**Area: ****Finding the area of common shapes**

**Levels 5-7**

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The objectives of this unit are to find the area of shapes involving:

* rectangles;

* triangles;

* parallelograms;

* trapeziums.

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The area of a shape is measured
in square units, such as cm^{2}, m^{2}, or mm^{2}. It is important that all lengths on a shape
are converted to the same unit before its area can be found.

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**Area of a rectangle**

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** **Area of a rectangle = length × width

**Example 1:**

Find the area of this shape:

__Solution__: We can split this shape up into 2
rectangles. One way of doing this is as
follows:

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**Area of a parallelogram**:

A parallelogram has an area equal to a rectangle with the same base and height measurements:

Therefore: Area of a parallelogram = base × (perpendicular) height.

Note: The measurements used for the base and the height must be perpendicular (i.e. at right angles) to each other.

**Example 2:**

Find the area of this parallelogram-

__Solution__: To find the area, we need two measurements
that are **perpendicular** to each
other. Here we see that the base is 6.8
cm and the height is 4.3 cm.

So area = 6.8 × 4.3 = 29.24 cm^{2}.

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**Example 3**:

Work out the length of the missing side.

__Solution__:

The measurements 12cm and 4cm are perpendicular to each other and so can be used to find the area of the parallelogram:

area
= 12 × 4 = 48cm^{2}

Another way to find the area of the parallelogram would be to multiply 8cm by the unknown height.

So

8 × ? = 48

? = 48 ÷ 8

? = 6cm

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**Area of a triangle**

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A triangle has an area that is half the area of the surrounding rectangle:

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So: area of a triangle = ½ × base × height or area =

**Example 4:**

Find the area of the triangle shown:

__Solution__:

Area =

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**Example 5:**

The triangle shown has an area of
36 cm^{2}. Find the height of
the triangle.

__Solution__:

Recall that a triangle has an area that is half of the surrounding rectangle.

Therefore the area of the
rectangle shown below must be 72cm^{2}.

The rectangle has length 9cm and width ?cm.

So 72 = 9 × ?

? = 72 ÷ 9 = 8cm

The triangle has height 8 cm.

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**Area of a trapezium**

The formula for finding the area of a trapezium is:

Area =

In this formula, *a* and *b* represent the __lengths of the parallel sides__ and *h* represents the __height__ of the
trapezium.

Instead of using the formula, some people prefer to remember the steps involved in finding the area of a trapezium:

__Step
1__: Add together the parallel sides

__Step
2__: Multiply by the height

__Step
3:__ Divide by 2.

**Example 6:**

Find the area of the trapezium shown below:

__Solution__:

Using the formula:

So: (working out the value of the brackets)

So:

OR using the steps:

Step 1: Add together the parallels sides 13 + 17 = 30

Step 2: Multiply by the height: 30 × 6.5 = 195

Step 3: Divide by 2: 195
÷ 2 = 97.5 cm^{2}.

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**Example 7:**

Find the area of the shape shown here:

__Solution__:

Area of the trapezium is

=

Area of the triangle is

So total area of the shape is 152
+ 96 = 248cm^{2}

^{ }

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**Important
points for area work**

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1) Remember to put units on your answers.

2) Remember to show your working out.

3) Learn the formulae!