3. Semicircle Formulas
A semicircle is exactly half of a circle. It has one curved boundary and one straight boundary, which is the diameter of the original circle.
In mensuration questions, the most important things to calculate for a semicircle are its area and perimeter.
What Is a Semicircle?
Curved part + straight diameter = complete boundary of a semicircle
A semicircle has two different parts in its boundary:
- The curved half of the circle
- The straight diameter
This is especially important when calculating the perimeter of a semicircle.
1. Area of a Semicircle
Since a semicircle is half of a circle, its area is half the area of a complete circle.
The area of a complete circle is:
Therefore, the area of a semicircle is:
Where:
| Symbol | Meaning |
|---|---|
| A | Area |
| r | Radius |
| π | Pi, commonly taken as 22/7 or 3.14 |
Example 1: Area of a Semicircle
Find the area of a semicircle whose radius is 7 cm.
Formula:
Substitute r = 7 cm and π = 22/7:
A = ½ × 22/7 × 7 × 7
A = ½ × 22 × 7
A = 77 cm²
Answer: 77 cm²
Example 2: Radius = 14 cm
Find the area of a semicircle with radius 14 cm.
A = ½ × 22/7 × 14 × 14
A = 308 cm²
Answer: 308 cm²
2. Area of a Semicircle When Diameter Is Given
If the question gives the diameter instead of the radius, first find the radius.
Then use:
Example
The diameter of a semicircular garden is 28 m. Find its area.
Step 1: Find radius
r = d/2
r = 28/2
r = 14 m
Step 2: Find area
A = ½ × 22/7 × 14 × 14
A = 308 m²
Answer: 308 m²
3. Perimeter of a Semicircle
The perimeter of a semicircle is different from the circumference of a complete circle.
A semicircle's complete boundary contains:
- Half of the circumference of the original circle
- The diameter
Half of the circumference is:
The diameter is:
Therefore:
Alternative Form
Example 1: Perimeter of a Semicircle
Find the perimeter of a semicircle whose radius is 7 cm.
Take π = 22/7.
P = 22/7 × 7 + 2 × 7
P = 22 + 14
P = 36 cm
Answer: 36 cm
Example 2: Radius = 14 cm
Find the perimeter of a semicircle whose radius is 14 cm.
P = 22/7 × 14 + 2 × 14
P = 44 + 28
P = 72 cm
Answer: 72 cm
4. Curved Length of a Semicircle
Sometimes a question asks only for the curved portion of a semicircle and does not include the diameter.
The curved portion is half the circumference of the complete circle.
Example
Find the curved length of a semicircle with radius 7 cm.
= 22/7 × 7
= 22 cm
Answer: 22 cm
Curved length does NOT include the diameter.
Perimeter includes both the curved length and the diameter.
5. Perimeter When Diameter Is Given
Since:
the perimeter can also be written as:
Therefore:
Example
The diameter of a semicircular track is 14 m. Find its perimeter.
Using:
P = 22/7 × 14/2 + 14
P = 22 + 14
P = 36 m
Answer: 36 m
6. Area vs Perimeter of a Semicircle
| Quantity | Formula | Unit |
|---|---|---|
| Area | ½πr² | cm², m², etc. |
| Curved Length | πr | cm, m, etc. |
| Perimeter | πr + 2r | cm, m, etc. |
| Diameter | 2r | cm, m, etc. |
7. Common Mistakes in Semicircle Questions
Mistake 1: Using πr as the perimeter
πr is only the curved length.
For the complete boundary of a semicircle:
Mistake 2: Forgetting the ½ in area
The area of a complete circle is πr², but a semicircle is half of it.
Mistake 3: Confusing diameter and radius
Always remember:
Mistake 4: Mixing units
Convert all measurements into the same unit before applying the formula.
8. Real-Life Example
Suppose a school has a semicircular garden with a radius of 7 metres.
If the school wants to know the amount of land covered by the garden, we calculate the area.
If the school wants to put a fence around the entire semicircular garden, we calculate its perimeter.
9. Practice Questions
Question 1: Find the area of a semicircle whose radius is 7 cm.
Question 2: Find the perimeter of a semicircle whose radius is 14 cm.
Question 3: The diameter of a semicircle is 28 cm. Find its area.
Question 4: Find the curved length of a semicircle with radius 21 cm.
Question 5: A semicircular garden has a diameter of 14 m. Find the length of fencing required to cover its complete boundary.
10. Semicircle Formula Quick Revision
11. Mensuration Exam Tip
For competitive exams, remember this simple rule:
Half of Circle Area → ½πr²
Half of Circle Circumference → πr
Complete Semicircle Boundary → πr + 2r
Whenever the question says “boundary”, “fencing” or “perimeter”, remember to include the straight diameter.
In Part 4 — Square Formulas, we will cover the area, perimeter, diagonal, inradius and circumradius of a square, along with diagrams, explanations, solved examples and competitive-exam tips.