Step by step
The question
Question: find the area of the triangle with sides 14 and 9, and the angle 25° 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 9 as its longest side, and h across from the 25° angle.
SOH: sin is the opposite over the hypotenuse.
sin 25° = h/9.
To remove the 9 at the bottom, we multiply by 9.
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.
9/9 is 1.
So just h is left.
So h = 9 × sin 25°.
2 Write the rule
The area of a triangle is a half times the base times the height.
The base is 14, the height is 9 × sin 25°.
So the rule is ½ × b × c × sin A: 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 14 and 9, and sin 25°.
4 Work it out
14 times 9 is 126.
Half of 126 is 63.
The calculator gives sin 25°: about 0.4226.
Put the number in.
That is about 26.625.
To the nearest tenth, the area is about 26.6 square units.
5 Check
Let's check another way: use the side 9 as the base.
Its height is 14 times sin 25°: about 5.92.
A half times 9 times that is about 26.6.
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.