Area of parallelograms - KS3 Maths

Key points

A series of two images. Each image shows a same parallelogram. In the first image the bottom horizontal side of the parallelogram has been labelled as the base. A dashed arrow extends the length of the base. In the middle of the shape a dashed vertical arrow, perpendicular to the base, passing to the opposite side, has been drawn with an arrow. It has been labelled as the height. In the second image the diagonal side, on the left, has been labelled as the base. A dashed arrow extends the length of this side. A diagonal dashed arrow, perpendicular to this base, passing to the opposite side, has been drawn. It has been labelled as the height. The arrows and labels for the base are coloured orange. The arrows and labels for the height are coloured blue. The parallelograms are coloured yellow. — Image caption,
The base and perpendicular height are at right angles.

A can be cut and rearranged to form a .
The length and width of the rectangle are the base and height of the parallelogram. The area of a parallelogram is the base multiplied by the height.
The base of a parallelogram is one of the sides. The perpendicular height is the shortest distance between the base and the opposite side.

Find the area of a parallelogram by rearranging it to make a rectangle

Identifying the base and the perpendicular height of a parallelogram:
- The base and the perpendicular height are at right angles.
- One side of the parallelogram is the base.
- The perpendicular height is at right angles to the base.
- The height can be shown inside or outside the parallelogram.
- The base and height may be shown in a different orientation, the base is not necessarily a side of the parallelogram.
Showing that a parallelogram has the same area as a rectangle:
- Split the parallelogram into a right-angled triangle and a trapezium.
- Move the triangle to the other end of the shape, making a rectangle.
- The length of the rectangle is the base of the parallelogram. The width of the rectangle is the height of the parallelogram. The shapes have the same area.

The area of the rectangle is found by multiplying the base and the perpendicular height.

The for this is \(A = bh\)

Examples

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Identify the base and height of the parallelogram.
Image caption,
The base and the perpendicular height are at right angles. One side of the parallelogram is the base. The base does not have to be the bottom side. The perpendicular height is not a side. It is at right angles to the base and is the shortest distance between the base and the opposite parallel side.
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The perpendicular height of the parallelogram may be shown inside or outside the shape.
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The base and perpendicular height may be shown in a different orientation, the base is not necessarily a horizontal side of the parallelogram.
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Show that the parallelogram above has the same area as a rectangle of base 4 cm and height 5 cm.
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To do this, split the parallelogram into a right-angled triangle and a trapezium. A trapezium is a quadrilateral with one pair of parallel sides.
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Move the right-angled triangle so that the shape becomes a rectangle. The rectangle has the same area as the parallelogram, even though it is a different shape.
Image caption,
The base and height of the rectangle are the base and perpendicular height of the parallelogram. The area of the rectangle is the length multiplied by the width, 𝑨 = 𝒍𝒘. The area of the parallelogram is the base multiplied by the perpendicular height, 𝑨 = 𝒃𝒉.

Question

Decide which of the lettered lengths are the base and perpendicular height of the parallelogram.

An image of a parallelogram. Two of the sides are vertical. The two diagonal sides slope down to the right. One of the diagonal sides is labelled as, a. One of the vertical sides is labelled as, b. A dashed horizontal line, drawn between the two vertical sides is labelled as, c. The parallelogram is coloured yellow.

How to calculate the area of a parallelogram

To work out the area of a parallelogram:

Identify the base and the height of the .
- Check that the base and height are measured in the same units.
- If necessary, convert the measurements so that the units match. When working in millimetres, a 1۰2 cm measurement would be converted to 12 mm.
Substitute the values of the base, \(b\), and the perpendicular height, \(h\), into the formula \(A = bh\).
Multiply the base and the perpendicular height.
Write down the answer with the appropriate square units.

Examples

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Find the area of the parallelogram.
Image caption,
The base and the perpendicular height are at right angles to each other. The base is 6 cm and the perpendicular height is 7 cm. They are in the same units.
Image caption,
The area of a parallelogram is the base multiplied by the perpendicular height. Substitute the values for the base and the perpendicular height into the formula 𝑨 = 𝒃𝒉, so 6 × 7 = 42. The area of the parallelogram is 42 cm².
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Work out the area of the parallelogram.
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The base and perpendicular height are at right angles to each other. The base is 4 cm and the perpendicular height is 12 mm. They are not in the same units. They need to be converted so that the units match.
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The area must be worked out using the same units. This may be in mm² or in cm². 4 cm is 40 mm. 12 mm is 1۰2 cm.
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Work either in centimetres or in millimetres. Use the formula by substituting the base and perpendicular height values. In centimetres, 4 cm × 1۰2 cm = 4۰8 cm. The area is 4۰8 cm². In millimetres, 40 mm × 12 mm = 480 mm. The area is 480 mm².

