GCSE · Maths · Edexcel · Spec 1MA1 · Higher

Area of any triangle (½ab sin C)

You know area = ½ × base × height. But what if nobody gives you the height, only two sides and an angle? One sine step builds it for you.

Higher

Maths · Trigonometry

Where does the missing height come from?

Read the triangle like this: the green hypotenuse is side a (fixed at 10 units), the angle θ is our angle C, and the red opposite side is the perpendicular height P. Drag θ and watch P change.

Angle θ30°

10°80°

SIN θ

sin θ =oppositehypotenuse

= 5.0 / 10.0 = 0.50

For sin, the renderer highlights the side opposite θ and the hypotenuse. Adjacent fades.

Area of a triangle · the height you already trust

Height: what do you think?

The sine step is about to build a height for you, so check the height you already know is solid first.

Which of these is closest to what you think right now?

The base is the bottom edge, so the height has to go straight up from it.Any edge can be the base, as long as the height is perpendicular to it.The height always sits inside the triangle.A perpendicular dropped inside any triangle cuts the side it meets exactly in half.

How sure are you?

I'm sureI think soI'm guessing

The base is the bottom edge, so the height has to go straight up from it.

This is a common misconception

Easy to think, because most triangles in books sit flat on the page. But any edge can be the base. Pick one, then the height is the perpendicular to it. Turn the page round and nothing about the area changes.

Any edge can be the base, as long as the height is perpendicular to it.

This matches the evidence

Yes, exactly. The base and the height just have to be perpendicular to each other, and area = ½ × base × perpendicular height still works.

The height always sits inside the triangle.

This is a common misconception

Not when there is an obtuse angle. There, the perpendicular height is often drawn outside the triangle. It still measures the height, so ½ × base × height still works. Don't let a line hanging off the triangle put you off.

A perpendicular dropped inside any triangle cuts the side it meets exactly in half.

This is a common misconception

Only in an isosceles or equilateral triangle, and for an isosceles one only for the altitude on the line of symmetry (from the vertex between the two equal sides). That altitude splits the triangle into two congruent right-angled triangles, so Pythagoras gives the height. Try sides of 5 cm, 5 cm and a 6 cm base: half the base is 3 cm, so height² = 5² − 3² = 16 and the height is 4 cm. In other triangles, a perpendicular does not bisect the side it meets.

Choose any edge as the base, drop the perpendicular onto it, and use ½ × base × height. That perpendicular height is exactly what the sine step is about to build for you when nobody hands it over.

Higher

Maths · Algebra

Build the formula in five lines

Click through. Each line says what was done to get there.

GoalShow that the area of any triangle is ½ab sin C, where a and b are two sides and C is the angle between them.

Area = ½ × b × P

Start with the formula you trust: take side b as the base and P as the perpendicular height.

The snag: nobody has told us P.

Step 1 of 5

The snag: nobody has told us P.

Higher

Area of a triangle · what have you been given?

Can you use it?

An area question gives you this information. What's your move?

Still to sort

Use ½ab sin C now (0)

Two sides and the angle between them

Where the line is: The angle has to be between the two sides you use, not just somewhere in the triangle.

Find the angle between the sides first (0)

180° minus the other two angles

Use ½ × base × height (0)

A base and its perpendicular height

Can't use ½ab sin C (0)

The wrong angle, or not enough information

6 of 6 still to sort.

Pick a card, then pick where it goes. Commit first, and then read why.

Higher

Which sides? Which angle?

Problem

In triangle PQR, PQ = 8 cm, QR = 12 cm and angle PQR = 35°. Work out the area of the triangle, correct to 1 decimal place.

Higher

Area of a triangle · running it backwards

You know the area. Find the side.

A triangle has area 24 cm². One side is b = 8 cm, and the angle between b and the unknown side a is C = 30°. Find a. (sin 30° = 0.5)

Write ½ab sin C = area with the numbers you know: ½ × a × 8 × sin 30° = 24. Only a is unknown.
missing step

Which line is step 2?

Work out the numbers multiplying a: ½ × 8 × sin 30° = 4 × 0.5 = 2, so 2a = 24.Work out the numbers multiplying a: ½ × 8 × sin 30° = 4 × 30 = 120, so 120a = 24.Work out the numbers multiplying a: ½ × 8 × sin 30° = 4 + 0.5 = 4.5, so 4.5a = 24.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Area = ½ab sin C

