Area of a Semicircle and a Quarter Circle

Learn to calculate the area of partial circles

1. Area of a Semicircle

The Formula

A semicircle is half of a circle, so its area is half the area of a full circle.

Area of semicircle = $\frac{1}{2} \times \pi \times r^2$

Visual: The Semicircle

Example 1: Diameter Given $(\text{Take } \pi = \frac{22}{7})$

Find the area of the semicircle. The diameter is 42 cm.

Solution:

Step 1: Find the radius

Radius = 42 cm ÷ 2 = 21 cm

Step 2: Find the area of the full circle

Area of circle = $\frac{22}{7} \times 21 \times 21$ = 1386 cm²

Step 3: Find the area of the semicircle

Area of semicircle = 1386 ÷ 2 = 693 cm²

Example 2: Radius Given $(\text{Take } \pi = 3.14)$

Find the area of a semicircle with radius 10 m.

Solution:

Area of circle = 3.14 × 10 × 10 = 314 m²

Area of semicircle = 314 ÷ 2 = 157 m²

2. Area of a Quarter Circle

The Formula

A quarter circle is one quarter of a circle, so its area is one quarter of the area of a full circle.

Area of quarter circle = $\frac{1}{4} \times \pi \times r^2$

Visual: The Quarter Circle

Example: $(\text{Take } \pi = 3.14)$

Find the area of the quarter circle. The radius is 4 cm.

Solution:

Area of circle = 3.14 × 4 × 4 = 50.24 cm²

Area of quarter circle = 50.24 ÷ 4 = 12.56 cm²

3. Area of a Three-Quarter Circle

The Formula

A three-quarter circle is three quarters of a circle, so its area is three quarters of the area of a full circle.

Area of three-quarter circle = $\frac{3}{4} \times \pi \times r^2$

Visual: The Three-Quarter Circle

Example: $(\text{Take } \pi = \frac{22}{7})$

Find the area of the three-quarter circle. The radius is 14 cm.

Solution:

Step 1: Find the area of the full circle

Area of circle = $\frac{22}{7} \times 14 \times 14$ = 616 cm²

Step 2: Find the area of the three-quarter circle

Area of $\frac{3}{4}$ circle = $\frac{3}{4} \times 616$ = 462 cm²

4. Summary of Formulas

Full Circle

$\pi r^2$

Semicircle

$\frac{1}{2} \times \pi r^2$

Quarter Circle

$\frac{1}{4} \times \pi r^2$

Three-Quarter

$\frac{3}{4} \times \pi r^2$

Tip: Always find the area of the full circle first, then multiply by the fraction ( $\frac{1}{2}$ , $\frac{1}{4}$ , or $\frac{3}{4}$ ).

5. Practice Problems

Problem 1 $(\text{Take } \pi = \frac{22}{7})$

Find the area of the semicircle. Diameter = 28 cm.

ShowHide solution

Radius = 28 ÷ 2 = 14 cm

Area of circle = $\frac{22}{7} \times 14 \times 14$ = 616 cm²

Area of semicircle = 616 ÷ 2 = 308 cm²

Problem 2 $(\text{Take } \pi = 3.14)$

Find the area of the quarter circle. Radius = 22 cm.

ShowHide solution

Area of circle = 3.14 × 22 × 22 = 1519.76 cm²

Area of quarter circle = 1519.76 ÷ 4 = 379.94 cm²

Problem 3 $(\text{Take } \pi = \frac{22}{7})$

A slice of pizza is in the shape of a quarter circle of radius 14 cm. Find the area of the slice.

ShowHide solution

Area of circle = $\frac{22}{7} \times 14 \times 14$ = 616 cm²

Area of slice = 616 ÷ 4 = 154 cm²

Problem 4 $(\text{Take } \pi = 3.14)$

A semicircular garden has a radius of 8 m. Find the area of the garden.

ShowHide solution

Area of circle = 3.14 × 8 × 8 = 200.96 m²

Area of semicircle = 200.96 ÷ 2 = 100.48 m²

Key Takeaways

Area of semicircle = $\frac{1}{2} \times \pi r^2$ = Area of circle ÷ 2

Area of quarter circle = $\frac{1}{4} \times \pi r^2$ = Area of circle ÷ 4

Area of three-quarter circle = $\frac{3}{4} \times \pi r^2$ = Area of circle × $\frac{3}{4}$

Always find the radius first if the diameter is given!

Strategy: Calculate the area of the full circle first, then multiply by the appropriate fraction.

Area of a Semicircle and a Quarter Circle

1. Area of a Semicircle

The Formula

Visual: The Semicircle

Example 1: Diameter Given (Take π=227)(\text{Take } \pi = \frac{22}{7})(Take π=722​)

Example 2: Radius Given (Take π=3.14)(\text{Take } \pi = 3.14)(Take π=3.14)

2. Area of a Quarter Circle

The Formula

Visual: The Quarter Circle

Example: (Take π=3.14)(\text{Take } \pi = 3.14)(Take π=3.14)

3. Area of a Three-Quarter Circle

The Formula

Visual: The Three-Quarter Circle

Example: (Take π=227)(\text{Take } \pi = \frac{22}{7})(Take π=722​)

4. Summary of Formulas

5. Practice Problems

Problem 1 (Take π=227)(\text{Take } \pi = \frac{22}{7})(Take π=722​)

Problem 2 (Take π=3.14)(\text{Take } \pi = 3.14)(Take π=3.14)

Problem 3 (Take π=227)(\text{Take } \pi = \frac{22}{7})(Take π=722​)

Problem 4 (Take π=3.14)(\text{Take } \pi = 3.14)(Take π=3.14)

Key Takeaways

Example 1: Diameter Given $(\text{Take } \pi = \frac{22}{7})$

Example 2: Radius Given $(\text{Take } \pi = 3.14)$

Example: $(\text{Take } \pi = 3.14)$

Example: $(\text{Take } \pi = \frac{22}{7})$

Problem 1 $(\text{Take } \pi = \frac{22}{7})$

Problem 2 $(\text{Take } \pi = 3.14)$

Problem 3 $(\text{Take } \pi = \frac{22}{7})$

Problem 4 $(\text{Take } \pi = 3.14)$