Topics
10. 2D Measurement
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Stage 3

10.09 Other problems with area

Lesson
Practice
Lesson

Are you ready?

Can you find the area of a rectangle by using the length and width?

Examples

Example 1

Find the area of the rectangle shown.

A rectangle with a length of 12 centimetres and a width of 2 centimetres.
Worked Solution
Create a strategy

Use the area of a rectangle formula: \text{Area}=\text{Length} \times \text{Width}

Apply the idea

We can see that length is 12 \text{ cm} and the width is 2 \text{ cm}.

\displaystyle \text{Area}\displaystyle =\displaystyle \text{Length} \times \text{Width}Use the formula
\displaystyle =\displaystyle 12 \times 2 Substitute the length and width
\displaystyle =\displaystyle 24 \text{ cm}^2Double 12
Idea summary

The area of a rectangle is given by \text{Area}=\text{Length} \times \text{Width}.

When we calculate area, the unit of measurement is squared. If we have sides measured in centimetres \text{(cm)}, for example, the area will be \text{cm}^2.

Break up shapes to work out area

By breaking a shape into smaller rectangles, we can work out the area of those rectangles first. We can then add those two values together, to work out the total area of our shape.

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Examples

Example 2

Answer the following:

A shape made up of 2 rectangles. Ask your teacher for more information.
a

Find the area of the top rectangle.

Worked Solution
Create a strategy

Use the area of a rectangle formula.

Apply the idea

The shape is made up of two rectangles.

Small rectangle on top plus big rectangle at the bottom.

We need to calculate the area of top rectangle with a length of 8 \text{ cm} and width of 6 \text{ cm}.

\displaystyle \text{Top rectangle}\displaystyle =\displaystyle 6 \times 8Multiply the length by the width
\displaystyle =\displaystyle 48 \text{ cm}^2
b

Find the area of the bottom rectangle.

Worked Solution
Create a strategy

Use the area of a rectangle formula.

Apply the idea

We need to calculate area of the bottom rectangle with a length of 11 \text{ cm} and width of 5 \text{ cm}.

\displaystyle \text{Bottom rectangle}\displaystyle =\displaystyle 11 \times 5Multiply the length by the width
\displaystyle =\displaystyle 55 \text{ cm}^2
c

Find the total area.

Worked Solution
Create a strategy

Add the areas of top and bottom rectangles.

Apply the idea
\displaystyle \text{Total area}\displaystyle =\displaystyle 48 + 55Add the areas
\displaystyle =\displaystyle 103 \text{ cm}^2
Idea summary

Make sure to add all the smaller shapes together to get the total area of the shape.

Area of rectangles with distributive property

Once we know how to work out the area of rectangles, there are some handy things we can do. The distributive property of area means we can work out the area of a rectangle by breaking it into smaller rectangles.

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Examples

Example 3

The rectangle below has been split in to two rectangles. We want to work out the area.

Rectangle split into 2 that has 6 rows of 10 unit squares. The purple part has 2 columns while the green part has 8 columns.
a

What is the area of the purple rectangle on the left?

Worked Solution
Create a strategy

Use the area of a rectangle formula.

Apply the idea
Puple rectangle made up of 6 rows of 2 unit squares.

We can see that it is made up of 2 columns which will be the width, and 6 rows which will be the length in the formula.

\displaystyle \text{Area of purple rectangle}\displaystyle =\displaystyle 2 \times 6Multiply the length by the width
\displaystyle =\displaystyle 12 \text{ m}^2
b

What is the area of the green rectangle on the right?

Worked Solution
Create a strategy

Use the area of a rectangle formula.

Apply the idea
Green rectangle made up of 6 rows of 8 unit squares.

We can see that it is made up of 8 columns which will be the length, and 6 rows which will be width in the formula.

\displaystyle \text{Area of green rectangle}\displaystyle =\displaystyle 8 \times 6Multiply the length by the width
\displaystyle =\displaystyle 48 \text{ m}^2
c

What is the area of the whole rectangle?

Worked Solution
Create a strategy

Use the area of a rectangle formula.

Apply the idea

We can see that the whole rectangle is made up of 10 columns which will be the length, and 6 rows which will be width in the formula.

\displaystyle \text{Area}\displaystyle =\displaystyle 10 \times 6Multiply the length by the width
\displaystyle =\displaystyle 60 \text{ m}^2
Reflect and check

We could also add the areas of the two smaller rectangles:

\displaystyle \text{Area}\displaystyle =\displaystyle 12+48Add the two smaller areas
\displaystyle =\displaystyle 60 \text{ m}^2
Idea summary

Some ways to work out the area of a rectangle in a grid are:

  • Break the rectangle into two parts then find the area of each part and add.

  • Multiply the number of unit squares in each row by the number of rows.

Outcomes

MA3-10MG

selects and uses the appropriate unit to calculate areas, including areas of squares, rectangles and triangles

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