Space, shape and measurement: Solve problems by constructing and interpreting trigonometric models
Unit 5: Apply the area rule
Dylan Busa

Unit outcomes
By the end of this unit you will be able to:
- Apply the area rule in 2-D triangles.
What you should know
Before you start this unit, make sure you can:
- Calculate the area of a triangle using the formula area=12×b×h.
- Use a calculator to calculate the sine of a given angle. Refer to level 2 subject outcome 3.6 unit 2 if you need help with this.
- Use a calculator to calculate the angle from a given ratio for sine. Refer to level 2 subject outcome 3.6 unit 2 if you need help with this.
Introduction
Trigonometry is not just useful for finding the lengths of unknown sides or the sizes of unknown angles in right-angled triangles. It can also be put to work in non-right-angled triangles and it can be used to find measures other than the lengths of sides and the sizes of angles.
The area rule, for example, is a useful trigonometric identity (remember those from unit 3?) that can be used to find the area of any triangle, even triangles that have no 90∘ angles. But why do we need another way to calculate the area of a triangle if we already have the well-known formula Area=12×b×h? Well, sometimes, we might not know the length of the perpendicular height or the base. Instead, we might know the size of one of the angles inside the triangle.
In this unit, we are going to derive the area rule and see how to use it to find the areas of all sorts of different triangles in different contexts.
Derive the area rule
Deriving the area rule is not too difficult. You will be led through this process in Activity 5.1.

Activity 5.1: Derive the area rule
Time required: 15 minutes
What you need:
- a piece of paper
- a pen or pencil
- a ruler
What to do:
- On your piece of paper, draw any triangle ABC and label the sides in relation to the opposite vertices (see Figure 1). The triangle can be any size and shape.
- Now drop a perpendicular from B to the opposite side and call this perpendicular h (see Figure 2).
- Write the expression for the area of triangle ABC based on its base and perpendicular height.
- Write an expression for sinA.
- Write a new expression for the area of triangle ABC that includes the term sinA. Hint: look for a substitution.
- Now, write an expression for sinC and then write a similar expression for the area of the triangle in terms ofsinC.
- What do you think the area of triangle ABC would be in terms of sinB?
What did you find?
- Remember that your triangle ABC can be any size and shape. Figure 1 shows just one possible triangle.
Figure 1: Triangle ABC - Remember that dropping a perpendicular means that the new line meets the other line at 90∘. Figure 2 shows what your perpendicular line may look like.
Figure 2: Triangle ABC with perpendicular h - The area of triangle ABC is given by A=12×b×h where the base is side b and the perpendicular height is h. Therefore, area of ΔABC is 12×b×h.
- sinA=oppositehypotenuse=hc.
- sinA=hc∴h=c×sinA. But area ΔABC is 12×b×h. Therefore, area ΔABC=12×b×c×sintextA=12bcsinA.
- sinC=oppositehypotenuse=ha. Therefore h=a×sinC and area ΔABC=12×b×a×sinC=12absinC
- In each case above, the expression for the area of the triangle is the product of the length of two sides and the sine of the angle between these two sides. Therefore, area ΔABC=12acsinB.
The area rule states that in any ΔABC, the area is given by:

Take note!
The area rule works for any triangle for which you know the length of any two sides and the size of the included angle (the angle between the two known sides).
Use the area rule
Let’s look at some examples of how to apply the area rule.

Example 5.2
Find the area of ΔABC correct to two decimal places.
Solution
We are not given the lengths of two sides and the included angle, so we cannot use the area rule with the information we currently have. We need to find the size of ˆB. To do this, we can use the fact that AB=BC (given) and therefore, ˆA=ˆC=43∘ (angles opposite equal sides). Then we can use that fact that angles in a triangle are supplementary to find ˆB.
AB=BC (given)
∴ˆA=ˆC=43∘ (angles opposite equal sides)
∴ˆB=180∘−ˆA−ˆC=180∘−43∘−43∘=94∘ (angles in a triangle are supplementary)
Now we can use the area rule.
Area ΔABC=12acsinB=12×8×8×sin94∘=31.92 m2
Remember to round off your final answer and include the correct area units.

