Step by step
The question
Question: find the area of the triangle with sides 8 and 5, and the angle 30° between them.
1 Find the height
The area needs a height.
Draw it from the top corner straight down to the base, at a square corner: h.
The small right triangle on the left has the side 5 as its longest side, and h across from the 30° angle.
SOH: sin is the opposite over the hypotenuse.
sin 30° = 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 30°.
2 Write the rule
The area of a triangle is a half times the base times the height.
The base is 8, the height is 5 × sin 30°.
So the rule is ½ × a × b × sin C: 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 8 and 5, and sin 30°.
4 Work it out
8 times 5 is 40.
Half of 40 is 20.
The calculator gives sin 30°: exactly 0.5.
Put the number in.
That is 10.
So the area is 10 square units.
5 Check
Let's check another way: use the side 5 as the base.
Its height is 8 times sin 30°: 4.
A half times 5 times that is 10.
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.