Area of compound figures with triangles, semicircles and quarter circles

Key Notes :

🔹What are Compound Figures?
  • A compound figure (or composite figure) is a shape made up of two or more simple shapes.
  • Examples of simple shapes: triangle, square, rectangle, circle, semicircle, quarter circle, trapezium.
  • To find the area of a compound figure, we divide it into known shapes, calculate each area, and then add or subtract as needed.
🔹 Area Formulas to Remember
  • Triangle: Area=1/2×base×height
  • Circle: Area=πr2
  • Semicircle (half circle): Area=1/2πr2
  • Quarter Circle (one-fourth of circle): Area=1/4πr2
🔹Steps to Solve Compound Figures
  • Identify the shapes inside the figure.
  • Divide the figure into known shapes (triangle, semicircle, quarter circle, etc.).
  • Apply the correct area formulas for each part.
  • Add areas if shapes are joined together.
  • Subtract areas if a part is removed or cut out.
🔹 Example Problems

✨ Example 1: Triangle + Semicircle

A figure has a triangle of base 10 cm and height 6 cm. A semicircle of radius 5 cm is attached to one side of the triangle.

  • Area of triangle = 1/2×10×6=30 cm2
  • Area of semicircle = 1/2π(52)=1/2×3.14×25=39.25 cm2
  • Total area = 30+39.25=69.25 cm2

✨ Example 2: Square with Quarter Circles in Corners

A square of side 14 cm has quarter circles drawn in each corner (radius 7 cm). Find the area left inside the square.

  • Area of square = 14×14=196 cm2
  • Area of one quarter circle = 1/4π(72)=14×3.14×49=38.465 cm2
  • 4 quarter circles = 4×38.465=153.86 cm2
  • Remaining area = 196−153.86=42.14 cm2
✨ Key Points to Remember
  • Always draw and label the figure clearly.
  • Use the correct radius (r) for circles and base, height for triangles.
  • Add areas when shapes are combined.
  • Subtract areas when a part is cut out.
  • Use π = 3.14 (or 22/7) depending on the question.

Learn with an example

▶️ What is the area of this figure?

Write your answer using decimals. Use 3.14 for ​𝜋. square 

_______ centimetres

Divide the figure into separate shapes:

Start with rectangle A. Rectangle A is 24 centimetres wide and 7 centimetres tall.

Multiply:

24 . 7 = 168

The area of rectangle A is 168 cm2.

Look at rectangle C. Rectangle C is 4 centimetres wide and 5 centimetres tall.

Multiply:

4 . 5 = 20

The area of rectangle C is 20 cm2.

Look at quarter circle D. Quarter circle D has a radius of 8 centimetres.

Solve for the area of quarter circle D.

3.14 . 82 / 4 = 50 . 24

The area of quarter circle D is 50.24 cm2.

Now add the areas of all the shapes.

168 cm2 + 154 cm2 + 20 cm2 + 50.24 cm2 = 392.24 cm2

The area is 392.24 square centimetres.

▶️ What is the area of this figure?

Write your answer using decimals. Use 3.14 for ​𝜋. square 

______ millimetres

Divide the figure into separate shapes:

Start with rectangle A. Rectangle A is 11 millimetres wide and 22 millimetres tall.

Multiply:

11 . 22 = 242

The area of rectangle A is 242 mm2.

Look at rectangle B. Rectangle B is 3 millimetres wide and 7 millimetres tall.

Multiply:

3. 7 = 21

The area of rectangle B is 21 mm2.

Look at rectangle C. Rectangle C is 4 millimetres wide and 3 millimetres tall.

Multiply:

4 . 3 = 12

The area of rectangle C is 12 mm2.

Look at semicircle D. Semicircle D has a diameter of 18 millimetres, which gives it a radius of 9 millimetres.

Solve for the area of semicircle D.

3 . 14 . 92 / 2 = = 127.17

The area of semicircle D is 127.17 mm2.

Now add the areas of all the shapes.

242 mm2 + 21 mm2 + 12 mm2 + 127.17 mm2 = 402.17 mm2

The area is 402.17 square millimetres.

▶️ What is the area of this figure?

Write your answer using decimals. Use 3.14 for ​𝜋. 

________ square kilometres

Divide the figure into separate shapes:

Start with rectangle A. Rectangle A is 8 kilometres wide and 22 kilometres tall.

Multiply:

8 . 22 = 176

The area of rectangle A is 176 km2.

Look at rectangle B. Rectangle B is 10 kilometres wide and 5 kilometres tall.

Multiply:

10 . 5 = 50

The area of rectangle B is 50 km2.

Look at semicircle C. Semicircle C has a diameter of 24 kilometres, which gives it a radius of 12 kilometres.

Solve for the area of semicircle C.

3.14 . 122 / 2 = 226.08

The area of semicircle C is 226.08 km2.

Now add the areas of all the shapes.

176 km2 + 50 km2 + 226.08 km2 = 452.08 km2

The area is 452.08 square kilometres.

Let’s practice!🖊️