Example 5.3
A parallelogram has adjacent sides of 16 cm and 23 cm. The angle between them is 43∘. Calculate the area of the parallelogram correct to two decimal places.
Solution
If no diagram is given, it is always best to draw your own using the given information. This does not have to be accurate.
We know that ABCD is a parallelogram. Therefore, if we can find the area of ΔACD we can multiply this by 2 to find the area of ABCD.
In ΔACD, we know two sides and the included angle so we can use the area rule.
Area ΔADC=12acsinD=12×23×16×sin43∘=125.49 cm2
The area of parallelogram ABCD=2×125.49 cm2=250.98 cm2.

Example 5.4
If ΔEFG has an area of 167.43 cm2 and e=17 cm and f=24.35 cm, what are the two possible sizes of ˆG, correct to two decimal places.
Solution
This question gives us the area and asks us to calculate the size of ˆG.
Area ΔEFG=12efsinG∴167.43=12×17×24.35×sinG∴sinG=53.99∘
We have found that ˆG=53.99∘ but the question mentions two possible solutions. Remember that sine is also positive in the second quadrant, in other words, for angles 90∘≤θ≤180∘. Therefore, ˆG=180∘−53.99∘=126.01∘ as well. But does this make physical sense? Does such a triangle exist? Figure 3 shows both possible triangles with these dimensions.


Exercise 5.1
- Calculate the area of ΔABC to two decimal places.
- Calculate the area of ΔABC given a=10 cm, c=8 cm and ˆB=35∘, to three decimal places.
- Determine the area of equilateral ΔPQR with p=14 cm to two decimal places.
- Determine, to two decimal places, the area of a parallelogram in which two adjacent sides are 10 cm and 13 cm and the angle between them is 55∘.
- Determine the length of AC (to one decimal place) if the area of ΔABC=16.18 m2.
The full solutions are at the end of the unit.
Summary
In this unit you have learnt the following:
- What the area rule is.
- How to use the area rule to find the area of any triangle where two sides and the included angle are known.
Suggested time to complete: 40 minutes
- Calculate the area of ΔKLM to two decimal places.
- Calculate, to two decimal places, the area of ΔABC given b=19 cm, c=18 cm and ˆA=49∘.
- Determine, to two decimal places, the area of ΔPQR with p=r=9 m.
- Determine the area of a parallelogram PQRS to two decimal places.
- Determine two possible values for ˆX (to one decimal place) if the area of ΔXYZ=26.72 m2.
The full solutions are at the end of the unit.
Exercise 5.1
- .
Area ΔABC=12absinC=12×12×9×sin25∘=22.82 u2
Note: No units were given so we include u2 for units squared. - .
Area ΔABC=12acsinB=12×10×8×sin35∘=22.943 cm2 - ΔPQR is an equilateral triangle. Therefore, all three sides have length 14 cm and all three angles are 60∘. We can choose any combination of sides and angle.
Area ΔABC=12pqsinR=12×14×14×sin60∘=84.87 cm2 - .
Area ΔACD=12acsinD=12×10×13×sin55∘=53.24 cm2
Area of parallelogram ABCD=2×53.24 cm2=106.49 cm2 - .
Area ΔABC=16.18 m2∴16.18=12×AC×8×sin54∘∴AC=4.045sin54∘∴AC=3.3 m
Unit 5: Assessment
- .
Area ΔKLM=12klsinM=12×12×4×sin103∘=23.38 u2 - .
Area ΔABC=12bcsinA=12×19×18×sin49∘=129.06 cm2 - PQ=RQ (given)
∴ˆR=ˆP=34∘ (angles opposite equal sides)
∴ˆQ=180∘−R−P=180∘−34∘−34∘=112∘ (angles in a triangle are supplementary)
Area ΔPQR=12prsinQ=12×9×9×sin112∘=37.55 m2 - PQRS is a parallelogram (given)
∴QR=PS=22 (opposite sides of parallelogram)
Area ΔQRS=12qssinR=12×15×22×sin135∘=116.67 u2
Area PQRS=2×116.67=233.35 u2(rounding off in the final step) - Area ΔXYZ=26.72 m2
∴26.72=12yzsinX=12×8×8×sinX∴sinX=0.835∴ˆX=56.6∘ or ˆX=180∘−56.62∘=123.4∘
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