Triangle area formula for non-right triangles

IB Mathematics: Applications and Interpretation SL· 3.6 Geometry and trigonometry· 12 min read

1. The $ rac{1}{2}ab \sin C$ Area Formula★★☆☆☆⏱ 4 min

The standard area formula $r a c 12 \times base \times height$ requires a perpendicular height, which is often unknown for non-right triangles. If you know two sides and the included angle between them, you can use this derived formula instead.

📘 Definition

Area of a triangle (two sides + included angle)

For sides $a, b$ with included angle $C$

$Area = \frac{1}{2} ab sin C$

Example:

$a = 3, b = 4, C = 3 0^{\circ}$ , Area = $\frac{1}{2} (3) (4) sin 3 0^{\circ} = 3$

🔬 Derivation

Goal:

Derive $r a c 12 ab sin C$ from the base-height formula

Starting from:

Triangle $A B C$ , with side $a$ opposite $A$ , side $b$ opposite $B$ , angle $C$ between sides $a$ and $b$

1
Draw perpendicular height $h$ from $A$ to the line containing side $a$
2
From right triangle trigonometry, $h = b sin C$
3
Substitute into base-height formula, with base = $a$ :
4
$\text{Area} = \frac{1}{2} \times a \times h = \frac{1}{2}a(b \sin C) = \frac{1}{2}ab \sin C$

Result:

This works for all triangles, including obtuse triangles, since $ (180^\circ - \theta) = \sin \theta$.

📐 Worked Example

Find the area of a triangle with sides 5 cm and 7 cm, with an included angle of 55°. Give answer to 3 significant figures.

1
We have two sides and the included angle, so use $r a c 12 ab sin C$ directly:
2
$a = 5, \ b = 7, \ C = 55^\circ$
3
Substitute values into the formula:
4
$\text{Area} = \frac{1}{2}(5)(7)\sin(55^\circ) = 17.5 \times 0.81915...$
5
Calculate and round to 3 significant figures:
6
$\text{Area} \approx 14.3 \text{ cm}^2$

Exam tip:

This formula is not given in the IB AI SL formula booklet, so you must memorize it.

2. Heron's Formula (Area from Three Sides)★★★☆☆⏱ 5 min

When you know all three side lengths of a non-right triangle but no angles, Heron's formula lets you calculate the area directly, without needing to first find an angle with the cosine rule.

📘 Definition

Heron's Formula

Side lengths $a, b, c$ , semi-perimeter $s = \frac{a + b + c}{2}$

$Area = s (s - a) (s - b) (s - c)$

📐 Worked Example

Find the area of a triangle with side lengths 4 m, 6 m, and 8 m. Give answer to 2 decimal places.

1
First calculate the semi-perimeter $s$ :
2
$s = \frac{4 + 6 + 8}{2} = 9$
3
Substitute into Heron's formula:
4
$\text{Area} = \sqrt{9(9-4)(9-6)(9-8)} = \sqrt{9 \times 5 \times 3 \times 1} = \sqrt{135}$
5
Calculate and round:
6
$\text{Area} \approx 11.62 \text{ m}^2$

3. Solving Applied Area Problems★★★☆☆⏱ 3 min

Most IB exam questions on this topic are applied, often involving land surveying, navigation, or irregular shape area. You will usually need to find missing sides/angles first with the sine or cosine rule before calculating area.

📐 Worked Example

A triangular parking lot has sides of length 40 m, 50 m, and 65 m. Find the area of the lot to 3 significant figures.

1
We know all three sides, so Heron's formula is the most efficient approach.
2
Calculate semi-perimeter:
3
$s = \frac{40 + 50 + 65}{2} = 77.5$
4
Substitute into Heron's formula:
5
$\text{Area} = \sqrt{77.5(77.5-40)(77.5-50)(77.5-65)} = \sqrt{77.5 \times 37.5 \times 27.5 \times 12.5} = \sqrt{998574.21875}$
6
Round to 3 significant figures:
7
$\text{Area} \approx 1000 \text{ m}^2 = 1.00 \times 10^3 \text{ m}^2$

✓ Quick check

Check you can select the right formula:

You have all three sides of a triangle. Which formula do you use?
- $\frac{1}{2} ab sin C$
- Heron's formula
- Either works equally well
Reveal answer
1 —
Correct! Heron's formula lets you calculate area directly from three sides.
You have two sides and a non-included angle. What do you do first?
- Use $\frac{1}{2} ab sin C$ directly
- Find the included angle with sine rule first
- Use Heron's formula directly
Reveal answer
1 —
Correct! $\frac{1}{2} ab sin C$ requires the angle between the two sides, so find the included angle first.

4. Common Pitfalls

Wrong move:

Using $r a c 12 ab sin C$ with a non-included angle instead of the included angle

Why:

The formula is only derived for the angle between the two known sides, so a non-included angle will give the wrong result

Correct move:

If you only have two sides and a non-included angle, first find the included angle using the sine rule and angle sum before applying the area formula

Wrong move:

Using the full perimeter instead of the semi-perimeter in Heron's formula

Why:

Heron's formula is defined for semi-perimeter (half the total perimeter), so using full perimeter gives a result off by a large factor

Correct move:

Always calculate $s = \frac{a + b + c}{2}$ first, and check that $s$ is larger than any individual side

Wrong move:

Using calculator radian mode instead of degree mode for $ C$

Why:

IB AI SL exam questions almost always give angles in degrees, so radian mode gives the wrong sine value

Correct move:

Always confirm your calculator is set to degree mode before starting any trigonometry calculation

Wrong move:

Rounding intermediate values when calculating area

Why:

Early rounding introduces error that can make your final answer wrong to the required significant figures

Correct move:

Keep full unrounded values in your calculator during intermediate steps, only round the final answer

5. Quick Reference Cheatsheet

Scenario	Formula	Notes
Two sides + included angle	$\frac{1}{2} ab sin C$	Must memorize for IB AI SL
All three sides known	$s (s - a) (s - b) (s - c), s = \frac{a + b + c}{2}$	Given in formula booklet
Base + perpendicular height known	$\frac{1}{2} \times base \times height$	Always works for any triangle
Two angles + one side	Find missing side with sine rule first, then use either formula	Always confirm you have the required inputs for your chosen formula

When this came up on past exams

AI-estimated based on syllabus patterns — cross-check with official past papers for accuracy. Use only as revision-focus signals.

2021 · 1
Calculate area of scalene triangle
2023 · 2
Land survey area problem

What's Next

Calculating the area of non-right triangles is a foundational skill for many higher-level topics in IB AI SL. You will use this skill to find the area of irregular polygons by splitting them into triangles, solve applied problems in surveying and navigation, and work through 3D trigonometry problems that require finding the area of triangular cross-sections or faces. Mastery of this topic also helps you recognize which formula to use given different sets of information, a key skill for Paper 2 exam questions.