What Is the Formula for Area and Perimeter of a Half Circle
You might have observed that your forehead forms a half circle, or if you look at the sky at night, the moon appears like half of the circle. What is this half circle? Well, it is a semi-circle. You can form this shape by taking a circle and cutting it into two halves, where each half has the same half of the area as that of the original circle.
There are various parameters to express the property of a half circle; these parameters include the diameter, area, and perimeter that can be calculated using semi-circle formulas that we will discuss here.
Besides this, the diameter of a circle divides the circle into two equal semicircles. The area of any semicircle is half of the area of a circle. Now, let us understand the semicircle formulas using solved examples.
Formula of a Half Circle
The below image shows the properties of a semicircle and that’s how we will determine the perimeter (circumference) and the area of a semicircle:
A Half circle
Area of a Semicircle Formula
From the above text, we got to know that a semicircle is half a circle, which means that the semicircle area is half of the area of a circle. So, the area of a circle is πR2 where R is the radius of the circle. The area of a semi-circle refers to the region or inner space of the semi-circle.
Now, we know that the radius of a circle is ‘r’, and the area of a circle is:
A = πr2
Now, the area of semicircle becomes:
A = (πr2)/2
where r = R is the radius of a semicircle and π(pi) is \[\frac{22}{7} \] or 3.142 approximately.
Perimeter of a Semicircle Formula
We use the perimeter of a semicircle formula to calculate the perimeter of a semicircle. For this, we must know either the diameter or radius of a circle along with the length of the arc.
Now, to evaluate the length of the arc of the semicircle, we must calculate the circumference of a circle, which is as follows:
The circumference of a circle is C = πd or C = 2πr.
Further, using the value of C, we can frame the formula for the perimeter of a semicircle which is equal to the sum of half of the circumference of the circle and diameter of a circle.
The perimeter of a semicircle formula is given by,
=(πR + 2R) or R(π + 2) units.
Here,
R = radius of a semicircle
d = Diameter of a semicircle.
Circumference of a Semicircle Formula
The circumference of a semicircle is the length of the arc around the semicircle. It is half of the circumference of a circle.
Please note that the difference between the circumference and perimeter of a semicircle is that the circumference is only the length of the arc which is the curved portion on the boundary, while the perimeter of the semicircle includes circumference and diameter both.
Hence, the formula to find the circumference of a semicircle is = πR units.
Here,
R = radius of the semicircle
Now, let us go through the solved example for using semicircle formulas.
Solved Example on Semicircle Formulas
Example 1. Determine the area of a semicircle whose diameter is 14 cm.
Solution: We are given with d = 14 cm, so the radius becomes \[\frac{14}{2} \] cm or 7 cm.
Now, putting the value in the formula, we get:
A = \[\frac{1}{2}\] (πr2)
= \[\frac{1}{2} \] (π x 72) = 3.14 (\[\frac{1}{2} \] x 72) = 3.14 x \[\frac{49}{2} \]
On solving, we get the area of the semicircle as 76.93 cm2.
Example 2. Calculate the circumference of a semi-circle whose diameter is 18 units.
Solution: We know the circumference of a semicircle of radius r. Now, putting these values, we have:
C= πr = 3.14 x \[\frac{18}{2} \]
We get the circumference as 28.26 cm.
Example 3. We are given the radius of a semicircle as 14 units. Now, using the semicircle formula, find its perimeter.
Solution: We have a radius of semicircle as 14 units.
Now, using the perimeter of a semicircle formula, we have πr + d = πr + 2r
= (14 x (22/7) + 28) units
= (44 + 28) units
= 72 units
The perimeter of the given semicircle is 72 units.
So, we got to know that the circumference of a circle is r and the perimeter of a semicircle is (πr + d). We also got to know how to use the semicircle formula in our questions.
FAQs on Half Circle Explained with Area and Perimeter Concepts
1. What is a half circle in Maths?
A half circle (or semicircle) is half of a full circle formed by cutting a circle along its diameter. It has:
- A curved edge equal to half the circumference of a circle
- A straight edge called the diameter
- A central angle of 180°
2. What is the formula for the area of a half circle?
The area of a half circle is (1/2)πr², where r is the radius. Since the area of a full circle is πr², dividing by 2 gives the semicircle’s area.
- Formula: Area = (1/2)πr²
- Example: If r = 4 cm, Area = (1/2) × π × 4² = 8π ≈ 25.13 cm²
3. What is the perimeter of a half circle?
The perimeter of a half circle is πr + 2r, which includes the curved arc and the diameter. Half the circumference is πr, and the diameter is 2r.
- Formula: Perimeter = πr + 2r
- Example: If r = 7 cm, Perimeter = 7π + 14 ≈ 35.99 cm
4. How do you find the area of a half circle with diameter?
To find the area of a half circle using diameter, first divide the diameter by 2 to get the radius, then apply (1/2)πr².
- Step 1: r = diameter ÷ 2
- Step 2: Use Area = (1/2)πr²
- Example: If diameter = 10 cm, r = 5 cm
- Area = (1/2) × π × 5² = 12.5π ≈ 39.27 cm²
5. What is the difference between a circle and a half circle?
The main difference is that a circle is a complete round shape, while a half circle is exactly half of it. Key differences include:
- Area: Circle = πr², Half circle = (1/2)πr²
- Angle at center: Circle = 360°, Half circle = 180°
- Boundary: Circle has only a curved edge; Half circle has one curved edge and one straight diameter
6. What is the curved surface length of a half circle?
The curved length (arc length) of a half circle is πr. Since the full circumference is 2πr, half of it equals πr.
- Formula: Arc length = πr
- Example: If r = 6 cm, Arc length = 6π ≈ 18.85 cm
7. How do you calculate the radius of a half circle from its area?
To find the radius from the area of a half circle, use r = √(2A/π). This comes from rearranging (1/2)πr² = A.
- Step 1: Multiply area by 2
- Step 2: Divide by π
- Step 3: Take the square root
- Example: If A = 50 cm², r = √(100/π) ≈ 5.64 cm
8. What are the properties of a half circle?
A half circle has specific geometric properties related to its shape and angles.
- Central angle is 180°
- Area is (1/2)πr²
- Perimeter is πr + 2r
- Any angle formed in a semicircle is a right angle (90°)
9. Why is the angle in a half circle 90 degrees?
The angle in a half circle is 90° because of Thales’ theorem, which states that an angle subtended by a diameter at the circumference is a right angle. In a semicircle:
- The diameter acts as the base
- The vertex lies on the curved arc
- The angle formed is always 90°
10. Can you give a real-life example of a half circle?
A half circle appears in many real-life objects such as arches, windows, and protractors. Examples include:
- A semicircular window with radius 3 m → Area = (1/2)π × 3² = 4.5π ≈ 14.14 m²
- A 180° protractor used to measure angles
- Architectural arches and bridge designs