½ × base × height in disguise, and it works for any triangle

What you need to know

Area of a triangle = ½ × base × perpendicular height. The base and height just need to be perpendicular to each other.
The base needn't be the bottom edge, and with an obtuse angle the height is often drawn outside the triangle.
Have a goSam says a triangle balanced on its tip has no 'bottom edge', so ½ × base × height is off the table. Is Sam right?
No. Any edge can be the base.
Base and height only have to be perpendicular to each other, so you choose an edge and drop the perpendicular onto it.
In an isosceles or equilateral triangle, the altitude on the line of symmetry makes two congruent right-angled triangles, so Pythagoras gives the height.
With sides a and b around angle C, sin C = P ÷ a. So the perpendicular height is P = a sin C.
Have a goSide a is 8 cm and angle C is 30° (sin 30° = 0.5). What is the perpendicular height P?
P = 8 × 0.5 = 4 cm
P = a sin C, so you multiply a by sin C. Dividing by it would give 16 cm, taller than side a itself.
Swap P = a sin C into ½ × base × height, with b as the base, and out pops area = ½ab sin C.
C must be the angle between the two sides you use. Any two sides will do, whatever they're called.
An angle that isn't between your two sides won't work. Two sides alone, or one side and one angle, fit many triangles.
If you know the other two angles, the angle between two sides is 180° minus their sum.
Have a goTwo corners of a triangle measure 38° and 82°. The two sides meeting at the third corner are your a and b. What is C?
C = 60°
C sits between the two sides, and it is what's left of 180° after the other two angles are taken away: 180° − 38° − 82°.
At C = 90°, sin 90° = 1, so ½ab sin C becomes ½ab. The formula works for right-angled triangles too.
Given the area, put what you know into ½ab sin C = area and rearrange for the missing side or angle.

The big picture

The area of any triangle is ½ab sin C, where a and b are two sides and C is the angle between them. It's ½ × base × height in disguise, because the height is a sin C. Use the angle between your two sides, and run the formula backwards when you're given the area.

Key points

1Area = ½ × base × perpendicular height, with the base and height at right angles to each other.

2The height from a sine: in a triangle with sides a and b and angle C between them, P = a sin C.

3Area of any triangle = ½ab sin C, where a and b are two sides and C is the angle between them.

4Check that C sits between your two sides before you substitute. Any two sides will do, whatever they're labelled.

5Two sides alone, or one side and one angle, is not enough information, because many different triangles fit.

6Given the area, substitute into ½ab sin C = area and rearrange to find a missing side or angle.

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Worked example

Problem

Triangle ABC is isosceles, with AB = AC = 9 cm and angle ABC = angle ACB = 50°. Work out its area, correct to 1 decimal place.

⚠ Watch out

Grabbing the nearest angle. ½ab sin C only works with the angle between the two sides you use, so check that C sits in the corner where your two sides meet. And don't assume two sides alone are enough, because lots of different triangles fit them.

🧠

Memory hook

C is the corner where sides a and b meet: the filling in the sandwich. If the angle isn't between the two sides, it isn't in the sandwich, so it can't be C.

✓

Check yourself

Sides 6 cm and 7 cm, with a 30° angle between them (sin 30° = 0.5). Find the area. Answer: ½ × 6 × 7 × sin 30° = 10.5 cm².

Flashcards

(13)

What is the area of a triangle in terms of a base and its height?

½ × base × perpendicular height. The base and the height just have to be perpendicular to each other.

Where is the perpendicular height drawn when the triangle has an obtuse angle?

Often outside the triangle. It is still the perpendicular height, so ½ × base × height still works.

When does a perpendicular dropped in a triangle cut the side it meets exactly in half?

Only in an isosceles or equilateral triangle (for an isosceles one, the altitude on the line of symmetry).

How do you find the perpendicular height of an isosceles triangle?

The altitude on the line of symmetry makes two congruent right-angled triangles, so use Pythagoras' theorem in one of them.

Sides a and b with angle C between them: what is the perpendicular height P?

P = a sin C, because sin C = P ÷ a in the right-angled triangle the height makes.

What is the formula for the area of any triangle using sine?

Area = ½ab sin C, where a and b are two sides and C is the angle between them.

You've picked your two sides. Which angle do you pair them with?

The one at the corner where they meet, the angle between them. The sides needn't be labelled a, b and C.

Can you use ½ab sin C with two sides and an angle that is not between them?

No. The formula needs the angle between the two sides.

Why isn't two sides alone, or one side and one angle, enough to find the area?

Many different triangles fit that information.

You know the other two angles of a triangle. How do you get the angle between two sides?

Take their sum away from 180°.

Why does ½ab sin C still work for a right-angled triangle?

sin 90° = 1, so the formula becomes ½ab, with a and b perpendicular.

You know the area of a triangle. How do you find a missing side or angle?

Substitute the known values into ½ab sin C = area, then rearrange to solve.

You know the area and the perpendicular height h. How do you find the base?

Use ½ × base × h = area, which gives the unknown directly.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Where does the missing height come from?

Height: what do you think?

Build the formula in five lines

Can you use it?

You know the area. Find the side.

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