Area of Composite Figures | Open Textbooks

What is a Composite Figure?

A Composite Figure is Made of Multiple Shapes

A composite figure (or composite shape) is a shape made up of two or more basic geometric shapes combined together.

In real life, most shapes we see are not simple rectangles or triangles. They are combinations of shapes! Think about the floor plan of a house, an L-shaped pool, or even a puzzle piece. To find the area of these shapes, we need to break them apart.

Examples of Composite Figures

These complex shapes can all be broken into rectangles and triangles.

The Strategy: Decompose and Calculate

To find the area of a composite figure, follow these steps:

Steps to Find Area of Composite Figures

Step 1: Identify the shapes - Look at the composite figure and decide how to break it into simpler shapes (rectangles, triangles, squares)
Step 2: Draw dividing lines - Sketch lines to separate the figure into parts
Step 3: Find dimensions - Label the length and width (or base and height) of each part
Step 4: Calculate each area - Use the appropriate formula for each shape
Step 5: Add the areas - Sum all the individual areas to get the total

Pro Tip: Sometimes it is easier to find the area of a larger shape and then SUBTRACT a missing piece. For example, a rectangle with a corner cut out can be solved by finding the full rectangle's area, then subtracting the cut-out piece.

Essential Area Formulas

Formulas You Need to Know

Rectangle

A = l x w

length times width

Square

A = s x s

side times side (or s squared)

Triangle

A = 1/2 x b x h

half of base times height

Parallelogram

A = b x h

base times height

Two Methods: Addition vs. Subtraction

+

Addition Method

Break the shape into smaller pieces and add all their areas together. Best for L-shapes, T-shapes, and joined figures.

-

Subtraction Method

Start with a larger simple shape, then subtract the missing pieces. Best for shapes with cutouts or holes.

Addition Method Example

Break an L-shape into two rectangles, find each area, then add them together.

Subtraction Method Example

Find the area of the full rectangle, then subtract the cutout piece.

Worked Examples

Let us solve some composite figure problems step by step.

Example 1: L-Shaped Room

Find the area of this L-shaped room.

1

Identify the shapes:
We can split this L-shape into two rectangles by drawing a vertical line.

2

Find the dimensions:
Rectangle 1 (left): 8 ft wide x 16 ft tall
Rectangle 2 (top right): 8 ft wide x 6 ft tall

3

Calculate each area:

Rectangle 1: A = 8 x 16 = 128 sq ft
Rectangle 2: A = 8 x 6 = 48 sq ft

4

Add the areas:

Total Area = 128 + 48 = 176 sq ft

Answer: The area of the L-shaped room is 176 square feet.

Example 2: T-Shaped Garden

A garden is shaped like a T. Find its total area.

1

Identify the shapes:
Split the T into a horizontal rectangle (top bar) and a vertical rectangle (stem).

2

Find the dimensions:
Top bar: 20 m wide x 4 m tall
Stem: 4 m wide x 12 m tall

3

Calculate each area:

Top bar: A = 20 x 4 = 80 sq m
Stem: A = 4 x 12 = 48 sq m

4

Add the areas:

Total Area = 80 + 48 = 128 sq m

Answer: The area of the T-shaped garden is 128 square meters.

Example 3: House-Shaped Sign

A sign is shaped like a house (rectangle with a triangle on top). Find its area.

1

Identify the shapes:
Rectangle (the house body) + Triangle (the roof)

2

Find the dimensions:
Rectangle: 14 in wide x 10 in tall
Triangle: base = 14 in, height = 8 in

3

Calculate each area:

Rectangle: A = 14 x 10 = 140 sq in
Triangle: A = 1/2 x 14 x 8 = 1/2 x 112 = 56 sq in

4

Add the areas:

Total Area = 140 + 56 = 196 sq in

Answer: The area of the house-shaped sign is 196 square inches.

Example 4: Rectangle with Corner Cut Out

Find the area of this shape (a rectangle with a square cut from one corner).

1

Identify the method:
Use subtraction! Find the area of the full rectangle, then subtract the cutout.

2

Find the dimensions:
Full rectangle: 20 cm x 14 cm
Cutout rectangle: 8 cm x 6 cm

3

Calculate each area:

Full rectangle: A = 20 x 14 = 280 sq cm
Cutout: A = 8 x 6 = 48 sq cm

4

Subtract the cutout:

Total Area = 280 - 48 = 232 sq cm

Answer: The area of the shape is 232 square centimeters.

Practice Problems

Try these problems on your own. Select the correct answer!

Problem 1: L-Shaped Patio

An L-shaped patio has the following dimensions: The main section is 12 ft by 8 ft. A smaller section extends 4 ft by 6 ft. What is the total area?

Problem 2: House Sign

A sign shaped like a house has a rectangular base that is 10 inches wide and 8 inches tall. The triangular roof has a base of 10 inches and a height of 6 inches. What is the total area?

Problem 3: Carpet with Cutout

A rectangular carpet is 15 feet by 12 feet. A square area of 3 ft by 3 ft is cut out for a floor vent. What is the remaining area of carpet?

Problem 4: T-Shaped Pool

A T-shaped swimming pool has a top section that is 18 m long and 4 m wide. The stem section is 6 m long and 8 m wide. What is the total surface area of the pool?

Check Your Understanding: Composite Figures Challenge

Test your skills with this 6-question challenge!

Composite Figures Challenge

Score: 0 / 6

Question 1 of 6

Challenge Complete!

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Next Steps

Key Takeaways

A composite figure is made of two or more basic shapes combined
Use the Addition Method: break into parts, find each area, then add
Use the Subtraction Method: find the whole area, then subtract cutouts
Always identify the shapes first and label all dimensions
Remember your formulas: Rectangle = l x w, Triangle = 1/2 x b x h

Practice identifying composite figures in real life (floor plans, logos, signs)
Try drawing your own composite figures and calculating their areas
Move on to learn about surface area in the next lesson