Worked examples
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.
sin 30° = h5
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 × sin 30° = 5 × h5
5/5 is 1.
So just h is left.
So h = 5 × sin 30°.
h = 5 × sin 30°
2 Write the rule
The area of a triangle is a half times the base times the height.
A = 12 × base × 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.
A = 12 × a × b sin C
Careful!
The angle must be the one between the two sides we use.
Another angle gives a wrong height.
Careful: the angle BETWEEN the two sides
3 Put the numbers in
Now put the numbers in: the two sides 8 and 5, and sin 30°.
A = 12 × 8 × 5 × sin 30°
4 Work it out
8 times 5 is 40.
Half of 40 is 20.
A = 20 × sin 30°
The calculator gives sin 30°: exactly 0.5.
sin 30° = 0.5
Put the number in.
A = 20 × 0.5
That is 10.
So the area is 10 square units.
A = 10
5 Check
Let's check another way: use the side 5 as the base.
Its height is 8 times sin 30°: 4.
other base 5: height = 8 × sin 30° = 4
A half times 5 times that is 10.
The same area!
So it is right.
A = 12 × 5 × 4 = 10
Let's look back.
First: find the height.
Then: write the rule.
Then: put the numbers in.
Then: work it out.
Last: check.
▶ Watch it solved
The question
Question: find the area of the triangle with sides 9 and 7, and the angle 50° 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 7 as its longest side, and h across from the 50° angle.
SOH: sin is the opposite over the hypotenuse.
sin 50° = h/7.
sin 50° = h7
To remove the 7 at the bottom, we multiply by 7.
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.
7 × sin 50° = 7 × h7
7/7 is 1.
So just h is left.
So h = 7 × sin 50°.
h = 7 × sin 50°
2 Write the rule
The area of a triangle is a half times the base times the height.
A = 12 × base × height
The base is 9, the height is 7 × sin 50°.
So the rule is ½ × a × c × sin B: two sides and the sine of the angle between them.
A = 12 × a × c sin B
Careful!
The angle must be the one between the two sides we use.
Another angle gives a wrong height.
Careful: the angle BETWEEN the two sides
3 Put the numbers in
Now put the numbers in: the two sides 9 and 7, and sin 50°.
A = 12 × 9 × 7 × sin 50°
4 Work it out
9 times 7 is 63.
Half of 63 is 31.5.
A = 31.5 × sin 50°
The calculator gives sin 50°: about 0.7660.
sin 50° ≈ 0.7660
Put the number in.
A ≈ 31.5 × 0.7660
That is about 24.13.
To the nearest tenth, the area is about 24.1 square units.
A ≈ 24.1
5 Check
Let's check another way: use the side 7 as the base.
Its height is 9 times sin 50°: about 6.89.
other base 7: height = 9 × sin 50° ≈ 6.89
A half times 7 times that is about 24.1.
The same area!
So it is right.
A = 12 × 7 × 6.89 ≈ 24.1
Let's look back.
First: find the height.
Then: write the rule.
Then: put the numbers in.
Then: work it out.
Last: check.
▶ Watch it solved
The question
Question: find the area of the triangle with sides 6 and 4, and the angle 120° 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 4 as its longest side.
Its angle at the bottom is 180° − 120° = 60°.
SOH: sin is the opposite over the hypotenuse.
sin 60° = h/4.
sin 60° = h4
To remove the 4 at the bottom, we multiply by 4.
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.
4 × sin 60° = 4 × h4
4/4 is 1.
So just h is left.
So h = 4 × sin 60°.
h = 4 × sin 60°
From the unit circle: an angle and 180° minus it have the same sine.
So sin 60° = sin 120°.
sin 60° = sin 120°
2 Write the rule
The area of a triangle is a half times the base times the height.
A = 12 × base × height
The base is 6, the height is 4 × sin 120°.
So the rule is ½ × b × c × sin A: two sides and the sine of the angle between them.
A = 12 × b × c sin A
Careful!
The angle must be the one between the two sides we use.
Another angle gives a wrong height.
Careful: the angle BETWEEN the two sides
3 Put the numbers in
Now put the numbers in: the two sides 6 and 4, and sin 120°.
A = 12 × 6 × 4 × sin 120°
4 Work it out
6 times 4 is 24.
Half of 24 is 12.
A = 12 × sin 120°
The calculator gives sin 120°: about 0.8660.
sin 120° ≈ 0.8660
Put the number in.
A ≈ 12 × 0.8660
That is about 10.392.
To the nearest tenth, the area is about 10.4 square units.
A ≈ 10.4
5 Check
Let's check another way: use the side 4 as the base.
Its height is 6 times sin 120°: about 5.2.
other base 4: height = 6 × sin 120° ≈ 5.2
A half times 4 times that is about 10.4.
The same area!
So it is right.
A = 12 × 4 × 5.2 ≈ 10.4
Let's look back.
First: find the height.
Then: write the rule.
Then: put the numbers in.
Then: work it out.
Last: check.
▶ Watch it solved
