The area of a triangle is A = (1/2) × base × height. Multiply the base by the perpendicular height, then take half. For a triangle with base 10 and height 6, the area is (1/2)(10)(6) = 30 square units.
Below we work two full examples, break the method into steps, and flag the mistakes students make most often.
Worked example: base 10, height 6
Problem. A triangle has a base of 10 m and a perpendicular height of 6 m. Find its area.
- Write the formula: A = (1/2)bh.
- Substitute b = 10 and h = 6: A = (1/2)(10)(6).
- Multiply base and height first: 10 × 6 = 60.
- Take half: A = (1/2)(60) = 30.
- Attach square units: A = 30 m².
Notice we used the perpendicular height — not the slanted side of the triangle. That distinction is where most errors start.
The triangle area formula
For any triangle:
A = (1/2) b h, where b is the length of any side you choose as the base and h is the perpendicular distance from that base to the opposite vertex.
A triangle is exactly half of a parallelogram with the same base and height, which is why the formula has the factor of 1/2.
Steps to find the area step by step
- Pick a base. Any side works. Example: in a triangle with sides 13, 13, 10, pick the side of length 10.
- Identify the matching height. The height must be perpendicular to the base you chose. Example: for that same triangle, the height to the base of 10 is 12.
- Multiply base × height. Example: 10 × 12 = 120.
- Take half. Example: (1/2)(120) = 60.
- Write square units. Example: 60 square units, or 60 cm² if the sides were in cm.
Right triangles: use the two legs
In a right triangle, the two legs are already perpendicular, so they act as the base and height directly. For legs 5 and 12, A = (1/2)(5)(12) = 30 square units — no separate height needed.
Worked example: a right triangle with legs 8 and 15
Problem. A right triangle has legs of 8 in and 15 in. Find its area.
- The legs meet at the right angle, so they are perpendicular. Use one as the base and the other as the height.
- Write the formula: A = (1/2)bh.
- Substitute b = 8, h = 15: A = (1/2)(8)(15).
- Multiply: 8 × 15 = 120.
- Take half: A = (1/2)(120) = 60.
The area is 60 in². (You do not need the hypotenuse of 17 for the area — it plays no role in the formula.)
Common mistakes to avoid
- Using a slanted side as the height. The height must be perpendicular to the base. The slant length of a triangle is not the height unless the triangle is a right triangle and you are using the two legs.
- Forgetting the 1/2. A = bh is the parallelogram formula. A triangle is half of that, so leaving out the 1/2 doubles the answer.
- Mismatching base and height. If you switch to a different side as the base, you must switch to the height perpendicular to that new base.
- Using the hypotenuse of a right triangle as a leg. In a right triangle, the two legs serve as base and height — not the hypotenuse.
- Writing linear units instead of square units. Area is always in square units: cm², m², in², ft².
Keep practicing with a guided tutor
Once the formula clicks, the tricky part is spotting the right height in messier problems — obtuse triangles, coordinate-plane triangles, or figures where only side lengths are given (Heron's formula territory). Sign up for LernOS to work through triangle-area problems with a Socratic tutor that asks the right questions until you can find the area on your own.