Area of Semicircles and Quarter Circles

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

Intermediate25 minLesson
Printable worksheet

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.

Interactive Sandbox

Expression Calculator

Try these:

History

No calculations yet

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 three-point arc in basketball, penalty areas in soccer
  • 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: Basketball Three-Point Area

The three-point area in basketball includes quarter-circle corners at the baseline.

Example:

Each baseline corner is a quarter circle with radius 1.22 meters. Area = π×1.2224≈1.17\frac{\pi \times 1.22^2}{4} \approx 1.17 square meters.

2Try It Yourself

A basketball court has two quarter-circle corners at each baseline (4 total), each with radius 1.22 meters.

What is the combined area of all four quarter-circle regions?

Step 1: Write the mathematical expression

Four quarter circles = one full circle:

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, formed by two perpendicular radii
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

More in This Topic

Parts of a Circle and Circumference

Intermediate25 min

Arc Length

Intermediate25 min

Area of Circles

Intermediate25 min

Area of Annulus (Ring)

Intermediate20 min