How to Find the Area and Perimeter of a Semi Circle Easily
A semi circle is a vital geometric concept in mathematics, forming the basis for topics like mensuration, geometry, and coordinate geometry. Understanding semi circles helps students solve area and perimeter problems in school exams and competitive tests. At Vedantu, we make learning shapes like the semi circle simple and engaging for students of all grades.
What is a Semi Circle?
A semi circle is exactly half of a full circle. If you cut a circle along its diameter, each half forms a semi circle. The diameter is the straight line passing through the center of the circle, connecting two points on the circle’s boundary. The distance from the center to any point on the curved boundary is the radius. The straight edge of a semi circle is the diameter, and the curved edge is called the arc.
Semi Circle Meaning and Properties
- A semi circle always has an angle of 180° at its center.
- The diameter is both the straight edge and the longest chord of the semicircle.
- The arc length of a semicircle is half the circumference of the original circle.
- It has exactly one line of symmetry (the diameter).
- If the circle’s radius is r, the diameter is 2r.
A semi circle is also called a half circle, and is a special type of sector with a central angle of 180°.
Formulas for Semi Circle
| Formula | Expression | Where... |
|---|---|---|
| Area of a Semi Circle | \(\frac{1}{2}\pi r^2\) | \(r\) = radius |
| Perimeter of a Semi Circle (with diameter) | \(\pi r + 2r\) or \(\frac{1}{2}\pi d + d\) |
\(d = 2r\) = diameter |
Remember, the perimeter of a semicircle includes both the curved edge and the straight edge (diameter).
Step-by-Step Worked Examples
Example 1: Find the Area of a Semi Circle
Find the area of a semi circle with radius 14 cm. Use \(\pi = 22/7\).
- Write the area formula: Area = \( \frac{1}{2} \pi r^2\)
- Plug in the values: Area = \( \frac{1}{2} \times \frac{22}{7} \times 14^2 \)
- Solve step-by-step:
- 14 × 14 = 196
- Area = \( \frac{1}{2} \times \frac{22}{7} \times 196 \)
- \( \frac{196}{7} = 28 \)
- Area = \( \frac{1}{2} \times 22 \times 28 \) = \( \frac{1}{2} \times 616 \) = 308 cm²
- Final Answer: 308 cm²
Example 2: Find the Perimeter of a Semi Circle
A semi circle has a radius of 7 cm. Calculate its perimeter. Use \(\pi = 22/7\).
- Use the perimeter formula: Perimeter = \(\pi r + 2r\)
- Plug in the values: \( = 22/7 \times 7 + 2 \times 7 = 22 + 14 = 36 \)
- Final Answer: 36 cm
Practice Problems
- Find the area of a semi circle with diameter 20 cm. (Use \(\pi = 3.14\))
- If the perimeter of a semi circle is 50 cm, what is its radius? (Use \(\pi = 3.14\))
- A window has a semi-circular arch with a radius of 1.5 m. Calculate the length of the curved edge.
- What is the difference between the area of a full circle of radius 10 cm and its semi circle?
- Find the perimeter of a semi circle with diameter 12 cm.
Common Mistakes to Avoid
- Forgetting to add the diameter to the perimeter calculation. (Always add curved edge + straight edge)
- Mixing up radius and diameter. (Diameter is twice the radius)
- Confusing area and perimeter formulas.
- Leaving the answer in terms of \(\pi\) instead of calculating the numerical value when required.
- Using wrong units—ensure area is in cm²/m² and perimeter in cm/m.
Real-World Applications
Semi circles are found in architecture (arches, bridges), design (half-rounded tables or domes), playgrounds, and engineering (cross-sections of tunnels or pipes). Understanding semi circle area and perimeter is useful when planning constructions, arts and crafts, or working with objects in real life.
For more on related geometric shapes, visit Area of a Circle or Perimeter and Area of Plane Figures on Vedantu!
In this topic, you learned the meaning, formulas, and applications of a semi circle. Mastering this concept improves your geometry and helps with practical problem-solving in mathematics and real-life situations. For more easy-to-understand lessons and practice, explore other geometry topics at Vedantu.
FAQs on Semi Circle: Area, Perimeter & Solved Maths Examples
1. What is a semi circle called?
A semicircle is simply half of a circle. It's formed by cutting a circle in half along its diameter, resulting in an arc measuring 180°.
2. Is a semicircle always 180°?
Yes, a semicircle always subtends an angle of 180° at the center of the original circle. This is because it represents exactly half of a full circle's 360°.
3. What is the formula for the area and perimeter of a semicircle?
The area of a semicircle is calculated as ½πr², where 'r' is the radius. The perimeter of a semicircle is πr + 2r; this includes the curved arc length (πr) and the diameter (2r).
4. What is semicircle class 9?
In class 9 mathematics, a semicircle is introduced as a fundamental geometric shape. Understanding its properties and formulas is crucial for solving problems related to area, perimeter, and mensuration.
5. Does the perimeter of a semicircle include the diameter or only the arc?
The perimeter of a semicircle includes both the curved arc (πr) and the straight diameter (2r).
6. What is the meaning of the term 'semicircle'?
A semicircle is defined as half of a circle. It's a plane figure bounded by a diameter and half of the circumference of the circle. Key features include a 180° arc and a diameter.
7. How do you calculate the area of a semicircle?
The area of a semicircle is half the area of a full circle with the same radius. The formula is: Area = ½πr², where 'r' represents the radius. Remember to use the correct value of π (approximately 3.14159 or 22/7).
8. What is the equation of a semicircle?
The equation of a semicircle depends on its orientation. For a circle centered at (h,k) with radius 'r', the upper semicircle is defined by y - k = √(r² - (x - h)²), and the lower semicircle is defined by y - k = -√(r² - (x - h)²).
9. How is a semicircle different from a half-circle?
The terms semicircle and half-circle are often used interchangeably. However, strictly speaking, a semicircle emphasizes the geometric shape defined by the diameter and half the circumference, while 'half-circle' is a more general description.
10. Where are semicircles used in real life?
Semicircles appear frequently in architecture (e.g., arches, windows), engineering (e.g., bridge designs), and even in everyday objects. They are also useful in various mathematical applications and problem-solving scenarios.
11. What are some real-world examples of semicircles?
Examples include the shape of a rainbow, the cross-section of a cylindrical tunnel, the design of certain types of furniture, and many architectural features. The applications are diverse and appear in both natural phenomena and man-made constructions.
