091e. Area of Composite Shapes

Learning Intentions

• To understand what a composite shape is
• identify simple shapes that make up a composite shape
• find the area of a composite shape

Pre-requisite Summary

• Know that a composite shape is a shape made by joining two or more simple shapes
• Know that common simple shapes include rectangles, squares, triangles and parallelograms
• Know that area measures the amount of surface inside a two-dimensional shape
• Know that area is measured in square units such as ${\text{cm}}^2$ and ${\text{m}}^2$
• Know how to find the area of a rectangle using $A=l\times w$
• Know how to find the area of a triangle using $A={\displaystyle \frac{1}{2}}bh$
• Know how to find the area of a parallelogram using $A=bh$
• Know that the total area of a composite shape can be found by adding the areas of its parts

Worked Examples

Worked Example 1

A shape is made by joining two rectangles.
One rectangle measures $8\text{ cm}$ by $3\text{ cm}$ and the other measures $5\text{ cm}$ by $2\text{ cm}$.
Find the total area.

Worked Example 2

An L-shaped figure can be split into two rectangles.
The first rectangle measures $10\text{ cm}$ by $4\text{ cm}$ and the second measures $6\text{ cm}$ by $3\text{ cm}$.
Find the area of the composite shape.

Worked Example 3

A composite shape is made from a rectangle and a triangle.
The rectangle measures $12\text{ cm}$ by $5\text{ cm}$.
The triangle has base $12\text{ cm}$ and height $4\text{ cm}$.
Find the total area.

Worked Example 4

A composite shape is made from a parallelogram and a rectangle.
The parallelogram has base $9\text{ cm}$ and height $4\text{ cm}$.
The rectangle measures $9\text{ cm}$ by $3\text{ cm}$.
Find the total area.

Worked Example 5

A shape is made from three rectangles.
Their dimensions are $7\text{ cm}\times 2\text{ cm}$, $4\text{ cm}\times 5\text{ cm}$ and $3\text{ cm}\times 2\text{ cm}$.
Find the total area.

Worked Example 6

A garden bed is made from a rectangle and a triangle.
The rectangle is $8\text{ m}$ by $6\text{ m}$, and the triangle has base $8\text{ m}$ and height $3\text{ m}$.
Find the total area.

Problems

Problem 1

A shape is made by joining two rectangles.
One rectangle measures $6\text{ cm}$ by $4\text{ cm}$ and the other measures $3\text{ cm}$ by $5\text{ cm}$.
Find the total area.

Problem 2

An L-shaped figure can be split into two rectangles.
The first rectangle measures $9\text{ cm}$ by $5\text{ cm}$ and the second measures $4\text{ cm}$ by $3\text{ cm}$.
Find the area of the composite shape.

Problem 3

A composite shape is made from a rectangle and a triangle.
The rectangle measures $10\text{ cm}$ by $6\text{ cm}$.
The triangle has base $10\text{ cm}$ and height $5\text{ cm}$.
Find the total area.

Problem 4

A composite shape is made from a parallelogram and a rectangle.
The parallelogram has base $8\text{ cm}$ and height $5\text{ cm}$.
The rectangle measures $8\text{ cm}$ by $4\text{ cm}$.
Find the total area.

Problem 5

A shape is made from three rectangles.
Their dimensions are $5\text{ cm}\times 2\text{ cm}$, $6\text{ cm}\times 4\text{ cm}$ and $2\text{ cm}\times 3\text{ cm}$.
Find the total area.

Problem 6

A garden bed is made from a rectangle and a triangle.
The rectangle is $7\text{ m}$ by $5\text{ m}$, and the triangle has base $7\text{ m}$ and height $4\text{ m}$.
Find the total area.

Exercises

Understanding and Fluency

1. Decide whether each shape is a composite shape.
   a) A shape made from two joined rectangles
   b) A single square
   c) A shape made from a triangle and a rectangle

2. Identify the simple shapes that make up each composite shape.
   a) An L-shape split into two rectangles
   b) A house-shaped figure split into a rectangle and a triangle
   c) A shape split into a parallelogram and a rectangle

3. Find the total area of each composite shape.
   a) Two rectangles: $4\text{ cm}\times 3\text{ cm}$ and $5\text{ cm}\times 2\text{ cm}$
   b) Two rectangles: $9\text{ cm}\times 2\text{ cm}$ and $3\text{ cm}\times 4\text{ cm}$
   c) Two rectangles: $6\text{ m}\times 5\text{ m}$ and $2\text{ m}\times 3\text{ m}$

