circumference of the semicircle is 360 cm.
The formula for the circumference of a semicircle is r(π+2) = 360 cm.
Therefore, The circumference will be r(\(\frac {22} {7}\)+ 2)
r(\(\frac {22 + 14} {7}\))=360
r(\(\frac {36} {7}\))=360
r=360 X (\(\frac {7} {36}\))
r=70 cm.
Ques. Calculate the radius of a semicircle with an area of 77 cm2. (2 marks)
Ans. The area of the semicircle is 77 cm2
The formula for the area of a semicircle is \(\frac {1} {2}\) X π r2 = 77 cm2
\(\frac {1} {2}\) x \(\frac {22} {7}\) x r2 = 77
\(\frac {11} {7}\) X r2 = 77
r2 = 49
r=7 cm.
Ques. What is the circumference of the semicircle with a diameter of 84 cm. (2 marks)
Ans. The radius of the semicircle is 84/2 = 42 cm
The formula for the circumference of a semicircle is r(π+2)
42 ( 22 / 7 + 2 )
42 ( 36 / 7 ) = 36 x 6
Therefore, the circumference of the semicircle is 216 cm.
Ques. The area of a semicircle is the same as the area of the circle. Find the radius of the semicircle. (2 marks)
Ans. The radius of the semicircle is 84/2 = 42 cm
The formula for the area of a circle is π r2
The formula for the area of a semicircle is \(\frac {1} {2}\) x π s2
\(\frac {1} {2}\) x π s2 = π r2
s2 = 2 r2
s = 2r
Ques. A farmer has agricultural land in the shape of a semicircle. Find the total cost incurred by the farmer if the cost of ploughing 1 m2 of land is 14 ∏. The radius of the land is 350 m. (3 marks)
Ans. The formula for the area of a semicircle is \(\frac {1} {2}\) x π r2
\(\frac {1} {2}\)x \(\frac {22} {7}\) x 350 x 350 = 11 x 50 x 350
= 192500 m2.
The total cost for ploughing 192500 m2 of land at the cost of 14 ∏ per m2 = 192500 x 14
= ∏ 26,95,000
Ques. Find the perimeter of the semicircle if the diameter of the semicircle is 20cm. (3 marks)
Ans. Given, d = 4 cm
- Perimeter of Semicircle(P) = (πr + d)
- r = d / 2 = 20/2 = 10 cm
- P = (2π + 10) cm
- P = 2(π + 5) cm
- P = 16.28 cm
Ques. Find the area of a semicircle with radius 2 cm. (3 marks)
Ans. Given, Radius = 2 cm
- Area of Semicircle(A) = [πr2] / 2
- A = [π(2)2] / 2
- A = [2π] / 2
- A = 6.28 cm2
Ques. Find the circumference of the semicircle if the diameter of the semicircle is 10 cm. (3 marks)
Ans. Diameter(d) = 10 cm
- Circumference of Semicircle(C) = πr
- Radius of semicircle(r) = d / 2 = 10 / 2 = 5 cm
- C = π(5) cm
- C = 15.7 cm
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