Area of a Right Triangle Calculator
Find the area of any right triangle using whichever two values you know: the two legs, one leg and the hypotenuse, one leg and an acute angle, or the hypotenuse and an acute angle. Pick a mode, enter your values, and get the area, missing side, altitude to the hypotenuse, and perimeter instantly.
Formula
Worked example
A right triangle with legs a = 3 m and b = 4 m: Area = (3 x 4) / 2 = 6 m^2. The hypotenuse is sqrt(9+16) = 5 m, the altitude to the hypotenuse is 2x6/5 = 2.4 m, and the perimeter is 3+4+5 = 12 m.
The right-triangle area formula
A right triangle has one 90-degree angle and two shorter sides called legs (a and b) that meet at that right angle. The longest side is the hypotenuse (c), which lies opposite the right angle. The area is always half the product of the two legs: Area = (a x b) / 2. This works because a right triangle is exactly half of a rectangle with sides a and b. All four solving modes in this calculator reduce to the same underlying formula once the missing leg or angle is resolved.
Four solving modes explained
If you know both legs, plug them directly into Area = a*b/2. If you know one leg and the hypotenuse, the missing leg is sqrt(c^2 - a^2) by the Pythagorean theorem, then apply the base formula. If you know one leg and an acute angle alpha, the second leg is a*tan(alpha), giving Area = a^2*tan(alpha)/2. If you know the hypotenuse and an acute angle, the two legs are c*sin(alpha) and c*cos(alpha), so Area = c^2*sin(alpha)*cos(alpha)/2. In every case the altitude from the right-angle vertex to the hypotenuse equals h = 2*Area/c, a useful check.
Special right triangles
Two right triangles appear so often they are worth memorising. The 45-45-90 (isosceles right) triangle has legs of equal length a, hypotenuse a*sqrt(2), and area a^2/2. The 30-60-90 triangle has legs a and a*sqrt(3), hypotenuse 2a, and area a^2*sqrt(3)/2. Pythagorean triples such as 3-4-5, 5-12-13, and 8-15-17 are right triangles whose three sides are all whole numbers. Any multiple of a triple (e.g. 6-8-10 or 9-12-15) is also a right triangle, and the area scales as the square of the multiplier.
Altitude to the hypotenuse
The altitude h is the perpendicular segment from the right-angle vertex to the hypotenuse. Its length is h = (a*b)/c = 2*Area/c. This altitude divides the original triangle into two smaller triangles, each of which is similar to the whole. This geometric mean relationship is used in many proofs of the Pythagorean theorem and appears frequently in coordinate geometry and optics problems.
Common right-triangle formulas by known inputs
| Known values | Area formula | Find hypotenuse |
|---|---|---|
| Both legs a, b | Area = a x b / 2 | c = sqrt(a^2 + b^2) |
| Leg a, hypotenuse c | Area = a x sqrt(c^2-a^2) / 2 | Already known |
| Leg a, angle alpha | Area = a^2 x tan(alpha) / 2 | c = a / cos(alpha) |
| Hyp. c, angle alpha | Area = c^2 x sin(alpha) x cos(alpha) / 2 | Already known |
| Isosceles (45-45-90) | Area = a^2 / 2 | c = a x sqrt(2) |
| 30-60-90 triangle | Area = (a^2 x sqrt(3)) / 2 | c = 2a (short leg a) |
Use the row that matches what you already know. All angles are acute (0 to 90 degrees); alpha and beta are complementary (alpha + beta = 90 degrees).
Frequently asked questions
What is the formula for the area of a right triangle?
The standard formula is Area = (a x b) / 2, where a and b are the two legs (the sides that meet at the right angle). If you do not know both legs, you can use equivalent forms: Area = a x sqrt(c^2 - a^2) / 2 when you know one leg and the hypotenuse, Area = a^2 x tan(alpha) / 2 when you know one leg and the adjacent acute angle, or Area = c^2 x sin(alpha) x cos(alpha) / 2 when you know the hypotenuse and one acute angle.
How do I find the area if I only know the hypotenuse and one angle?
Use the formula Area = c^2 x sin(alpha) x cos(alpha) / 2. For a 30-60-90 triangle with hypotenuse 10, alpha = 30 degrees: Area = 100 x sin(30°) x cos(30°) / 2 = 100 x 0.5 x 0.866 / 2 = 21.65 square units. Select the "Hypotenuse and angle" mode in this calculator and it does the trigonometry for you.
What is the altitude to the hypotenuse?
The altitude to the hypotenuse (h) is the perpendicular distance from the right-angle vertex to the hypotenuse. It equals h = (a x b) / c, which is the same as 2 x Area / c. For a 3-4-5 triangle, h = (3 x 4) / 5 = 2.4. This segment divides the original triangle into two smaller triangles that are both similar to the original.
What are Pythagorean triples and why do they matter?
A Pythagorean triple is a set of three positive integers (a, b, c) satisfying a^2 + b^2 = c^2. The most famous is (3, 4, 5): 9 + 16 = 25. Others include (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any whole-number multiple of a triple is also valid, such as (6, 8, 10). These are useful in construction and engineering because all three sides are exact integers with no rounding error. The area of a 3-4-5 triangle is 6 square units; a 5-12-13 triangle has area 30 square units.
What is the area of a 45-45-90 isosceles right triangle?
In a 45-45-90 triangle both legs are equal, so if each leg is a, the area is a^2 / 2. The hypotenuse is a x sqrt(2). For example, legs of 5 cm give area = 25 / 2 = 12.5 cm^2 and hypotenuse = 5*1.414 = 7.07 cm.
How is the perimeter of a right triangle calculated?
The perimeter is the sum of all three sides: P = a + b + c. Once you know all three sides from any of the four solving modes, just add them. For a 3-4-5 triangle the perimeter is 3 + 4 + 5 = 12 units.