4. Find the total area of each composite shape.
   a) Rectangle $8\text{ cm}\times 4\text{ cm}$ and triangle with base $8\text{ cm}$, height $3\text{ cm}$
   b) Rectangle $10\text{ cm}\times 6\text{ cm}$ and triangle with base $10\text{ cm}$, height $2\text{ cm}$
   c) Rectangle $7\text{ m}\times 5\text{ m}$ and triangle with base $7\text{ m}$, height $4\text{ m}$

5. Find the total area of each composite shape.
   a) Parallelogram with base $6\text{ cm}$ and height $4\text{ cm}$, and rectangle $6\text{ cm}\times 3\text{ cm}$
   b) Parallelogram with base $9\text{ m}$ and height $5\text{ m}$, and rectangle $9\text{ m}\times 2\text{ m}$
   c) Parallelogram with base $12\text{ cm}$ and height $3\text{ cm}$, and rectangle $12\text{ cm}\times 4\text{ cm}$

6. An L-shaped figure is split into two rectangles. Find the area.
   a) $7\text{ cm}\times 4\text{ cm}$ and $3\text{ cm}\times 2\text{ cm}$
   b) $12\text{ cm}\times 5\text{ cm}$ and $4\text{ cm}\times 3\text{ cm}$
   c) $9\text{ m}\times 6\text{ m}$ and $2\text{ m}\times 4\text{ m}$

7. A shape is made from three rectangles. Find the total area.
   a) $3\text{ cm}\times 2\text{ cm}$, $5\text{ cm}\times 4\text{ cm}$, $2\text{ cm}\times 2\text{ cm}$
   b) $8\text{ cm}\times 3\text{ cm}$, $4\text{ cm}\times 5\text{ cm}$, $2\text{ cm}\times 3\text{ cm}$
   c) $6\text{ m}\times 2\text{ m}$, $7\text{ m}\times 3\text{ m}$, $3\text{ m}\times 2\text{ m}$

8. Find the total area of each composite shape.
   a) Rectangle $9\text{ cm}\times 4\text{ cm}$, triangle with base $9\text{ cm}$ and height $5\text{ cm}$
   b) Rectangle $11\text{ cm}\times 3\text{ cm}$, triangle with base $11\text{ cm}$ and height $2\text{ cm}$
   c) Rectangle $15\text{ m}\times 4\text{ m}$, triangle with base $15\text{ m}$ and height $6\text{ m}$

Reasoning

9. Explain why splitting a composite shape into simple shapes helps to find its area.

10. A student says the area of a composite shape is found by adding all the side lengths. Explain why this is incorrect.

11. A composite shape is made from a rectangle and a triangle. Explain why the total area is found by adding the two areas, not multiplying them together.

12. A student splits the same composite shape in two different valid ways and gets the same total area. Explain why this happens.

Problem-solving

13. A patio is made from two rectangles. One is $6\text{ m}$ by $4\text{ m}$ and the other is $3\text{ m}$ by $2\text{ m}$. Find the total area.

14. A sign is made from a rectangle $12\text{ cm}\times 5\text{ cm}$ with a triangle on top of base $12\text{ cm}$ and height $4\text{ cm}$. Find the total area.

15. A floor plan shows an L-shaped room split into two rectangles measuring $8\text{ m}\times 5\text{ m}$ and $3\text{ m}\times 2\text{ m}$. Find the area of the room.

16. A garden bed is made from a parallelogram with base $10\text{ m}$ and height $4\text{ m}$ and a rectangle $10\text{ m}\times 3\text{ m}$. Find the total area.

17. A composite tile design is made from three rectangles with areas $18{\text{ cm}}^2$, $24{\text{ cm}}^2$ and $10{\text{ cm}}^2$. Find the total area.

18. A playground section is made from a rectangle $14\text{ m}\times 6\text{ m}$ and a triangle with base $14\text{ m}$ and height $5\text{ m}$. How many square metres does it cover?

Potential Misunderstandings

• A student may think a composite shape is a completely new type of shape rather than a shape made from simple shapes
• A student may not identify all of the simple shapes needed to split the figure correctly
• A student may confuse area with perimeter and add outside side lengths instead of areas
• A student may use the wrong formula for one of the simple shapes
• A student may forget to divide by $2$ when finding the area of a triangle
• A student may use a slanted side instead of the perpendicular height
• A student may multiply or add measurements that belong to different parts of the shape incorrectly
• A student may find the areas of the parts correctly but then combine them incorrectly
• A student may omit square units in the final answer

Next: 092. Volume of Rectangular Prisms