8.01 Area
Units
The units for length are commonly written as millimetres, centimetres, metres and kilometres. When calculating the area of any shape it is important that all the dimensions are written using the same units of length. This results in the following units for area:
square millimetres = mm2
(picture a square with side lengths of $1$1 mm each - this one is pretty small!)
square centimetres = cm2
(picture a square with side lengths of $1$1 cm each - about the size of a fingernail)
square metres = m2
(picture a square with side lengths of $1$1 m each - what do you know that is about this big?)
square kilometres = km2
(picture a square with a side length of $1$1km - I wonder how many of these your town or city is?)
Areas of rectangles
A rectangle is a shape which has the specific geometric properties:
- opposite sides are equal in length
- opposite sides are parallel
- adjacent sides meet at right angles.
$\text{Area of a rectangle }=\text{length }\times\text{width }$Area of a rectangle =length ×width
$A=lw$A=lw
Knowing the area of a rectangle is a very powerful thing! As you will see in this section, areas of rectangles will help us find the areas of triangles, parallelograms, trapeziums and even more!
Area of a square
A square is a shape which has the specific geometric properties:
- opposite sides are parallel
- adjacent sides meet at right angles
- all sides are of equal length.
$\text{Area of a square }=\text{Side length}\times\text{Side length}$Area of a square =Side length×Side length
$A=s^2$A=s2
Practice questions
Question 1
Find the area of the rectangle shown.
Question 2
Find the area of the square shown.
Area of a triangle
Did you know a rectangle can be drawn around every single triangle. See these:
But what is really amazing about these constructions is that the area of the triangle is exactly equal to the area of the extra triangles we added to make the each rectangle!
Why does this matter?
Well we already know how to find the area of a rectangle ($A=lw$A=lw)
This means that the following formula can be found to calculate the area of a triangle:
$\text{Area of a triangle }=\text{Half of the area of the rectangle with base and height the same as the triangle }$Area of a triangle =Half of the area of the rectangle with base and height the same as the triangle
$\text{Area of a triangle }=\frac{1}{2}\times\text{base }\times\text{height }$Area of a triangle =12×base ×height
$A=\frac{1}{2}\times bh$A=12×bh
Practice questions
Question 3
Find the area of the triangle shown.
A right-angled triangle is illustrated with its base measures $8$8 m and its height measures $7$7 m. The hypotenuse, side opposite to the right angle indicated by a small box, is not labeled.
Question 4
Find the value of $h$h in the triangle if its area is $120$120 cm2.
Other quadrilaterals
A quadrilateral is a shape with 4 straight sides. As we already know how to find the area of rectangles and triangles, one way to find the area of parallelograms, trapeziums, kites or rhombuses, would be to break up these shapes into smaller components comprising of rectangles and triangles, or by manipulating the shapes to look like rectangles.
Area of parallelograms
A parallelogram is a shape which has the specific geometric properties:
- opposite sides are equal in length
- opposite sides are parallel
From the interactive above can you see how easily a parallelogram turns into a rectangle? The height is perpendicular to the base.
Given we know the $\text{Area of a rectangle }=lw$Area of a rectangle =lw, we can calculate the area of any parallelogram.
$\text{Area of a parallelogram }=\text{base }\times\text{height }$Area of a parallelogram =base ×height
$A=bh$A=bh
Worked example
Example 1
Find the area of this parallelogram
Think: Identify the values for the base and height
Do:
| A | $=$= | $b\times h$b×h |
| $=$= | $32\times14$32×14 mm2 | |
| $=$= | $30\times14+2\times14$30×14+2×14 mm2 | |
| $=$= | $420+28$420+28 mm2 | |
| $=$= | $48$48 mm2 |
Practice question
Question 5
Find the area of the parallelogram shown.
A parallelogram is shown. The base measures 21cm. The height measures 20 cm, as represented by a vertical dashed line inside the parallelogram which is drawn perpendicularly from the top side to the bottom right vertex.
Area of a trapezium
A trapezium (sometimes called trapezoid) is a shape which has the specific geometric properties:
- 1 pair of opposite sides that are parallel
All of these are trapeziums
From the interactive you can see how to flip the trapezium over and turn it into a parallelogram, and we already know that a parallelogram has the same area as a rectangle! This new shape is made up of two trapeziums so we start with the parallelogram formula then halve it.
$\text{Area of a trapezium}=\frac{1}{2}\times\left(\text{base 1 }+\text{base 2 }\right)\times\text{height }$Area of a trapezium=12×(base 1 +base 2 )×height
$A=\frac{1}{2}\times\left(a+b\right)\times h$A=12×(a+b)×h
Worked example
Example 2
A new chocolate bar is to be made with the following dimensions, the graphic artist needs to know the area of the trapezium to begin working on a wrapping design. Find the area.
