# Triangle area formula for non-right triangles

> IB Mathematics: Applications and Interpretation SL · Unit 3: Geometry and Trigonometry
> Source: https://www.owlsprep.com/study/ib-math-ai-sl-u3-triangle-area-formula-for-non/

This module covers two formulas to find the area of non-right (oblique) triangles, when you do not know the base and perpendicular height. You will learn when to use each formula, avoid common exam traps, and solve applied problems.

**Prerequisites:** [Right triangle trigonometry (SOH-CAH-TOA)](https://www.owlsprep.com/study/ib-math-ai-sl-u3-right-triangle-trigonometry/); [Sine and cosine rules for non-right triangles](https://www.owlsprep.com/study/ib-math-ai-sl-u3-sine-cosine-rules/)

## Learning objectives

- Derive and recall the $\frac{1}{2}ab \sin C$ area formula for non-right triangles
- Apply Heron's formula to find area from three known sides
- Solve applied problems involving area of oblique triangles
- Select the correct formula based on given information

## The $\frac{1}{2}ab \sin C$ Area Formula

The standard area formula $\frac{1}{2} \times \text{base} \times \text{height}$ 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.

**Area of a triangle (two sides + included angle)** — $\text{Area} = \frac{1}{2}ab \sin C$

*Notation:* For sides $a, b$ with included angle $C$

*Example:* $a=3, b=4, C=30^\circ$, Area = $\frac{1}{2}(3)(4)\sin 30^\circ = 3$

**Derivation:** Derive $\frac{1}{2}ab \sin C$ from the base-height formula

*Starting from:* Triangle $ABC$, 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. $$\text{Area} = \frac{1}{2} \times a \times h = \frac{1}{2}a(b \sin C) = \frac{1}{2}ab \sin C$$

*Conclusion:* 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 $\frac{1}{2}ab \sin C$ 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.

## Heron's Formula (Area from Three Sides)

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.

**Heron's Formula** — $\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}$

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

> **info**
>
> Heron's formula is given in the IB AI SL formula booklet, so you do not need to memorize it, just know how to apply it.

**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. $$\text{Area} \approx 11.62 \text{ m}^2$$

## Solving Applied Area Problems

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. $$\text{Area} \approx 1000 \text{ m}^2 = 1.00 \times 10^3 \text{ m}^2$$

**Check your understanding**

Check you can select the right formula:

1. 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

   *Answer:* Heron's formula

   *Why:* Correct! Heron's formula lets you calculate area directly from three sides.

2. 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

   *Answer:* Find the included angle with sine rule first

   *Why:* Correct! $\frac{1}{2}ab \sin C$ requires the angle between the two sides, so find the included angle first.

## Common pitfalls

- **Wrong:** Using $\frac{1}{2}ab \sin C$ with a non-included angle instead of the included angle
  - Why it fails: The formula is only derived for the angle between the two known sides, so a non-included angle will give the wrong result
  - Correct: 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:** Using the full perimeter instead of the semi-perimeter in Heron's formula
  - Why it fails: 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: Always calculate $s = \frac{a+b+c}{2}$ first, and check that $s$ is larger than any individual side
- **Wrong:** Using calculator radian mode instead of degree mode for $
 C$
  - Why it fails: IB AI SL exam questions almost always give angles in degrees, so radian mode gives the wrong sine value
  - Correct: Always confirm your calculator is set to degree mode before starting any trigonometry calculation
- **Wrong:** Rounding intermediate values when calculating area
  - Why it fails: Early rounding introduces error that can make your final answer wrong to the required significant figures
  - Correct: Keep full unrounded values in your calculator during intermediate steps, only round the final answer

## Cheatsheet

| Scenario | Formula | Notes |
| --- | --- | --- |
| Two sides + included angle | $\frac{1}{2}ab \sin C$ | Must memorize for IB AI SL |
| All three sides known | $\sqrt{s(s-a)(s-b)(s-c)}, \ s = \frac{a+b+c}{2}$ | Given in formula booklet |
| Base + perpendicular height known | $\frac{1}{2} \times \text{base} \times \text{height}$ | 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 |

## 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.

- [Statistics and Probability](https://www.owlsprep.com/study/ib-math-ai-sl-u4-overview/)
- [Discrete and continuous data types](https://www.owlsprep.com/study/ib-math-ai-sl-u4-discrete-and-continuous-data-types/)
- [Data representation: histograms, box plots, cumulative frequency](https://www.owlsprep.com/study/ib-math-ai-sl-u4-data-representation-histograms-box-plots/)

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