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2D Trigonometry

2D Trigonometry

Area of a triangle and multi-step problems

Area Of Triangle Extended 6 min read

On this page
  1. In this note
  2. Formula bank
  3. Area = ½ab sin C
  4. Multi-step problems
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 6.3: area of a triangle and multi-step problems
  • 6.3Use area = ½ab sin C for any triangle.
  • 6.3Work backwards from an area to a side or an angle.
  • 6.3Solve multi-step problems: split shapes into triangles and chain the rules.

Before you start: The sine and cosine rules.

Formula bank

Area of a triangleOn the formula sheet\(\text{Area}=\tfrac12ab\sin C\)C is the angle BETWEEN sides a and b.Use it when two sides and the included angle.
Angle from an areaLearn it\(\sin C=\dfrac{2\times\text{Area}}{ab}\)can give two angles: C and 180° − C.Use it when area given, angle wanted.
Side from an areaLearn it\(a=\dfrac{2\times\text{Area}}{b\sin C}\)Use it when area given, side wanted.

Area = ½ab sin C

Area = ½ab sin C: two sides and the angle between themCab
The angle must be between the two sides.

'Half base times height' needs the height. The trig formula finds it for you: with sides a and b and the angle C between them, the height is \(b\sin C\), so the area is \(\tfrac12ab\sin C\).

General methodHow to find the area of any triangle

  1. Find two sides and the angle BETWEEN them.
  2. Not between? Use the sine or cosine rule first to get the right angle or side.
  3. Area \(=\tfrac12ab\sin C\).

Example Two sides are 8 cm and 11 cm with 40° between them. Find the area.

Two sides and the included angle.

  1. \(\tfrac12(8)(11)\sin40^\circ=28.28\)

Answer 28.3 cm²

Example A triangle has sides 8 cm and 10 cm and area 30 cm². Find the angle between those sides.

Work backwards.

  1. \(\tfrac12(8)(10)\sin C=30\)
  2. \(\sin C=0.75\)
  3. \(C=48.6^\circ\) or \(180^\circ-48.6^\circ=131.4^\circ\)

Answer 48.6° (or 131.4° if the angle is obtuse)

Try it yourself

1
Sides 5 cm and 6 cm with 130° between them. Area (2 d.p.)?
Show answer

Answer 11.49

2
Area 50 cm², \(b=12\) cm, \(C=56^\circ\). Find a (2 d.p.).
Hint
\(a=100\div(12\sin56^\circ)\).
Show answer

Answer 10.05

Multi-step problems

Quadrilateral ABCD split by the diagonal BDABCD70°6857
Split along a diagonal into two triangles.

Exam questions chain the rules. Split the shape into triangles, find the shared side (often a diagonal) in the first triangle, then use it in the second.

General methodHow to answer a multi-step triangle question

  1. Split the shape into triangles; mark everything you know.
  2. Find a triangle where you can already use a rule. Work out the missing piece (often the shared side).
  3. Carry it, unrounded, into the next triangle.
  4. Areas of a quadrilateral: add the two triangles.

Example ABCD: AB = 6, AD = 8, angle BAD = 70°, BC = 5, CD = 7 (cm). Find the area of ABCD.

Triangle ABD first (SAS), then BCD needs its angle at C.

  1. area ABD \(=\tfrac12(6)(8)\sin70^\circ=22.55\)
  2. \(BD^{2}=36+64-96\cos70^\circ=67.17\)
  3. \(\cos C=\dfrac{25+49-67.17}{70}\Rightarrow C=84.4^\circ\)
  4. area BCD \(=\tfrac12(5)(7)\sin84.4^\circ=17.42\)
  5. total \(=39.97\)

Answer 40.0 cm²

Ways they test it

Type 1 Area directly

"Calculate the area of triangle ABC." (2 marks)

Spot it two sides and the angle between.

  1. \(\tfrac12ab\sin C\).

Type 2 Area needs a first step

"Calculate the area." (5 marks)

Spot it the known angle is NOT between the known sides.

  1. Sine/cosine rule for the missing piece.
  2. Then the area.

Type 3 Area given

"The area is 30 cm². Find angle C." (3 marks)

Spot it an area in the question.

  1. Area formula = value.
  2. \(\sin^{-1}\); obtuse? 180 − it.

For example Sides 8, 10, area 30

  1. \(\sin C=0.75\)

Answer 48.6°

Type 4 Field / quadrilateral

"ABCD is a field... Calculate its area." (6-10 marks)

Spot it a 4-sided shape with a diagonal.

  1. Two triangles; shared diagonal.
  2. Add the areas.

Mistakes to avoid

  • ✗ Using an angle that is not between the two sides✓ find the included angle first
  • ✗ Rounding the diagonal before the second triangle✓ keep it on the calculator (ANS)Early rounding loses the accuracy mark.
  • ✗ Forgetting the ½✓ \(\tfrac12ab\sin C\)

Summary

  • Area \(=\tfrac12ab\sin C\), C between a and b.
  • Area given? Rearrange; an angle could be acute or obtuse.
  • Big shapes: split into triangles, chain the rules, keep full accuracy.

If you see …

  • 'area of triangle' with no height\(\tfrac12ab\sin C\).
  • a quadrilateralsplit along a diagonal.

Practice

Practice: easy, medium, hard

Easy

1
Sides 8 cm and 11 cm with 40° between. Area (2 d.p.)?
Show answer

Answer 28.28

2
Sides 7 cm and 9 cm with 48° between. Area (2 d.p.)?
Show answer

Answer 23.41

3
Area of an equilateral triangle of side 12 cm (2 d.p.)?
Hint
Every angle is 60°.
Show answer

Answer 62.35

Medium

1
Sides 8 and 10, area 30. The acute angle between them (1 d.p.)?
Guide me

What is it asking? an angle

What do we know? \(40\sin C=30\)

Plan \(\sin^{-1}0.75\)

Show working
  1. 48.6

Answer 48.6

2
Same triangle, but the angle is obtuse (1 d.p.)?
Guide me

What is it asking? an angle

What do we know? acute 48.6°

Plan 180 − 48.6

Show working
  1. 131.4

Answer 131.4

3
Area 50, \(b=12\), \(C=56^\circ\). Find a (2 d.p.).
Guide me

What is it asking? a side

What do we know? \(6a\sin56^\circ=50\)

Plan \(a=50\div(6\sin56^\circ)\)

Show working
  1. 10.05

Answer 10.05

Hard

1
ABCD: AB = 6, AD = 8, angle BAD = 70°. Find BD (2 d.p.).
Guide me

What is it asking? a side

What do we know? SAS: cosine rule

Plan \(36+64-96\cos70^\circ\)

Show working
  1. 67.17
  2. 8.20

Answer 8.20

PartAnswer / workingMarksGuidance
\(36+64-96\cos70\)M1
8.20A1
Total: 2 marks
2
Then BC = 5, CD = 7. Find angle BCD (1 d.p.).
Guide me

What is it asking? an angle

What do we know? SSS with BD² = 67.17

Plan \(\cos C=(25+49-67.17)/70\)

Show working
  1. 84.4

Answer 84.4

PartAnswer / workingMarksGuidance
\(\frac{25+49-67.17}{70}\)M1
84.4A1
Total: 2 marks
3
So the area of ABCD (1 d.p.)?
Guide me

What is it asking? an area

What do we know? 22.55 + 17.42

Plan add

Show working
  1. 39.97

Answer 40.0

PartAnswer / workingMarksGuidance
\(\tfrac12(6)(8)\sin70+\tfrac12(5)(7)\sin84.4\)M2
40.0A1
Total: 3 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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