Area of Semicircle

A semicircle is half of a circle created by cutting a circle straight through the middle. To find how much space a semicircle takes up, we take the area of a full circle and divide it by two. This helps us measure things like curved gardens, arches, and other semicircular structures in real life.

Table of Contents

• What is the Area of a Semicircle?
• Area of a Semicircle Formula
• Derivation of Area of a Semicircle Formula
• How to Find the Area of a Semicircle?
• Solved Examples on Area of a Semicircle
• Practice Questions on Area of a Semicircle

What is the Area of a Semicircle?

when you look at a semicircle's area, you just trying to figure out how much space is really inside that half circle. You cut a pizza right down the middle, and now you have two semicircles on your hands. The area's what actually tells you the size of each piece. Here's the thing, though, a semicircle isn't the same as a full circle. You got one side that curves nicely, and then one side that's completely flat, which we call the diameter. You actually see this shape everywhere in real life, from semicircular decks that need painting to gardens you're planning to plant. The bottom line is that we always measure area in square units. If you start with meters, you get square meters back as your answer.

Area of a Semicircle Formula

The formula for finding the area of a semicircle is actually quite simple once you understand it. Since a semicircle is half of a circle, you take the circle formula and divide it by two.

The formula is: Area = πr² ÷ 2

π (pi) is a special number that comes up whenever you work with circles. It's about 3.14, but sometimes people use 22/7 as a fraction instead. You'll find it on most calculators, and using 3.14 works fine for most situations.

r stands for radius. The radius is the distance from the very center point to the curved edge. It's always half of the diameter (the straight line across the widest part).

r² means you multiply the radius by itself. So if your radius is 5, you'd calculate 5 times 5, which equals 25.

÷ 2 is the division by two, which represents that we're only dealing with half a circle, not a whole one.

You might see this formula written in different ways, and they all mean the same thing:

• Area = πr² ÷ 2
• Area = (πr²)/2
• Area = 0.5 × πr²
• Area ≈ 1.57 × r² (since π/2 equals about 1.57)

Derivation of Area of a Semicircle Formula

Understanding where this formula comes from makes it stick in your mind much better than just memorizing it. The good news is that the derivation is really straightforward and doesn't require anything complicated.

Step 1: The Full Circle Formula

We start with something everyone learns about circles: the area of a circle is πr². This formula has been around for thousands of years and is the foundation for everything we do with semicircles. Every circle, no matter how big or small, uses this same formula.

Step 2: What Makes a Semicircle

A semicircle happens when you take a circle and cut it perfectly in half. You do this by drawing a straight line through the center of the circle. That line is called the diameter. When you cut along this line, you get two identical pieces, and each piece is a semicircle.

Step 3: Simple Division

Here's the key insight: since you've cut the circle into two equal parts, each part must have exactly half the area of the original circle. So if the whole circle has area πr², then one semicircle has area = πr² ÷ 2.

Step 4: Different Ways to Write It

Once you have πr² ÷ 2, you can write it in whatever way makes most sense for your situation:

• Keep it as a fraction: (πr²)/2
• Write it as multiplication: 0.5 × πr²
• Use a decimal: 1.57 × r²
• Leave π in the answer: (πr²)/2

How to Find the Area of a Semicircle?

Finding the area of a semicircle is straightforward when you follow a simple process. Here's the practical method:

• Identify the radius: The radius is the distance from the centre of the semicircle to the curved edge. Sometimes you're given the radius directly, and sometimes you need to find it from other measurements. If you have the diameter (the straight line across the full width), simply divide it by 2 to get the radius.
• Square the radius: Multiply the radius by itself. For example, if your radius is 5 cm, then r² = 5 × 5 = 25 cm².
• Multiply by π: Take your squared radius and multiply it by pi (π ≈ 3.14). Using our example: 25 × 3.14 = 78.5
• Divide by 2: Finally, divide your result by 2 to account for the semicircle being half a circle. In our example: 78.5 ÷ 2 = 39.25 cm²

Different Values of π:

• Using π = 3.14: This is the most practical for everyday calculations
• Using π = 3.14159: This gives more precision
• Using π = 22/7: This fractional approach works well when you're working with fractions

Also use: Semicircle Area Calculator

Solved Examples on Area of a Semicircle

Example 1: Finding area with a given radius

Problem: A semicircular garden has a radius of 7 meters. What is its area?

Solution: Given: r = 7 m

Step 1: Square the radius r² = 7 × 7 = 49 m²

Step 2: Multiply by π 49 × 3.14 = 153.86 m²

Step 3: Divide by 2 153.86 ÷ 2 = 76.93 m²

The area of the semicircular garden is approximately 76.93 square meters.

Example 2: Using a fractional radius

Problem: Find the area of a semicircle with radius 10 cm.

Solution: Given: r = 10 cm

Step 1: r² = 10 × 10 = 100 cm²

Step 2: 100 × π = 100π cm² (we can leave it in terms of π for exactness)

Step 3: (100π) ÷ 2 = 50π cm²

If we use π ≈ 3.14159: 50 × 3.14159 = 157.08 cm²

The exact answer is 50π cm², and the approximate decimal value is 157.08 cm².

Example 3: Finding area from diameter

Problem: A semicircular arch has a diameter of 12 feet. Calculate its area.

Solution: Given: d = 12 feet, so r = 12 ÷ 2 = 6 feet

Step 1: r² = 6 × 6 = 36 ft²

Step 2: 36 × 3.14 = 113.04 ft²

Step 3: 113.04 ÷ 2 = 56.52 ft²

The area of the semicircular arch is 56.52 square feet.

Example 4: Real world application

Problem: A semi-circular swimming pool has a radius of 8 meters. How much pool surface area needs to be covered with a shade?

Solution: Given: r = 8 m

Step 1: r² = 8 × 8 = 64 m²

Step 2: 64 × 3.14159 = 201.06 m²

Step 3: 201.06 ÷ 2 = 100.53 m²

The pool owner needs shade material to cover approximately 100.53 square meters of the pool surface.

Practice Questions on Area of a Semicircle

Try solving these problems. Work through each one carefully, and check your answers afterward.

1. A semicircular piece of cardboard has a radius of 5 inches. What is its area?

Hint: Use the formula with radius = 5 inches.

2. The diameter of a semicircular mat is 12 feet. Calculate the area of the mat.

Hint: Remember to divide the diameter by 2 first to get the radius.

3. You're building a semicircular deck with a radius of 10 meters. How many square meters of deck are you building?

Hint: Follow the four steps: square the radius, multiply by π, divide by 2.

4. A semicircular garden path has an area of 24.5 square meters. What is the radius of this path?

Hint: Work backwards from the area. Multiply by 2, divide by π, then find the square root.

5. Compare two semicircular tables. One has a radius of 4 feet, and the other has a radius of 6 feet. What is the difference in their areas?

Hint: Calculate the area of each table separately, then subtract.