How to Find the Area of a Triangle
Half base times height covers almost every triangle you will meet, as long as you use a height that is perpendicular to the base you chose. The other two methods exist for when that perpendicular height is not given.
The standard formula
Area = (1/2) x base x height, where the height is the perpendicular distance from the base to the opposite vertex. Any of the three sides can be the base, as long as the height you use is measured perpendicular to that side.
Right triangles are easier
In a right triangle the two legs are already perpendicular, so one is the base and the other is the height. No extra construction is needed, and the hypotenuse is never used in the area formula.
When you have two sides and the angle between them
- Area = (1/2) x a x b x sin C, where C is the angle between sides a and b.
- The angle must be the one between the two sides, not any other angle.
- This is the version to use when no perpendicular height is given.
Worked examples
Example 1
Question
A triangle has a base of 12 cm and a perpendicular height of 5 cm. Find its area.
Answer
30 cm squared
Steps
- 1Use area = (1/2) x base x height.
- 2Substitute: (1/2) x 12 x 5.
- 3(1/2) x 12 = 6, and 6 x 5 = 30.
- 4Area = 30 cm squared. The units are squared because two lengths were multiplied.
Example 2
Question
A right triangle has legs of 7 cm and 24 cm and a hypotenuse of 25 cm. Find its area.
Answer
84 cm squared
Steps
- 1The two legs are perpendicular to each other, so they serve as base and height.
- 2Area = (1/2) x 7 x 24.
- 37 x 24 = 168.
- 4Half of 168 is 84, so the area is 84 cm squared.
The 25 is a distractor. Using the hypotenuse as a base without its own perpendicular height is the standard trap in this question.
Example 3
Question
Two sides measure 8 cm and 10 cm with an angle of 30 degrees between them. Find the area.
Answer
20 cm squared
Steps
- 1No perpendicular height is given, so use area = (1/2) x a x b x sin C.
- 2Here a = 8, b = 10, and C = 30 degrees, the angle between them.
- 3sin 30 degrees = 0.5.
- 4Substitute: (1/2) x 8 x 10 x 0.5.
- 5(1/2) x 80 = 40, and 40 x 0.5 = 20 cm squared.
Common mistakes
Using a slanted side as the height.
The height must be perpendicular to the base. A slanted side is longer than the perpendicular height and inflates the area.
Forgetting the half and reporting base times height.
A triangle is half of the rectangle around it. Missing the half doubles every answer.
Using the hypotenuse as the base in a right triangle without its own height.
Use the two legs. They are already perpendicular, which is exactly what the formula needs.
Giving the answer in cm instead of cm squared.
Area is always in squared units. Marks are lost for this more often than for the arithmetic.
Practice it
Work these out first, then open the answer. Stuck halfway? Select the problem and ask Solvecoon for the step you are missing.
1. Base 9 cm, perpendicular height 6 cm. Find the area.
Show answer
27 cm squared. Half of 9 x 6 = 54.
2. A triangle has area 48 cm squared and base 16 cm. Find its height.
Show answer
6 cm. From 48 = (1/2)(16)h, so 48 = 8h.
3. A right triangle has sides 5, 12, and 13. Find its area.
Show answer
30. The legs are 5 and 12, so the area is half of 60.
Questions people ask
- Which side should I choose as the base?
- Whichever one has a perpendicular height you actually know. The area comes out the same for all three choices.
- What if I only know the three sides?
- Use Heron's formula, which works from the three side lengths and the semi-perimeter without needing a height.
- Does the sine formula need radians?
- Only if your calculator is in radian mode. Check the mode before trusting the answer, because the wrong mode gives a plausible but wrong number.
- Why is the area half the base times the height?
- Because two copies of a triangle fit together into a parallelogram with the same base and height, and the parallelogram's area is base times height.
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