We know how to find the area of right-angled triangles - we just multiply the two short sides together, and halve the result:
The area of this triangle is $\frac{1}{2}ab$12ab, half the area of the rectangle.
If the triangle is not right-angled we can still find the area, as long as we know two sides and the angle between them:
The area of this triangle is the base $a$a times the height, and then halved, just like for right-angled triangles. But what is the height? It isn't $b$b in this case, but we can use $b$b and the angle $C$C to find it.
Proof of the area sine rule
Here we have made a small right-angled triangle within our larger triangle, with hypotenuse $b$b and short side $h$h, the height of our large triangle. Using trigonometric ratios, the value of $\sin C$sinC is the opposite side, $h$h, divided by the hypotenuse, $b$b.
So:
| $\sin(c)$sin(c) | $=$= | $\frac{h}{b}$hb |
And hence:
| $h$h | $=$= | $b\sin(C)$bsin(C) |
Putting this all together with the area formula:
| $Area$Area | $=$= | $\frac{1}{2}bh$12bh |
We obtain the formula:
| $Area$Area | $=$= | $\frac{1}{2}ab\sin(C)$12absin(C) |
It really doesn't matter what you call the sides as long as you have two sides and the included angle. It's worth noting that we always label the sides with lower case letters, and the angles directly opposite the sides with a capital of the same letter.
The formula is most commonly written as follows:
$Area=\frac{1}{2}ab\sin C$Area=12absinC
Where $a$a and $b$b are the known side lengths, and $C$C is the given angle between them, as per the diagram above.
Note that the formula can also be written:
$Area=\frac{1}{2}bc\sin A$Area=12bcsinA
and,
$Area=\frac{1}{2}ac\sin B$Area=12acsinB
Practice questions
Question 1
Find the area of $\triangle ABC$△ABC correct to two decimal places.
Question 2
We wish to find the area of this triangle.
Find $\angle BAC$∠BAC, giving your answer correct to one decimal place.
Using the value of the angle found in the previous part, calculate the area of the triangle.
(Give your answer correct to 2 decimal places.)
Question 3
Use the diagram below to calculate the following:
The angle $x$x, rounded to the nearest minute.
The area of the triangle to one decimal place.