GM Lesson 028 Composite Area Problems

Learning Intentions

By the end of this lesson, students will be able to:

• Decompose composite shapes into standard two-dimensional objects.
• Calculate composite areas by adding or subtracting areas.
• Communicate area solutions using clear working and square units.

Prerequisites

Students should already be able to:

• Calculate the area of a rectangle using .
• Calculate the area of a triangle using .
• Calculate the area of a parallelogram using .
• Calculate the area of a trapezium using .
• Calculate the area of a circle using .
• Calculate the area of a sector using .
• Use square units such as , and .

Key Idea Summary

A composite shape is made by joining or removing standard shapes.

To calculate a composite area:

1. Break the shape into familiar parts.
2. Label any missing lengths.
3. Calculate each area separately.
4. Add areas when shapes are joined.
5. Subtract areas when a section is removed.
6. State the answer using square units.

The key structure is:

or

Direct Instruction and Worked Examples

Time Allocation

Time Allocation

• Introduction, warmup and vocabulary: 5 minutes
• Direct instruction: 15 minutes
• Understanding checks: 5 minutes
• Exercises: 20 minutes
• Homework: 20 to 30 minutes outside the lesson it was taught in.

Link to original

Worked Example 1: Adding Rectangles

A garden bed is shaped like an L. It can be split into two rectangles.

• Rectangle A has length and width .
• Rectangle B has length and width .

Find the total area.

Rectangle A:

Rectangle B:

Total area:

Therefore, the area of the composite shape is:

Worked Example 2: Subtracting a Removed Rectangle

A rectangular piece of cardboard is by . A rectangular corner measuring by is cut out.

Find the remaining area.

Area of the large rectangle:

Area removed:

Remaining area:

Therefore, the remaining area is:

Worked Example 3: Rectangle with a Semicircle

A window is made from a rectangle with a semicircle on top.

• The rectangle is wide and high.
• The semicircle has diameter .

Find the total area, correct to decimal place.

Area of the rectangle:

The semicircle has diameter , so the radius is:

Area of the semicircle:

Total area:

Therefore, the area of the window is approximately:

Worked Example 4: Trapezium and Triangle

A park section is made from a trapezium joined to a triangle.

The trapezium has parallel sides and , with perpendicular height .

The triangle has base and perpendicular height .

Find the total area.

Area of the trapezium:

Area of the triangle:

Total area:

Therefore, the total area is:

Understanding Checks

Check 1

A composite shape is made from a rectangle of area and a triangle of area .

What is the total area?

Check 2

A large rectangle has area . A smaller rectangle of area is removed.

What is the remaining area?

Check 3

A student calculates the area of a composite shape by adding a rectangle and a triangle. The rectangle has dimensions by . The triangle has base and height .

Write the calculation needed to find the total area.

Check 4

A circle has radius . What expression gives the area of a semicircle with this radius?

Check 5

A sector has angle and radius .

What fraction of the full circle is the sector?

Exercises

Simple Familiar Exercises

Exercise 1

A composite shape is made from two rectangles.

• Rectangle A measures by .
• Rectangle B measures by .

Find the total area.

Exercise 2

A large rectangle measures by . A smaller rectangle measuring by is removed.

Find the remaining area.

Exercise 3

A composite shape is made from a rectangle and a triangle.

• The rectangle measures by .
• The triangle has base and perpendicular height .

Find the total area.

Exercise 4

A shape is made from a square with side length and a semicircle attached to one side.

The semicircle has diameter .

Find the total area, correct to decimal place.

Exercise 5

A trapezium has parallel sides and with perpendicular height . A rectangle measuring by is attached to it.

Find the total area.

Homework Problems

Problem 1

A composite shape is made from a rectangle measuring by and a triangle with base and height .

Find the total area.

Problem 2

A rectangular courtyard measures by . A square garden bed of side length is removed from one corner.

Find the remaining courtyard area.

Problem 3

A window is made from a rectangle with a semicircle on top.

• The rectangle is wide and high.
• The semicircle has diameter .

Find the total area, correct to decimal place.

Problem 4

A shaded region is formed by subtracting a circle of radius from a square of side length .

Find the shaded area, correct to decimal place.

Problem 5

A garden bed is made from a trapezium joined to a semicircle.

• The trapezium has parallel sides and with perpendicular height .
• The semicircle has diameter .

Find the total area, correct to decimal place.

Next: GM Lesson 029 Surface Area of Rectangular Prisms