Area of Semicircles and Quarter Circles

Learn how to calculate the area of semicircles and quarter circles using the circle area formula.

Intermediate25 min16 problems
Practice Problems

Definition

A semicircle is half of a circle, and a quarter circle is one-fourth of a circle.
Since the area of a full circle is A=πr2A = \pi r^2, we can find the area of parts of a circle:
Semicircle (half circle):
Asemicircle=πr22A_{\text{semicircle}} = \frac{\pi r^2}{2}
Quarter circle:
Aquarter=πr24A_{\text{quarter}} = \frac{\pi r^2}{4}
The key is to first calculate the full circle's area, then take the appropriate fraction.

Try it now

What is the formula for the area of a semicircle?

Worked Examples

A semicircular window has a radius of 4 meters. What is its area?
1

Identify the formula for a semicircle

A=πr22A = \frac{\pi r^2}{2} → Semicircle formula
2

Substitute the radius

A=π×422A = \frac{\pi \times 4^2}{2} → A=π×162A = \frac{\pi \times 16}{2}
3

Simplify

A=16π2=8πA = \frac{16\pi}{2} = 8\pi → A=8πA = 8\pi square meters
4

Calculate the decimal value

A=8×3.14159≈25.13A = 8 \times 3.14159 \approx 25.13 → A≈25.13A \approx 25.13 square meters

Common Mistakes

Forgetting to divide by 2 for semicircles or by 4 for quarter circles

Why it's wrong: Students calculate the full circle area but forget to take the fraction.

Correct: Always ask: What fraction of a circle is this? Semicircle = divide by 2, quarter circle = divide by 4.

Confusing diameter with radius in the formula

Why it's wrong: Sometimes problems give the diameter, but the formula uses radius.

Correct: If given diameter, divide by 2 first to get the radius before using the formula.

Squaring pi instead of the radius

Why it's wrong: Writing π2r\pi^2 r instead of πr2\pi r^2.

Correct: Remember: only the radius is squared. Pi is just multiplied: π×r2\pi \times r^2.

Interactive Visual

Area and perimeter

r = 5

Area39.27

A=πr22=π⋅522≈39.27A = \dfrac{\pi r^2}{2} = \dfrac{\pi \cdot 5^2}{2} \approx 39.27

Perimeter25.71

P=πr+2r=π⋅5+10≈25.71P = \pi r + 2r = \pi \cdot 5 + 10 \approx 25.71

Half a circle: half the area, and the perimeter also includes the straight diameter.

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Practice Problems

16 problems
Problem 1 of 16
Easy
What is the formula for the area of a semicircle?

Why It Matters

Semicircles and quarter circles appear everywhere in design and architecture:
  • Architecture: Arched windows and doorways are often semicircular
  • Sports: The corner arcs of a soccer field are quarter circles, and the halfway line cuts the center circle into two semicircles
  • Design: Logos, pizza slices, and pie charts use circle fractions
  • Engineering: Tunnels, bridges, and domes often incorporate semicircular shapes
Understanding these shapes helps in calculating materials, painting areas, and designing spaces.

Real World Applications

Architecture: Arched Windows

Architects use semicircular windows to add elegance to buildings. Calculating the area helps determine glass costs.

Example:

A cathedral window has a semicircular top with radius 2 meters. The glass area is π×42=2π≈6.28\frac{\pi \times 4}{2} = 2\pi \approx 6.28 square meters.

1Try It Yourself

A Romanesque church has 8 semicircular windows, each with a radius of 1.5 meters.

What is the total glass area for all windows?

Step 1: Write the mathematical expression

Calculate one window, then multiply by 8:

Sports: Soccer Field Markings

A soccer field has a quarter circle in each corner and a center circle that the halfway line splits into two semicircles.

Example:

Each corner arc has a radius of 1 meter, so the grass inside it covers π×124≈0.79\frac{\pi \times 1^2}{4} \approx 0.79 square meters.

2Try It Yourself

The center circle of a soccer field has a radius of 9.15 meters. The halfway line splits it into two semicircles, one in each team's half.

What is the area of the semicircle in one team's half? Round to two decimal places.

Step 1: Write the mathematical expression

Use the semicircle formula with r=9.15r = 9.15:

Key Takeaways

  • 1A semicircle is half a circle: A=πr22A = \frac{\pi r^2}{2}
  • 2A quarter circle is one-fourth of a circle: A=πr24A = \frac{\pi r^2}{4}
  • 3First calculate the full circle area (πr2\pi r^2), then take the appropriate fraction
  • 4If given diameter, divide by 2 to find radius before calculating

Frequently Asked Questions

Divide the diameter by 2 to get the radius, then use the formula. For example, if a semicircle has diameter 10 cm, the radius is 5 cm, and the area is π×252=12.5π\frac{\pi \times 25}{2} = 12.5\pi square cm.
Divide the diameter by 2 to get the radius, then use the formula. For example, if a semicircle has diameter 10 cm, the radius is 5 cm, and the area is π×252=12.5π\frac{\pi \times 25}{2} = 12.5\pi square cm.
Yes, 3.14 is a common approximation. For more precision, use 3.14159 or the pi button on your calculator. Leave answers in terms of pi when exact values are needed.
The same principle applies! For a third of a circle, divide by 3: A=πr23A = \frac{\pi r^2}{3}. For any fraction, multiply the full circle area by that fraction.

Glossary

Semicircle
Half of a circle, formed by cutting a circle along its diameter
Quarter circle
One-fourth of a circle, bounded by two perpendicular radii and the arc between them
Radius
The distance from the center of a circle to any point on its edge
Diameter
The distance across a circle through its center (twice the radius)
Pi (π)
The ratio of a circle's circumference to its diameter, approximately 3.14159

Formula Card

Full circle

A=πr2A = \pi r^2

Area of a complete circle

Semicircle

A=πr22A = \frac{\pi r^2}{2}

Half of a full circle's area

Quarter circle

A=πr24A = \frac{\pi r^2}{4}

One-fourth of a full circle's area

Today's challengeOne problem, three tries

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