How to Find the Area of a Triangle (Step by Step)
The area of a triangle is half the base times the height - but what if you don't know the height? Here are three reliable methods, each with a worked example.
The area of a triangle is ½ × base × height: pick a side as the base, measure the perpendicular height straight up from it to the opposite corner, multiply the two, and halve the result. That single formula solves most triangle problems - and when you're not given the height, two other methods fill the gap.
This guide covers all three: the basic base-and-height formula, Heron's formula (for when you know all three sides), and the sine formula (for two sides and the angle between them). Each comes with a worked example, and a quick table at the end shows which to reach for.
Method 1: ½ × base × height
The workhorse formula. Any side can be the base; the height is the perpendicular distance from that base to the opposite vertex. Multiply them and take half:
For example, a triangle with a base of 8 cm and a perpendicular height of 5 cm:
The part students get wrong: what counts as the height
The height is the perpendicular distance from the base to the opposite corner - not the length of a slanted side. If you multiply the base by a sloping edge instead of the true vertical height, your answer will be too big. In a right triangle the height is easy (one leg is already perpendicular to the other), but in an obtuse triangle the height can fall outside the triangle entirely. Always look for the right-angle mark, or drop your own perpendicular line, before you multiply.
Method 2: Heron's formula (when you know all three sides)
If you know the three side lengths but no height, Heron's formula gives the area directly. First find the semi-perimeter s (half the perimeter), then plug it in:
Take a triangle with sides 5, 6 and 7. The semi-perimeter is (5 + 6 + 7) ÷ 2 = 9, and the rest follows:
Method 3: two sides and the included angle (½ab·sin C)
When you know two sides and the angle between them (the 'included' angle), use the sine formula. It's especially common in trigonometry problems:
For sides of 6 and 8 with a 30° angle between them (and sin 30° = 0.5):
Which formula should you use?
Match the method to what the problem gives you:
| When you know… | Use | Formula |
|---|---|---|
| Base and perpendicular height | Basic formula | A = 12bh |
| All three side lengths | Heron's formula | A = √(s(s−a)(s−b)(s−c)) |
| Two sides and the angle between them | Sine formula | A = 12ab·sin C |
| A right triangle (two legs) | Basic formula | A = 12(leg1)(leg2) |
Common mistakes (and how to avoid them)
- Using a slanted side instead of the perpendicular height - always measure the height at a right angle to the base.
- Forgetting to halve - the base × height product is a rectangle; a triangle is half of it.
- Mixing up units - if the base and height are in centimetres, the area is in square centimetres (cm²).
- Using ½ab·sin C with the wrong angle - it must be the angle between the two sides a and b, not another angle.
- Rounding too early in Heron's formula - keep full precision until the final square root.
How a tutor makes geometry click
Geometry rewards seeing the figure clearly, and that's exactly what a tutor helps with. Working live on a shared whiteboard, a tutor can draw each triangle, mark the true perpendicular height, and show which area method a problem is really asking for - then drill it until choosing the right approach is automatic. Our online geometry tutoring pairs you with a specialist who makes the formulas make sense, and connects them to the rest of the course.
Because area is a building block for so much of geometry - from coordinate geometry to surface area and beyond - getting it solid now pays off all year. Your first trial lesson is free.
The bottom line
Every triangle's area comes down to three tools: ½ × base × height when you have a height, Heron's formula when you have all three sides, and ½ab·sin C when you have two sides and the angle between them. Identify what you're given, pick the matching formula, watch your units, and the answer follows every time.
Frequently asked questions
What is the formula for the area of a triangle?+
The area of a triangle is A = ½ × base × height - multiply the base by the perpendicular height (the straight-line distance from the base to the opposite corner) and take half. This works for every triangle, as long as the height is measured at a right angle to the base, not along a slanted side.
How do you find the area of a triangle without the height?+
Use one of two methods. If you know all three side lengths, use Heron's formula: find s = (a + b + c) / 2, then A = √(s(s−a)(s−b)(s−c)). If you know two sides and the angle between them, use A = ½ × a × b × sin(C). Both give the exact area without ever measuring a height.
What is the height of a triangle?+
The height (or altitude) is the perpendicular distance from a chosen base to the opposite vertex - the length of a straight line that meets the base at a right angle. It is not the length of a slanted side. In an obtuse triangle the height can even fall outside the triangle, which is where many students go wrong.
How do you find the area of a right triangle?+
In a right triangle the two legs (the sides that meet at the right angle) are already perpendicular, so one leg is the base and the other is the height. The area is simply A = ½ × leg₁ × leg₂. There's no need to find a separate height because the legs provide it.
What is Heron's formula?+
Heron's formula finds a triangle's area from its three side lengths alone. First compute the semi-perimeter s = (a + b + c) / 2, then A = √(s(s−a)(s−b)(s−c)). It's the go-to method when you know all three sides but no height or angle.
Can a tutor help me get better at geometry?+
Yes - geometry is highly visual, and a tutor working live on a shared whiteboard can draw each triangle, mark the true perpendicular height, and coach which area method a problem calls for until it's automatic. That guided practice is what turns geometry from a set of half-remembered formulas into a reliable skill.
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