Areas of Shapes
Areas of Shapes Revision
Areas of Shapes
The area of a 2D shape is the amount of space it takes up in 2 dimensions, and its units are always squared, e.g
\text{cm}^2,\hspace{1mm}\text{m}^2
You need to know the formulas to calculate the areas of some common shapes and be able to rearrange them. Revising rearranging formulae will help with this topic.
![how to calculate the area of a rectangle](https://mmerevise.co.uk/app/uploads/2020/09/Area-of-Rectangle.png)
Area of Rectangle
![how to calculate the area of a rectangle](https://mmerevise.co.uk/app/uploads/2020/09/Area-of-Rectangle.png)
The area of a rectangle is length \times width
\text{Area} = L \times W
![how to calculate the area of a parallelogram](https://mmerevise.co.uk/app/uploads/2020/09/Area-of-Parallelogram.png)
Area of a Parallelogram
![how to calculate the area of a parallelogram](https://mmerevise.co.uk/app/uploads/2020/09/Area-of-Parallelogram.png)
The area of a parallelogram is base \times vertical height
\text{Area} = b\times h
![how to calculate the area of a trapezium](https://mmerevise.co.uk/app/uploads/2020/09/Area-of-Trapezium.png)
Area of a Trapezium
![how to calculate the area of a trapezium](https://mmerevise.co.uk/app/uploads/2020/09/Area-of-Trapezium.png)
The formula to calculate the area of a trapezium is:
\text{Area} = \dfrac{1}{2}(a+b)h
where a and b are the lengths of the parallel sides and h is the vertical height.
![how to calculate the area of a triangle half base times height](https://mmerevise.co.uk/app/uploads/2020/09/Area-of-Triangle.png)
Area of a Triangle 1
![how to calculate the area of a triangle half base times height](https://mmerevise.co.uk/app/uploads/2020/09/Area-of-Triangle.png)
The equation to calculate the area of a triangle is:
\text{Area} = \dfrac{1}{2} \times b \times h
Where b is the base width of the triangle and h is the vertical height.
![how to calculate the area of a triangle half a b sin c](https://mmerevise.co.uk/app/uploads/2020/09/Area-of-Triangle2.png)
Area of a Triangle 2
![how to calculate the area of a triangle half a b sin c](https://mmerevise.co.uk/app/uploads/2020/09/Area-of-Triangle2.png)
Another way to calculate the area of a triangle is as follows:
\dfrac{1}{2} \times a \times b \times \sin(C)
Where a and b are side lengths and C is the angle between the side lengths.
![area of a trapezium example question](https://mmerevise.co.uk/app/uploads/2020/09/Areas-of-Shapes-Example1.png)
Example 1: Finding the Area of a Trapezium
The shape is a trapezium with a perpendicular height of 4mm.
Calculate the area of the trapezium.
[2 marks]
![area of a trapezium example question](https://mmerevise.co.uk/app/uploads/2020/09/Areas-of-Shapes-Example1.png)
Formula: \text{Area}=\frac{1}{2}(a+b)h,
where a = 8,\hspace{1mm}b = 12.5, and h = 4 .
\begin{aligned}\text{Area } &= \dfrac{1}{2}(8+12.5) \times 4 \\ &=\dfrac{1}{2} \times 20.5 \times 4 \\ &= 41\text{ mm}^2 \end{aligned}
![area of a triangle example question](https://mmerevise.co.uk/app/uploads/2020/09/Areas-of-Shapes-Example2.png)
Example 2: Area of a Triangle
The triangle has a base of 6cm and an area equal to 24\text{ cm}^2.
Calculate its perpendicular height.
[2 marks]
![area of a triangle example question](https://mmerevise.co.uk/app/uploads/2020/09/Areas-of-Shapes-Example2.png)
Formula: \text{Area} = \dfrac{1}{2}\times b \times h
So we can add the numbers we know into the equation and solve for h:
24= \dfrac{1}{2}\times 6 \times h
24= 3 \times h
\dfrac{24}{3}= h
h = 8\text{cm}
Areas of Shapes Example Questions
Question 1: The triangle below has a base of 11.5 cm and a perpendicular height of 12 cm.
Calculate its area.
[2 marks]
The formula for the area is \frac{1}{2} \times b \times h, where b = 11.5 \text{ and } h = 12. So, we get:
\text{Area } = \dfrac{1}{2} \times 11.5 \times 12 = 69\text{cm}^2
Question 2: Below is a trapezium with sides of length 8cm, 5cm, and 5cm as shown below.
Calculate the perpendicular height and use it to find the total area.
[2 marks]
To work out the area, we will need to find the perpendicular height by forming a right-angled triangle,
The hypotenuse of the right-angled triangle is 5cm and the base is 3cm (8cm-5cm=3cm), so we get the perpendicular height of the trapezium as,
\text{Perpendicular height} = \sqrt{5^2 - 3^2} = \sqrt{16} = 4\text{cm}
Now we know the perpendicular height, we can calculate the area.
\text{Area} = \dfrac{1}{2}(a + b)h = \dfrac{1}{2}(5 + 8) \times 4 = 26\text{cm}^2
Question 3: Calculate the area of the parallelogram shown below.
[2 marks]
Area of a parallelogram is given by the formula,
\text{Area}=\text{base}\times\text{height}.
Therefore,
\text{Area}=8 \times 15 =120\text{cm}^2
HIGHER ONLY
Question 4: The triangle below has area 1.47\text{m}^2.
Work out the length of x to 2 decimal places.
[3 marks]
As we need to find a missing side-length rather than the area, we’re going to have to set up an equation and rearrange it to find x. The formula for the area we’ll need here is
\dfrac{1}{2}ab \sin(C),
so, our equation is
\dfrac{1}{2} \times 2.15 \times x \times \sin(26) = 1.47
Simplifying the left-hand-side, we get
1.075 \sin (26) \times x = 1.47
Finally, dividing through by 1.075\sin(26), and putting it into a calculator, we get
x = \dfrac{1.47}{1.075\sin(26)} = 3.12\text{m (2 d.p.)}
Areas of Shapes Worksheet and Example Questions
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(NEW) Areas of Shapes Exam Style Questions - MME
Level 4-5Level 6-7GCSENewOfficial MMEAreas of Shapes Drill Questions
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Area and Volume - Drill Questions
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