Area of a Right Triangle Calculator
Our free triangle calculator solves area aright triangle problems. Get worked examples, visual aids, and downloadable results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Area of a Right Triangle Calculator
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Formula: Area = (1/2) x base x height = (1/2) x leg_a x leg_b
Worked example — Area = 54 sq cm | Hypotenuse = 15 cm | Perimeter = 36 cm | Altitude = 7.2 cm
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 Hypotenuse = sqrt(9^2 + 12^2) = sqrt(81 + 144) = sqrt(225) = 15 cm Perimeter = 9 + 12 + 15 = 36 cm Altitude 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 Leg b = 20 x cos(35) = 20 x 0.8192 = 16.383 m Area = (1/2) x 11.472 x 16.383 = 93.969 sq m Perimeter = 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.
What are the properties of a right triangle that make area calculation easier?
Right triangles have several properties that simplify area calculations. First, the two legs are perpendicular, eliminating the need to compute a separate height. Second, the Pythagorean theorem lets you find any missing side from two known sides. Third, the acute angles are complementary (they add to 90 degrees), so knowing one angle gives you the other. Fourth, the circumradius always equals half the hypotenuse. Fifth, the inradius equals (a + b - c) / 2 where a and b are legs and c is the hypotenuse. These relationships allow multiple pathways to compute the area from various combinations of known values.
How do you find the area of a right triangle using trigonometry?
Trigonometry provides powerful tools for finding right triangle areas when you know a side and an angle. If you know the hypotenuse h and one acute angle theta, the area equals (1/2) times h squared times sin(theta) times cos(theta). If you know one leg a and the adjacent acute angle theta, the other leg equals a times tan(theta), so the area is (1/2) times a squared times tan(theta). If you know one leg a and the opposite acute angle theta, the other leg equals a / tan(theta), giving area = a squared / (2 times tan(theta)). Each formula derives from the basic definitions of sine, cosine, and tangent.
What are common right triangle side ratios used in practice?
Several right triangle side ratios appear frequently in mathematics and practical applications. The most famous Pythagorean triple is 3-4-5 (and its multiples like 6-8-10 and 9-12-15). Other common triples include 5-12-13, 8-15-17, 7-24-25, and 20-21-29. Special right triangles include the 45-45-90 triangle with ratios 1:1:sqrt(2) and the 30-60-90 triangle with ratios 1:sqrt(3):2. Carpenters and builders commonly use the 3-4-5 ratio to verify right angles during construction, a technique dating back to ancient Egyptian rope stretchers.
How is the area of a right triangle used in coordinate geometry?
In coordinate geometry, right triangle areas are fundamental for calculating distances and areas of polygons. When you have three points forming a triangle, you can often identify or construct right triangles using horizontal and vertical lines. The area formula using coordinates is Area = (1/2) times the absolute value of x1(y2 - y3) + x2(y3 - y1) + x3(y1 - y2). Additionally, the distance formula itself is derived from the Pythagorean theorem applied to a right triangle formed by horizontal and vertical components. Right triangles also underpin the concept of slope in linear equations.
What is the inradius and circumradius of a right triangle?
For a right triangle with legs a and b and hypotenuse c, the circumradius (radius of the circumscribed circle) is always exactly c/2, meaning the hypotenuse is the diameter of the circumscribed circle. The center of this circle is the midpoint of the hypotenuse. The inradius (radius of the inscribed circle) equals (a + b - c) / 2. For a 3-4-5 right triangle, the circumradius is 5/2 = 2.5 and the inradius is (3 + 4 - 5) / 2 = 1. These formulas make right triangles easier to work with than general triangles when computing circle-related measurements.
Can you find the area of a right triangle using only the perimeter?
You cannot uniquely determine the area of a right triangle from the perimeter alone, because multiple right triangles can have the same perimeter but different areas. However, if you know the perimeter P and one additional piece of information (such as the hypotenuse or one leg), you can find the area. For example, given perimeter P and hypotenuse c, the sum of legs is P - c, and the product of legs can be found from (P - c) squared minus c squared divided by 2, which gives you the area directly as half that product. The maximum-area right triangle for a given perimeter is the isosceles right triangle (45-45-90).
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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