Finding Area of Compound Figures

How to finding area of compound figures, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

Compound figures (also called composite figures) are shapes made by combining two or more simple shapes such as rectangles, triangles, or semicircles. To find the area of a compound figure, you break it into simpler pieces, find each area separately, and then add (or subtract) the results. This skill appears frequently on the GED Math test, and with a clear strategy it becomes completely manageable.

What Is a Compound Figure?

A compound figure is any flat shape that cannot be classified as a single standard polygon. An L-shape, a T-shape, or a rectangle with a triangular notch cut out are all examples. Because every compound figure is built from basic shapes, the formulas you already know for rectangles, triangles, circles, and parallelograms are all you need.

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How to Find the Area of Compound Figures

Method 1: Addition (Combining Shapes)

Divide the compound figure into non-overlapping simpler shapes, find the area of each, and add them together.

Total \(\color{blue}{\text{ Area } = \text{ Area }}\)₁ + Area₂ + …

Method 2: Subtraction (Removing a Piece)

Sometimes it is easier to start with a large simple shape and subtract the area of the missing piece.

Total \(\color{blue}{\text{ Area } = \text{ Area }}\) of large \(\color{blue}{\text{ shape } – \text{ Area }}\) of removed piece

Key Formulas

• Rectangle: \(\color{blue}{A = l \times w}\)
• Triangle: A = ½ × \(\color{blue}{b \times h}\)
• Circle / Semicircle: A = πr² (full circle) or ½πr² (semicircle)
• Parallelogram: \(\color{blue}{A = b \times h}\)

Step-by-Step Summary

1. Look at the compound figure and identify the simpler shapes inside it.
2. Decide whether to add (combining pieces) or subtract (removing a cutout).
3. Label all needed dimensions from the diagram.
4. Calculate the area of each simple shape using the correct formula.
5. Add or subtract the individual areas to get the total area.
6. State the answer with the correct square units.

Watch: Finding the Perimeter and Area of a Composite Shape (Video Lesson)

Math with Mr. J walks through an L-shaped example step by step:

Worked Examples

Example 1: Find the area of an L-shaped figure with outer dimensions 10 \(\color{blue}{\text{ ft } \times 8}\) ft, with a 4 \(\color{blue}{\text{ ft } \times 3}\) ft rectangle removed from the upper-right corner.

Step 1: Area of full \(\color{blue}{\text{ rectangle } = 10 \times 8 = 80}\) ft²
Step 2: Area of removed \(\color{blue}{\text{ piece } = 4 \times 3 = 12}\) ft²
Step 3: Total \(\color{blue}{\text{ area } = 80 – 12}\) = 68 ft²

Example 2: A figure is made of a 6 \(\color{blue}{m \times 4}\) m rectangle topped with a semicircle of diameter 4 m (radius 2 m).

Step 1: Area of \(\color{blue}{\text{ rectangle } = 6 \times 4 = 24}\) m²
Step 2: Area of semicircle = ½ × π × 2² ≈ 6.28 m²
Step 3: Total area ≈ \(\color{blue}{24 + 6.28}\) = 30.28 m²

Example 3: Find the area of a 12 \(\color{blue}{\text{ in } \times 5}\) in rectangle with a 3 \(\color{blue}{\text{ in } \times 2}\) in rectangular notch removed.

\(\color{blue}{\text{ Area } = 12 \times 5 – 3 \times 2 = 60 – 6}\) = 54 in²

Example 4: A T-shape has a top bar 8 \(\color{blue}{\text{ cm } \times 2}\) cm and a vertical stem 2 \(\color{blue}{\text{ cm } \times 5}\) cm.

\(\color{blue}{\text{ Area } = (8 \times 2) + (2 \times 5) = 16 + 10}\) = 26 cm²

More Practice: Area of Composite Figures (Video)

This video works through additional composite figure examples with clear step-by-step explanations:

Exercises

1. An L-shape has outer dimensions 9 \(\color{blue}{\text{ ft } \times 6}\) ft with a 3 \(\color{blue}{\text{ ft } \times 3}\) ft square removed from one corner. Find the area.
2. A rectangle 7 \(\color{blue}{m \times 5}\) m has a right triangle with base 3 m and height 4 m attached to one end. Find the total area.
3. A figure is a 10 \(\color{blue}{\text{ cm } \times 8}\) cm rectangle with a 4 \(\color{blue}{\text{ cm } \times 2}\) cm cutout. Find the area.
4. A compound shape consists of two rectangles: one 6 \(\color{blue}{\text{ in } \times 4}\) in and one 6 \(\color{blue}{\text{ in } \times 3}\) in placed end to end. Find the total area.
5. A window is a 12 \(\color{blue}{\text{ ft } \times 3}\) ft rectangle topped by a 4 \(\color{blue}{\text{ ft } \times 4}\) ft square. Find the total area.

Answers

1. \(\color{blue}{9 \times 6 – 3 \times 3 = 54 – 9}\) = 45 ft²
2. \(\color{blue}{7 \times 5}\) + ½ × \(\color{blue}{3 \times 4 = 35 + 6}\) = 41 m²
3. \(\color{blue}{10 \times 8 – 4 \times 2 = 80 – 8}\) = 72 cm²
4. \(\color{blue}{6 \times 4 + 6 \times 3 = 24 + 18}\) = 42 in²
5. \(\color{blue}{12 \times 3 + 4 \times 4 = 36 + 16}\) = 52 ft²

Frequently Asked Questions

How do you find the area of compound figures?

Split the compound figure into simple shapes that do not overlap, such as rectangles and triangles, find the area of each one, then add them. For example, an L-shape can be cut into a 6 × 4 rectangle (24) and a 3 × 2 rectangle (6), giving 24 + 6 = 30 square units. Sometimes it is quicker to find the area of a big rectangle and subtract the missing corner.

How do you find the area of compound figures with triangles?

Treat the triangle as its own piece. Find its area with A = ½ × base × height, find the area of the other shapes, and add everything together. A house shape made of a 10 × 6 rectangle with a triangle roof of base 10 and height 4 has area 60 + ½ × 10 × 4 = 60 + 20 = 80 square units. Make sure the triangle’s height is measured straight up from its base.

How do you find the area of compound figures made of rectangles?

Draw a line to cut the figure into two or more rectangles, then multiply length × width for each and add. Missing side lengths come from the sides you know: if the full bottom edge is 12 and the top part is 5 wide, the other piece is 12 − 5 = 7 wide. Cutting the shape a different way should always give the same total, which is a good way to check your work.

How do you find the area of composite figures with circles?

Use A = πr² for a full circle and half of that for a semicircle, then add or subtract it from the other pieces. A 10 × 4 rectangle with a semicircle of diameter 4 on one end has radius 2, so the semicircle is ½ × π × 2² ≈ 6.28. The total is about 40 + 6.28 = 46.28 square units. Remember to halve the diameter to get the radius first.

How do you find the area and perimeter of composite figures?

For area, break the shape into pieces and add their areas. For perimeter, ignore the cut lines and add only the lengths around the outside edge. An L-shape with sides 8, 3, 5, 4, 3 and 7 has a perimeter of 8 + 3 + 5 + 4 + 3 + 7 = 30 units. Area is in square units, while perimeter is in plain units like cm or ft.

Related Topics

• Area of a Triangle
• Area of a Parallelogram
• Area of a Trapezoid
• Polygons
• Rectangular Prisms
• Cubes