Think: We need to identify the 2 bases and the height. We can see these on the given diagram.
Do:
| $\text{Area of a trapezium}$Area of a trapezium | $=$= | $\frac{1}{2}\times\left(\text{base 1 }+\text{base 2}\right)\times\text{height }$12×(base 1 +base 2)×height |
| $=$= | $\frac{1}{2}\times\left(a+b\right)\times h$12×(a+b)×h | |
| $=$= | $\frac{1}{2}\times\left(4+8\right)\times3$12×(4+8)×3 | |
| $=$= | $\frac{1}{2}\times12\times3$12×12×3 | |
| $=$= | $18$18 cm2 |
Practice questions
Question 6
Find the area of the trapezium shown.
Question 7
Find the value of $x$x if the area of the trapezium shown is $65$65 cm2.
Start by substituting the given values into the formula for the area of a trapezium.
$A=\frac{1}{2}\left(a+b\right)h$A=12(a+b)h
Area of a kite
A kite is a 2D shape which has the specific geometric properties:
- 2 pairs of adjacent sides that are equal
- at least 1 pair of opposite angles which are equal.
Of course the kite you fly around on a windy day is named after the geometric shape it looks like.
Kites can taken on many different shapes and sizes. Try moving points $A$A, $O$O and $D$D on this mathlet to make many kinds of kites.
From the interactive you'll notice that if you copy the inner triangles of the kite and rearrange them you can create - you guessed it - a rectangle. To do this draw some diagonals from A to D and from E to F. The 2 pairs of resulting equal right-angled triangles can be arranged so to make 2 rectangles that share an equal side. But notice, the shared side length is only half the length of EF.
Because we know the $\text{Area of a rectangle }=l\times w$Area of a rectangle =l×w, then we can work out the area of any kite.
We tend to call the long diagonal $x$x and we call the short diagonal of the kite $y$y. These give us the length and width of the rectangle that the kite fits inside.
$\text{Area of a kite}=\frac{1}{2}\times\text{diagonal 1}\times\text{diagonal 2}$Area of a kite=12×diagonal 1×diagonal 2
$A=\frac{1}{2}\times x\times y$A=12×x×y
Worked example
Example 3
Find the area of this kite
Think: I need to identify the long diagonal length and the short diagonal length.
Do:
| $\text{Area of a kite }$Area of a kite | $=$= | $\frac{1}{2}\times x\times y$12×x×y |
| $=$= | $\frac{1}{2}\times4\times\left(2\times0.9\right)$12×4×(2×0.9) | |
| $=$= | $\frac{1}{2}\times4\times1.8$12×4×1.8 | |
| $=$= | $3.6$3.6 mm2 |
Practice questions
Question 8
Find the area of the kite shown.
Question 9
The area of a kite is $640$640 cm2 and one of the diagonals is $59$59 cm. If the length of the other diagonal is $y$y cm, what is the value of $y$y rounded to two decimal places?
Area of a rhombus
A rhombus is a 2D shape which has the specific geometric properties:
- all sides are equal in length
- opposite sides are parallel
- opposite angles are equal
- diagonals bisect each other
You can play with this mathlet to make many kinds of rhombuses, it also shows that you only need 1 side length and 1 angle to create one.
From this interactive you can see that as you copy the inner triangles of the rhombus and place them accordingly you can create - you guessed it - a rectangle. Of course, like the trapezium and kite, our original shape is only $\frac{1}{2}$12 of this rectangle.
Because we know the $\text{Area of a rectangle }=l\times w$Area of a rectangle =l×w, then we can work out the area of any rhombus.
Let's call the diagonals $x$x and $y$y. These give us the length and width of the rectangle that the rhombus fits inside.
$\text{Area of a rhombus }=\frac{1}{2}\times\text{diagonal 1}\times\text{diagonal 2}$Area of a rhombus =12×diagonal 1×diagonal 2
$A=\frac{1}{2}\times x\times y$A=12×x×y
Worked example
Example 4
A packing box with a square opening is squashed into the rhombus shown. What is the area of the opening of the box?
Think: I need to be able to identify the two diagonals.
Do:
| $\text{Area of a rhombus }$Area of a rhombus | $=$= | $\frac{1}{2}\times\text{diagonal 1 }\times\text{diagonal 2 }$12×diagonal 1 ×diagonal 2 |
| $=$= | $\frac{1}{2}\times x\times y$12×x×y | |
| $=$= | $\frac{1}{2}\times16\times4$12×16×4 | |
| $=$= | $32$32 cm2 |
Practice question
Question 10
Find the shaded area shown in the figure.
