Calculate the area, hypotenuse, perimeter, and angles of a right triangle given the base and height. Perfect for geometry, construction, and engineering.
Last updated: April 2026 | By Patchworkr Team
A right triangle is a triangle with one angle measuring exactly 90 degrees (a right angle). The side opposite the right angle is called the hypotenuse and is always the longest side. The other two sides are called legs or catheti.
Key properties of right triangles:
Right triangles are fundamental in trigonometry, construction, navigation, and physics. They appear whenever you have perpendicular measurements or need to calculate diagonal distances.
The base and height are the two sides that form the right angle (90°). They must be perpendicular to each other. In a right triangle, these are called the legs or catheti.
Why: The area formula specifically uses the two perpendicular sides. Using any other pair would give incorrect results. The right angle guarantees these sides are perpendicular.
Confirm that the 90-degree angle is where your base and height meet. The hypotenuse (longest side) should be opposite the right angle.
Why: Using the hypotenuse in the area formula would be mathematically incorrect. Verifying the right angle position prevents calculation errors and ensures you're using the correct formula variant.
Multiply the base by the height, then divide by 2. This gives the area of the triangle in square units.
Why: A right triangle is exactly half of a rectangle with the same base and height. The rectangle area is base × height, so a triangle is half that. This geometric relationship is fundamental to the formula.
Use the Pythagorean theorem to find the third side. This expands the information you have about the triangle and enables additional calculations.
Why: Knowing all three sides allows you to calculate perimeter, angles, altitude to hypotenuse, inradius, and circumradius. Hypotenuse is essential for many real-world applications.
Double-check that the area value is smaller than the rectangle with same base and height. Verify the hypotenuse is longer than both legs. Interpret the area in context of the real-world problem.
Why: Verification catches formula errors and unit mistakes. Interpreting results ensures the calculated area makes sense for the application (e.g., siding for a roof gable must be positive and reasonable in size).
Roof Gable Area
Area = ½ × base × height, where the base and height are the two sides that form the right angle (90°).
Only the two sides that meet at the right angle. You cannot use the hypotenuse as base or height in the area formula.
It states that c² = a² + b², where c is the hypotenuse and a and b are the two legs. It only works for right triangles.
Use the Pythagorean theorem. If you know both legs, c = √(a² + b²). If you know one leg and the hypotenuse, the other leg = √(c² - a²).
One angle is always 90°. The other two angles add up to 90° (they're complementary). Use trigonometry to find them: tan(θ) = opposite / adjacent.
Yes! Any triangle with sides in the ratio 3:4:5 (or multiples like 6:8:10) is a right triangle. This is called a Pythagorean triple.
Because a right triangle is exactly half of a rectangle with the same base and height. The diagonal cuts the rectangle into two equal right triangles.
No. The hypotenuse is always the longest side in a right triangle. If it's not, you don't have a right triangle.
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