Area of a rectangle
A rectangle is a shape with the specific geometric properties:
- opposite sides equal in length
- opposite sides parallel
- sides meet at right angles.
The area of a rectangle is the size of the space within its boundary. This is the number of unit squares it takes to cover the shape. For a rectangle with length $5$5 centimetres and width $3$3 centimetres, how many squares of size $1$1 cm2 would it take to cover it?
If we divide the shape into a grid of $1$1 cm2, we can quickly see that it would take $3$3 rows of $5$5 squares to cover the shape, and so the area is $15$15 cm2. We can always take this approach of dividing a rectangle into a grid - and in doing so, we obtain the formula for the area of a rectangle:
$\text{Area of a rectangle}=\text{length}\times\text{width}$Area of a rectangle=length×width
$A=L\times W$A=L×W
Area of a square
A square is a special kind of rectangle, because it meets all the requirements of a rectangle but has the extra property that all sides are equal length.
To find the area of a square, we use the rectangle rule,
$A=L\times W$A=L×W
Consider this example of a square with sides equal to 4 units:
 |
Area | $=$= | $4\times4$4×4 | |
| $=$= | $16$16 square units |
As the length and width are the same, the area of a square has a simpler rule that we can use:
$\text{Area of a square}=\text{length}\times\text{length}$Area of a square=length×length
$A=L^2$A=L2
Practice questions
Question 1
Find the area of the attached figure.
Question 2
Find the area of the rectangle shown.
Area of triangles
The area of a triangle is given by the formula $Area=\frac{1}{2}\text{base}\times\text{height}$Area=12base×height, which can be abbreviated to $A=\frac{1}{2}bh$A=12bh.
The height in this formula is referring to the height that is perpendicular to the base length. This may be one of the sides of the triangle (like in the right-angled triangle shown above), it may be within the triangle (like the second triangle above), or it may be outside of the triangle (like in the third triangle above).
Where does this formula come from? Notice we can draw a rectangle around any triangle. For example:
In each case, that the area of the triangle is exactly equal to the area of the extra triangles we added to make the each rectangle! So the area of a triangle is half the area of the rectangle with the same base and height.
Use the following applet to see a range of triangles and their related rectangles by moving the points and slider.
|
|
Worked example
Example 1
Find the area of the triangle pictured below.
Think: Identify the base and perpendicular height and use the formula. Here the base is $7$7 cm and the height perpendicular to this is $8$8 cm.
Do:
| $A$A | $=$= | $\frac{1}{2}bh$12bh |
Write the formula. |
| $=$= | $\frac{1}{2}\times7\times8$12×7×8 cm2 |
Substitute in known values and write down units. |
|
| $=$= | $\frac{1}{2}\times56$12×56 cm2 |
|
|
| $=$= | $28$28 cm2 |
|
So the area of this triangle is $28$28 cm2.
The rule for the area of a triangle is:
$Area=\frac{1}{2}\text{base}\times\text{height}$Area=12base×height
This can be abbreviated to:
$A=\frac{1}{2}bh$A=12bh
The base and height must be perpendicular, as shown in the diagrams below:
 |
 |
 |
Practice question
Question 3
Find the area of the triangle with base length $10$10 m and perpendicular height $8$8 m shown below.
Area of a parallelogram
A parallelogram is a shape with the geometric properties:
- opposite sides equal in length
- opposite sides parallel
Parallelogram exploration
Use the applet below to explore how a parallelogram can be broken up and turned into a rectangle.
We know that the area of a rectangle is given by $A=L\times W$A=L×W, so we can work out the area of any parallelogram.
We call the bottom side of the parallelogram the base, and we also need the height of the parallelogram. The height must be measured perpendicularly (at right angles) to the base, not along the sloping side, just like for the area of a triangle.
$\text{Area of a parallelogram}=\text{base}\times\text{height}$Area of a parallelogram=base×height
$A=b\times h$A=b×h
The calculation above is exactly the same as the area of a rectangle even though the rule uses different words for the length and width).
Worked example
Example 2
Find the area of the parallelogram shown:
Think: Identify the values for the base and height, then multiply.
