Right Triangle Calculator
Calculate right triangle instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
c = sqrt(a^2 + b^2), Area = (1/2)ab
Where a and b are the two legs (sides adjacent to the right angle) and c is the hypotenuse (side opposite the right angle). The area is half the product of the legs. Angles are found using inverse trigonometric functions.
Worked Examples
Example 1: Classic 3-4-5 Right Triangle
Problem:Solve the right triangle with legs a = 3 and b = 4 completely.
Solution:Hypotenuse c = sqrt(9 + 16) = sqrt(25) = 5\nAngle A = arctan(3/4) = 36.8699 degrees\nAngle B = arctan(4/3) = 53.1301 degrees\nArea = (1/2)(3)(4) = 6\nPerimeter = 3 + 4 + 5 = 12\nAltitude to hypotenuse = (3)(4)/5 = 2.4\nInradius = (3+4-5)/2 = 1\nCircumradius = 5/2 = 2.5
Result:Hypotenuse = 5, Angles: 36.87 and 53.13 deg, Area = 6, Perimeter = 12, Inradius = 1, Circumradius = 2.5
Example 2: Ladder Against a Wall
Problem:A 13-foot ladder rests against a wall with its base 5 feet from the wall. How high does it reach?
Solution:This is a right triangle with hypotenuse c = 13 and one leg a = 5.\nOther leg b = sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12 feet\nAngle with ground = arctan(12/5) = 67.38 degrees\nArea = (1/2)(5)(12) = 30 sq ft\nThis is a 5-12-13 Pythagorean triple
Result:The ladder reaches 12 feet up the wall at an angle of 67.38 degrees from the ground.
Frequently Asked Questions
What defines a right triangle?
A right triangle is a triangle that contains exactly one angle of 90 degrees, called the right angle. The side opposite the right angle is the hypotenuse, which is always the longest side, and the other two sides are called legs or catheti. The right angle is typically denoted by a small square in geometric diagrams. Right triangles are the foundation of trigonometry, as the six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) are originally defined as ratios of the sides of a right triangle. The Pythagorean theorem (a^2 + b^2 = c^2) applies exclusively to right triangles, providing the fundamental relationship between the three sides.
What are the trigonometric ratios in a right triangle?
The six trigonometric ratios relate angles to side lengths in a right triangle. For an acute angle A: sine(A) = opposite/hypotenuse, cosine(A) = adjacent/hypotenuse, tangent(A) = opposite/adjacent. The reciprocal functions are: cosecant(A) = hypotenuse/opposite, secant(A) = hypotenuse/adjacent, cotangent(A) = adjacent/opposite. These ratios are constant for a given angle regardless of the triangle's size, which is what makes trigonometry so powerful. For a 3-4-5 triangle, sin(A) = 3/5 = 0.6, cos(A) = 4/5 = 0.8, tan(A) = 3/4 = 0.75. Knowing any one trigonometric ratio for an angle is sufficient to determine the angle and all other ratios.
How is the inradius of a right triangle calculated?
The inradius of a right triangle has the elegant formula r = (a + b - c) / 2, where a and b are the legs and c is the hypotenuse. This is simpler than the general triangle formula and can be derived from the fact that the incircle touches the hypotenuse at a distance r from each leg. For a 3-4-5 triangle: r = (3 + 4 - 5) / 2 = 1. The incircle center is located at coordinates (r, r) from the right angle vertex. The inradius can also be expressed as r = area / s where s is the semi-perimeter: r = 6 / 6 = 1 (confirmed). An interesting property: the diameter of the incircle equals the sum of the legs minus the hypotenuse. The incircle is always entirely contained within the triangle and tangent to all three sides.
What is the circumradius of a right triangle?
The circumradius of a right triangle is always exactly half the hypotenuse: R = c/2. This is a direct consequence of Thales theorem, which states that any angle inscribed in a semicircle is a right angle. The converse means that for any right triangle, the hypotenuse is a diameter of the circumscribed circle. The circumcenter (center of the circumscribed circle) is therefore always located at the midpoint of the hypotenuse. For a 3-4-5 triangle: R = 5/2 = 2.5. This property means the circumradius is always greater than or equal to the inradius, with the ratio R/r = c / (a + b - c). For a 3-4-5 triangle: R/r = 2.5, and the minimum ratio for right triangles occurs for the isosceles right triangle where R/r = 1 + sqrt(2) approximately 2.414.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy