Step by step
The question
Question: find the area of the triangle with sides 13 and 5, and the angle 135° between them.
1 Find the height
The area needs a height.
Draw it from the top corner straight down to the bottom line.
The angle is more than 90°, so it lands outside the triangle.
The small right triangle outside has the side 5 as its longest side.
Its angle at the bottom is 180° − 135° = 45°.
SOH: sin is the opposite over the hypotenuse.
sin 45° = h/5.
To remove the 5 at the bottom, we multiply by 5.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
5/5 is 1.
So just h is left.
So h = 5 × sin 45°.
From the unit circle: an angle and 180° minus it have the same sine.
So sin 45° = sin 135°.
2 Write the rule
The area of a triangle is a half times the base times the height.
The base is 13, the height is 5 × sin 135°.
So the rule is ½ × a × c × sin B: two sides and the sine of the angle between them.
Careful!
The angle must be the one between the two sides we use.
Another angle gives a wrong height.
3 Put the numbers in
Now put the numbers in: the two sides 13 and 5, and sin 135°.
4 Work it out
13 times 5 is 65.
Half of 65 is 32.5.
The calculator gives sin 135°: about 0.7071.
Put the number in.
That is about 22.981.
To the nearest tenth, the area is about 23.0 square units.
5 Check
Let's check another way: use the side 5 as the base.
Its height is 13 times sin 135°: about 9.19.
A half times 5 times that is about 23.0.
The same area!
So it is right.
Let's look back.
First: find the height.
Then: write the rule.
Then: put the numbers in.
Then: work it out.
Last: check.