Area of a Right Triangle Calculator
Our free triangle calculator solves area aright triangle problems. Get worked examples, visual aids, and downloadable results.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
Area = (1/2) x base x height = (1/2) x leg_a x leg_b
In a right triangle, the two legs are perpendicular, so they serve as base and height. The area is half the product of the two legs. If you know the hypotenuse and an angle, use trigonometry: leg_a = hyp x sin(angle), leg_b = hyp x cos(angle).
Worked Examples
Example 1: Area from Two Known Legs
Problem:A right triangle has legs measuring 9 cm and 12 cm. Find the area, hypotenuse, perimeter, and altitude to the hypotenuse.
Solution:Area = (1/2) x 9 x 12 = 54 sq cm\nHypotenuse = sqrt(9^2 + 12^2) = sqrt(81 + 144) = sqrt(225) = 15 cm\nPerimeter = 9 + 12 + 15 = 36 cm\nAltitude to hypotenuse = (9 x 12) / 15 = 108 / 15 = 7.2 cm
Result:Area = 54 sq cm | Hypotenuse = 15 cm | Perimeter = 36 cm | Altitude = 7.2 cm
Example 2: Area from Hypotenuse and Angle
Problem:A right triangle has a hypotenuse of 20 m and one acute angle of 35 degrees. Find both legs and the area.
Solution:Leg a = 20 x sin(35) = 20 x 0.5736 = 11.472 m\nLeg b = 20 x cos(35) = 20 x 0.8192 = 16.383 m\nArea = (1/2) x 11.472 x 16.383 = 93.969 sq m\nPerimeter = 11.472 + 16.383 + 20 = 47.855 m
Result:Leg a = 11.472 m | Leg b = 16.383 m | Area = 93.969 sq m
Frequently Asked Questions
What is the formula for the area of a right triangle?
The area of a right triangle is calculated using the formula Area = (1/2) times base times height. In a right triangle, the two legs are perpendicular to each other, so one leg serves as the base and the other as the height. This simplifies the calculation because you do not need to separately find the height as you would with oblique triangles. If the two legs measure a and b, the area equals (a times b) / 2. This is derived from the general triangle area formula and works because the right angle guarantees perpendicularity between the two legs.
How do you find the area of a right triangle if you only know the hypotenuse?
You cannot find the area of a right triangle from the hypotenuse alone because infinitely many right triangles share the same hypotenuse length but have different leg measurements and therefore different areas. However, if you know the hypotenuse and one acute angle, you can use trigonometry: one leg equals hypotenuse times sin(angle) and the other equals hypotenuse times cos(angle). Then the area is (1/2) times hypotenuse squared times sin(angle) times cos(angle), which simplifies to (1/4) times hypotenuse squared times sin(2 times the angle). This approach requires at least one angle in addition to the hypotenuse.
What is the Pythagorean theorem and how does it relate to right triangle area?
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse equals the sum of the squares of the two legs: a squared plus b squared equals c squared. While this theorem directly calculates side lengths rather than area, it is essential for area calculations when you know one leg and the hypotenuse. You first use the theorem to find the missing leg (b = sqrt(c squared minus a squared)), then compute the area as (1/2) times a times b. The Pythagorean theorem is one of the most fundamental results in all of mathematics.
How do you find the altitude to the hypotenuse of a right triangle?
The altitude (height) drawn from the right angle vertex to the hypotenuse creates two smaller right triangles, each similar to the original. The altitude length equals the product of the two legs divided by the hypotenuse: altitude = (a times b) / c. This can also be derived from the area relationship: since Area = (1/2) times base times height, and using the hypotenuse as the base, the height equals 2 times Area divided by hypotenuse. For a 3-4-5 right triangle, the altitude to the hypotenuse is (3 times 4) / 5 = 2.4 units. This altitude is always the shortest distance from the right angle to the hypotenuse.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy