Area of Composite Shapes
Break complex shapes into simpler parts, calculate each area, and combine. The key skill for real-world measurement problems.
Printable Worksheets
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Worksheet
Download or print the worksheet to work through this lesson.
Q1: Imagine a rectangular room with a circular pillar in one corner. How would you find the area of carpet needed? Describe your strategy before learning any formulas.
Q2: A garden is shaped like an L (a large rectangle with a smaller rectangle removed from one corner). Would you find its area by adding pieces together or by subtracting a missing piece? Explain your choice.
Learning Intentions
Know
- Area formulas for triangles, rectangles, parallelograms, trapeziums, circles and sectors.
- The difference between addition and subtraction strategies for composite shapes.
Understand
- That a composite shape can be dissected into simpler shapes in more than one way.
- Why choosing the most efficient dissection saves time and reduces error.
Can Do
- Calculate the area of composite shapes using addition and subtraction.
- Select the most efficient dissection strategy for a given shape.
Success Criteria
- I can state the area formula for any basic shape (triangle, rectangle, parallelogram, trapezium, circle, sector).
- I can dissect a composite shape and calculate its total area by adding simpler areas.
- I can calculate the area of a shape with a hole or cut-out by subtraction.
Key Terms
Common Mistakes to Avoid
Wrong: Using the diagonal of a rectangle as its height when calculating area. The height must be perpendicular to the base.
Right: Area of a parallelogram = base $\times$ perpendicular height, not base $\times$ side length.
Wrong: Adding all side lengths to find the area of a composite shape. Perimeter and area are different measures.
Right: Label every dimension clearly. Check that all measurements use the same unit before calculating.
Before tackling composite shapes, ensure these formulas are fluent.
Unit consistency: All dimensions must be in the same unit before calculating. If a shape has sides in metres and centimetres, convert everything to one unit first.
What to write in your book
- Rectangle: $A = l \times w$
- Triangle: $A = \frac{1}{2}bh$
- Parallelogram: $A = b \times h$
- Circle: $A = \pi r^2$
Correct! $A = \frac{1}{2} \times 12 \times 5 = 30$ cm$^2$.
Not quite. $A = \frac{1}{2} \times 12 \times 5 = 30$ cm$^2$.
Split the composite shape into simpler shapes whose areas you can calculate, then add them together.
Steps:
- Draw a line (or lines) to split the shape into basic shapes.
- Label all dimensions. You may need to calculate missing lengths.
- Calculate the area of each part.
- Add the areas together.
What to write in your book
- Draw lines to split the shape into basic shapes
- Label all dimensions; calculate any missing lengths
- Calculate each part's area, then add them together
Correct! Rectangle = $9 \times 4 = 36$ m$^2$. Semicircle radius = 2 m, so area = $\frac{1}{2}\pi(2)^2 = 2\pi$ m$^2$.
Not quite. Rectangle = $9 \times 4 = 36$ m$^2$. Semicircle radius = 2 m, so area = $\frac{1}{2}\pi(2)^2 = 2\pi$ m$^2$.
When a shape has a hole, cut-out, or missing corner, it is often faster to find the area of the enclosing shape and subtract the missing part.
Steps:
- Identify the smallest rectangle (or other basic shape) that completely encloses the composite shape.
- Calculate the area of this enclosing shape.
- Calculate the area of the missing part(s).
- Subtract: Total area = Enclosing area $-$ Missing area.
What to write in your book
- Identify the smallest enclosing shape
- Calculate the enclosing area and the missing part(s)
- Subtract: Total = Enclosing area $-$ Missing area
Correct! Subtract the pond area from the field area. This avoids splitting the irregular remaining region into many parts.
Not quite. Subtract the pond area from the field area. This avoids splitting the irregular remaining region into many parts.
Interactive: Area Dissection Tool
Your Turn
Question 1: Find the area of a composite shape made from a rectangle 10 cm by 6 cm with a triangle of base 6 cm and height 4 cm on top.
Question 2: A square patio of side 7 m has a rectangular planter of 2 m by 1.5 m removed from one corner. Find the remaining patio area.
Question 3: Explain when the subtraction strategy is more efficient than the addition strategy.
Revisit Your Thinking
Look back at your Think First answer about the room with the pillar. Now calculate the exact carpetable area. Did your method match the subtraction strategy? Could you also solve it using the addition strategy? Which is more efficient here, and why?
Earlier you were asked: What was your first thought on this topic?
Now that you've worked through the lesson, write a fuller answer. What changed in your thinking?
What is the area of a triangle with base 12 cm and perpendicular height 5 cm?
A composite shape is made from a rectangle 9 m by 4 m with a semicircle of diameter 4 m on one end. What is the total area?
A square of side 10 cm has a smaller square of side 4 cm cut out from one corner. What is the remaining area?
Which strategy is most efficient for finding the area of a rectangular field with a circular pond in the middle?
The area of a sector with radius 6 cm and angle $60\degree$ is:
A garden bed is shaped as shown: a rectangle 8 m by 5 m with a right-angled triangle of base 5 m and height 3 m on one of the 5 m sides. Find the total area of the garden bed.
A rectangular deck measures 15 m by 10 m. A circular spa of radius 2.5 m is built into the deck.
(a) Find the area of the deck surrounding the spa. Give your answer to 1 decimal place. (3 marks)
(b) Explain why the subtraction strategy is more efficient than the addition strategy for this problem. (1 mark)
The floor plan of a room consists of a rectangle 12 m by 8 m joined to a trapezium with parallel sides 8 m and 5 m, and height 3 m. There is a rectangular column 1 m by 1 m in the corner where the two shapes meet.
(a) Find the area of the rectangle. (1 mark)
(b) Find the area of the trapezium. (1 mark)
(c) Find the total floor area, accounting for the column. (2 marks)
(d) If carpet costs $\$45$ per m$^2$, calculate the total cost to carpet the room. (1 mark)
Rectangle
$A = l \times w$
Triangle
$A = \frac{1}{2}bh$
Parallelogram
$A = b \times h$
Trapezium
$A = \frac{1}{2}(a+b)h$
Circle
$A = \pi r^2$
Sector
$A = \frac{\theta}{360}\pi r^2$
Real-Life Link
Land surveyors in Australia use composite area calculations daily when measuring irregular blocks of land. A typical suburban block might be a rectangle with a curved front boundary (a sector of a circle) and a corner cut off for a driveway. Council rates and land tax are calculated from these exact areas. In construction, painters and tilers must calculate wall and floor areas that include windows, doors and fixtures — all composite shape problems. Getting the area wrong means ordering too much or too little material, costing time and money.
Game Time!
Test your area skills in an interactive challenge.
Play Area Challenge