Question

Work out the area of the parallelogram.

An image of a parallelogram. Two of the sides are vertical. The two diagonal sides slope up to the right. One of the diagonal sides is labelled as, five metres. One of the vertical sides is labelled as, nine metres. A dashed line, perpendicular to the diagonal sides, is drawn between the two diagonal sides. It is labelled as, eight metres. The parallelogram is coloured pink.

How to calculate the base or perpendicular height, given the area

The area of a parallelogram is the base multiplied by the perpendicular height. To find either the base or the perpendicular height, use the inverse of multiplication, which is division.

To calculate the base of a parallelogram:
- Divide the area by the perpendicular height.
To calculate the perpendicular height of a parallelogram:
- Divide the area by the base.

Sometimes a maths problem may involve needing to work out the area first.

Examples

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Image caption,
The area of a parallelogram is the base multiplied by the perpendicular height. To find either the base or the perpendicular height, use the inverse operation. The inverse of multiply by the perpendicular height is divide by the perpendicular height. The base is the area divided by the perpendicular height. The inverse of multiply by the base is divide by the base. The perpendicular height is the area divided by the base.
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The parallelogram has an area of 24 m². Find the length of the base of the parallelogram.
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The base and the perpendicular height are at right angles. The perpendicular height is 3 m. The base multiplied by the perpendicular height gives the area of the parallelogram. The base multiplied by 3 is 24 m². The inverse of multiply by 3 is divide by 3. Divide the area by the perpendicular height. 24 ÷ 3 = 8. The length of the base of the parallelogram is 8 m.
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The parallelogram has an area of 65 mm². The base is 13 mm. Find the perpendicular height of the parallelogram.
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The base and the perpendicular height are at right angles to each other. The base is 13 mm. The base multiplied by the perpendicular height gives the area of the parallelogram. This is the same as the perpendicular height multiplied by the base. The perpendicular height multiplied by 13 is 65. The inverse of multiply by 13 is divide by 13. Divide the area by the base. 65 ÷ 13 = 5. The perpendicular height of the parallelogram is 5 mm.
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A parallelogram has a height which is one third of the length of the base. The area of the parallelogram is 12cm². Find the base and height of the parallelogram.
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The area of a parallelogram is the base multiplied by the height. The area of the parallelogram is 12 cm². Factor pairs of 12 are integer values that multiply to give 12. The factor pairs of 12 are 1 and 12, 2 and 6, and 3 and 4. The values of the base and height could be either of each pair of values. The base is the larger value so the factor pairs to choose between are 4 and 3, 6 and 2 and 12 and 1
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The height of the parallelogram is one third of its base, base ÷ 3 = height. This is the same as base = height × 3. 6 ÷ 3 = 2 and 6 = 2 × 3
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The parallelogram has an area of 12 cm² and a height that is one third of its base. 6 × 2 = 12. The area of the parallelogram is 12 cm². The base of the parallelogram is 6 cm, and its height is 2 cm.

Question

The area of the parallelogram is 200 mm². Work out the perpendicular height of the parallelogram.

An image of a parallelogram. Two sides are horizontal, and two sides are diagonal, sloping up to the left. One of the horizontal sides has been labelled as twenty five millimetres. Written inside the parallelogram: Area equals two hundred millimetres squared. The parallelogram is coloured green.

Practise finding the area of parallelograms

Practise finding the area of parallelograms with this quiz. You may need a pen and paper to help you with your answers.

Quiz

Real-life maths

An image of a tiled floor using, different coloured, congruent parallelograms. — Image caption,
Parallelogram-shaped tiles can produce designs that are really creative.

Tile manufacturers sell different types of tiles that are parallelogram-shaped.

The total area that the tiles contained in a pack will cover is given on its packaging. This helps a customer buying the tiles to know how many packs they should get in order to cover a wall or floor.

The manufacturer will need to know the area of each parallelogram-shaped tile to be able to give the customer this information.

Lots of arrangements that are possible with parallelogram-shaped tiles can produce designs that are really creative and often more unusual to look at than regular rectangular-shaped tiles.