Triangle area tutorial
There are two main ways to find the area of a triangle. If you know a base and its corresponding height, use the classic $A = \tfrac{1}{2}bh$ formula. If you know all three side lengths but not the height, use Heron's formula instead. This tutorial covers both methods with worked examples so you can handle any triangle area problem.
Core formulas
Base × Height: $A = \dfrac{1}{2}\,b\,h$
Heron's formula: $A = \sqrt{s(s-a)(s-b)(s-c)}$, where $s = \dfrac{a+b+c}{2}$
Method 1 — Base × Height
Pick any side of the triangle as the base ($b$). The height ($h$) is the perpendicular distance from the base to the opposite vertex. Because the height forms a right angle with the base, you can think of the triangle as exactly half of a rectangle with dimensions $b \times h$.
$$A = \frac{1}{2}\,b\,h$$
This formula works for every triangle — acute, right, or obtuse — as long as you measure $h$ perpendicular to $b$.
When to use base × height
- The problem directly states the base and height.
- The triangle is a right triangle — one leg is the base, the other is the height.
- You can calculate the height from other given information (e.g., coordinates on a grid).
Isolate b (know A and h)
$b = \dfrac{2A}{h}$
Isolate h (know A and b)
$h = \dfrac{2A}{b}$
Method 2 — Heron's formula
Heron's formula finds the area using only the three side lengths $a$, $b$, and $c$. First compute the semi-perimeter:
$$s = \frac{a + b + c}{2}$$
Then plug $s$ into:
$$A = \sqrt{\,s\,(s - a)\,(s - b)\,(s - c)\,}$$
This is especially useful when you know all three sides but the height is not given or is hard to compute.
Worked examples
Example 1: base × height with a right triangle
A right triangle has legs of length 6 and 8. Find its area.
One leg is the base and the other is the height because they meet at a right angle.
$A = \tfrac{1}{2} \times 6 \times 8 = \tfrac{48}{2} = \mathbf{24}$
Example 2: base × height with a non-right triangle
A triangle has a base of 14 and a height drawn to that base of 9. Find the area.
$A = \tfrac{1}{2} \times 14 \times 9 = \tfrac{126}{2} = \mathbf{63}$
It doesn't matter what the other two sides are — as long as you know the perpendicular height to the chosen base, this formula works.
Example 3: Heron's formula
A triangle has sides $a = 7$, $b = 8$, and $c = 9$. Find the area.
$s = \dfrac{7 + 8 + 9}{2} = \dfrac{24}{2} = 12$
$A = \sqrt{12 \times (12-7) \times (12-8) \times (12-9)}$
$\phantom{A} = \sqrt{12 \times 5 \times 4 \times 3} = \sqrt{720} \approx \mathbf{26.83}$
Example 4: Heron's formula with an equilateral triangle
All three sides equal 10. Find the area.
$s = \dfrac{10+10+10}{2} = 15$
$A = \sqrt{15 \times 5 \times 5 \times 5} = \sqrt{1875} \approx \mathbf{43.30}$
For equilateral triangles there is a shortcut — $A = \frac{\sqrt{3}}{4}\,s^2$ — but Heron's formula still works perfectly.
Choosing the right method
| Scenario | Best method | Why |
|---|---|---|
| Base and height are given | $\tfrac{1}{2}bh$ | Fastest — just one multiplication and a division |
| Right triangle with two legs | $\tfrac{1}{2}bh$ | The legs are the base and height |
| Three sides known, no height | Heron's | No need to compute the height separately |
| Coordinates of vertices | Either | Compute side lengths for Heron's, or find height via distance formula |
Common mistakes to avoid
- Forgetting to halve. A triangle is half a rectangle, not the full rectangle. Always divide by 2.
- Using a slant side as the height. The height must be perpendicular to the base. In an obtuse triangle the height can fall outside the triangle — it's still perpendicular to the base line extended.
- Miscomputing the semi-perimeter. Add all three sides then divide by 2. A common slip is dividing only one side.
- Negative under the radical. If $s(s-a)(s-b)(s-c)$ is negative, the three lengths can't form a triangle. Double-check the values.
- Rounding too early. Carry extra decimal places through intermediate steps and round only the final answer.
Refreshable sample
Try one triangle area question
Work through the problem before checking. The practice page generates different values each time, so you can keep drilling until the method feels automatic.
Ready to practise?
The practice page lets you choose base×height questions, Heron's formula questions, or a mix of both. Generate as many as you want, check all at once, and read the worked solution for every problem.
Open triangle area practice