Do:
| $A$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 questions
Question 4
Find the area of the parallelogram shown.
A parallelogram with the horizontal top side labeled 15 cm, suggesting the measurement of its base. A vertical dashed line is drawn beyond the parallelogram, connected to a horizontal dashed line perpendicular to the horizontal top side. This vertical dashed line is labeled 14 cm, suggesting the measurement of the parallelogram's height perpendicular to the base.
Area of a trapezium
A trapezium (sometimes called trapezoid) is a shape with:
- One pair of opposite sides that are parallel
All of these are trapeziums:
Trapezium exploration
The interactive below shows that two identical trapeziums can be joined to form a parallelogram. We can use this property to determine the area of a trapezium.
We can see in the diagrams below that two identical trapeziums make a parallelogram with an area:
$\text{Area }=\left(a+b\right)\times h$Area =(a+b)×h.
So a single trapezium must have half of the area of the parallelogram:
$\text{Area }=\frac{1}{2}\times\left(a+b\right)\times h$Area =12×(a+b)×h
|
$\text{Area }=\frac{1}{2}\times\left(a+b\right)\times\text{height }$Area =12×(a+b)×height |
|
$\text{Area of a trapezium}=\frac{1}{2}\times\text{sum of parallel sides}\times\text{height}$Area of a trapezium=12×sum of parallel sides×height
$A=\frac{1}{2}\times\left(a+b\right)\times h$A=12×(a+b)×h
Typically, the letters $a$a and $b$b are used for the length of the parallel sides (it doesn't matter which one is which)
Practice questions
Question 5
Find the area of the trapezium shown.
Question 6
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
Areas of kites and rhombuses
A kite is a shape with geometric properties:
- Two pairs of adjacent sides that are equal
- One pair of equal angles
Of course the kite you fly around on a windy day is named after the geometric shape it looks like.
A rhombus is a kite with the additional property that all sides are the same length. (A rhombus also meets all the conditions for a parallelogram.)
Kite exploration
The interactive below shows that a kite can be broken into triangles, which can be used to determine the area.
$\text{Area of a kite }=\frac{1}{2}\times\text{long diagonal }\times\text{short diagonal }$Area of a kite =12×long diagonal ×short diagonal
$A=\frac{1}{2}\times x\times y$A=12×x×y
Typically, the letters $x$x and $y$y are used for the length of the diagonals of the kite (it doesn't matter which way around).
Practice questions
Question 7
Find the area of the kite shown.
Question 8
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 circle
We already know that area is the space inside a two dimensional shape. We can find the area of a circle, but we will need a special rule.
The following investigation will demonstrate what happens when we unravel sectors of a circle.
When we realign the segments we end up with a parallelogram shape. We can use this to find the area of a circle based on our knowledge that the area of a parallelogram has formula $A=bh$A=bh. The parallelogram formed from a circle has a base length equal to half the circumference and the height is the radius.
$\text{Area of a circle}=\pi r^2$Area of a circle=πr2
Area of a sector
As we saw in our previous lesson, a sector is a fraction of a circle and the arc length is a fraction of the circumference. The same is true for area: the area of a sector is a fraction of the total area. For example, the area of a semicircle is half the area of the full circle.
Semicircle
$\text{Area of semicircle}=\frac{1}{2}\times\pi r^2$Area of semicircle=12×πr2
We can make a formula for the area of any sector depending on the angle $\theta$θ that subtends the arc at the centre.
A sector with contained angle $\theta$θ corresponds to a fraction $\frac{\theta}{360}$θ360 of a full circle and so its area is given by:
| $A$A | $=$= | $\frac{\theta}{360}\times\pi r^2$θ360×πr2 |
Practice questions
Question 9
If the radius of the circle is $5$5 cm, find its area.
Give your answer as an exact value.
Question 10
Find the area of the following sector of a circle.
Round your answer to one decimal place.
QUESTION 11
A goat is tethered to a corner of a fenced field (shown). The rope is $9$9 m long. What area of the field can the goat graze over?
Give your answer correct to two decimal places.