8.01 Areas of special quadrilaterals

A parallelogram is a four-sided shape where the opposite sides are parallel. It is similar to a rectangle, but doesn't necessarily have right angles. This means a rectangle is a special type of parallelogram.

The good news is, we can use the similarities between parallelograms and rectangles to calculate the area of a parallelogram.

The area of a parallelogram can be found by multiplying the base and the perpendicular height, same as a rectangle.

Because a parallelogram might not have right angles, we need to make sure we use the perpendicular height when we work out the area. It doesn't matter which side we use as the base, as long as the height is the perpendicular height with respect to the base. \text{Area of a parallelogram}=\text{base }\times \text{height }\text{or } A=bh

Examples

Example 1

By rearranging the parallelogram into a rectangle, find its area.

Worked Solution

Create a strategy

Use the formula to find the area of the rectangle.

Apply the idea

Any parallelogram can be rearranged to form a rectangle:

The width of the rectangle is equal to the perpendicular height of the parallelogram, and their base lengths are the same.

\displaystyle \text{Area }	\displaystyle =	\displaystyle \text{length} \times \text{width}	Use the rectangle area formula
	\displaystyle =	\displaystyle 10 \times 7	Substitute the dimensions
	\displaystyle =	\displaystyle 70\ \text{cm}^2	Evaluate the product

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A rhombus is a four-sided shape with four equal sides and opposite sides parallel. It is similar to a square, but doesn't necessarily have right angles. This means a square is a special type of rhombus, and a rhombus is a special type of parallelogram.

If we are given the base and perpendicular height, then we can just use the formula for the area of a parallelogram. However, we are usually given the lengths of the diagonals, so we will need another method for finding the area of a rhombus.

Once again, we can use the similarities between rhombuses and rectangles to find their area.

By turning a rhombus into a rectangle, we can see that we end up with a rectangle that has one dimension equal to the length of one of the diagonals while the other dimension will be half the other diagonal. Notice that it does not matter which diagonal we half.\text{Area of a rhombus}=\frac{1}{2}\times \text{diagonal }x\times \text{diagonal }\ y\ \text{or} \ A=\frac{1}{2}xy

Examples

Example 2

Find the area of the rhombus shown.

Worked Solution

Create a strategy

We can find the area of rhombus using the formula: A=\dfrac{1}{2}xy

Apply the idea

We can see from the diagram that the diagonals have lengths of 4 mm and 8 mm.

\displaystyle \text{Area }	\displaystyle =	\displaystyle \frac{1}{2}\times 4 \times 8	Substitute the values
	\displaystyle =	\displaystyle 16\ \text{mm}^2	Evaluate the product

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Trapeziums (or "trapezia") are related to parallelograms. That's important because it means we can use the area of a parallelogram to help us work out the area of a trapezium.

We could break it up into a rectangle, and either one or two triangles, depending on its starting shape, and calculate each of these shapes individually.

We can also turn a trapezium into a rectangle, as demonstrated in the image below. By cutting it in the appropriate position, we can rearrange it to form a rectangle. The length, l, will be the average length of the two parallel sides, 6 and 13, so 9.5 will be the length of the rectangle formed.

Area of a trapezium is given by: \text{Area} = \dfrac{1}{2}\times \left(\text{side }a+\text{side }b\right)\times \text{height }h or A=\dfrac{1}{2}(a+b)h.

The sides a and b are the lengths of the parallel sides, and h is the perpendicular height.

Examples

Example 3

Find the area of the trapezium shown.

Worked Solution

Create a strategy

We can find the area of trapezium using the formula:A= \dfrac{1}{2}(a+b)h

Apply the idea

The side lengths are a=5\ \text{cm} and b=8\ \text{cm}, and the height is h=3 cm.

\displaystyle A	\displaystyle =	\displaystyle \dfrac{1}{2}(a+b)h	Use the formula
	\displaystyle =	\displaystyle \frac{1}{2}\times \left(5+8\right)\times 3	Substitute the values
	\displaystyle =	\displaystyle 19.5\ \text{cm}^2	Evaluate the product

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Kites are made up of two isosceles triangles. It is similar to a rhombus, but is not necessarily made up of two identical isosceles triangles, like a rhombus. This means a rhombus is a special type of kite.

Area of a kite is given by: \text{Area of a kite}=\dfrac{1}{2}\times \text{diagonal }x\times \text{diagonal }y or A=\dfrac{1}{2}xy.

Examples

Example 4

Find the area of the kite below.

Worked Solution

Create a strategy

We can use the formula: A=\dfrac{1}{2}xy.

Apply the idea

\displaystyle \text{Area }	\displaystyle =	\displaystyle \frac{1}{2}xy	Use the formula
	\displaystyle =	\displaystyle \frac{1}{2}\times 18\times 10	Substitute the diagonal lengths
	\displaystyle =	\displaystyle 90\ \text{cm}^2	Evaluate

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Now that we are familiar with all the formulas, we can find a missing dimension if we know the area of the shape and the other dimensions needed to substitute into the area formula.

For instance, if we know the area of a rhombus and the length of one diagonal, we can substitute the known values in to the area formula and then rearrange to solve for the unknown dimension.

Examples

Example 5

The kite below has an area of 48 \text{ cm}^2. The length of one of its diagonals is 12\,cm.

If the other diagonal has a length of k cm, solve for the value of k.

Worked Solution

Create a strategy

Substitute the area, the known diagonal, and the pronumeral k into the formula for the area of a kite.

Apply the idea

\displaystyle \text{A}	\displaystyle =	\displaystyle \frac{1}{2}\times x\times y	Use the formula
\displaystyle 48	\displaystyle =	\displaystyle \frac{1}{2}\times 12\times k	Substitute the values and k
\displaystyle 48	\displaystyle =	\displaystyle 6\times k	Evaluate the product
\displaystyle k	\displaystyle =	\displaystyle \frac{48}{6}	Divide both sides by 6
	\displaystyle =	\displaystyle 8\, \text{cm}	Evaluate

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