Right Triangle Calculator
Calculate right triangle instantly with our math tool. Shows detailed work, formulas used, and multiple solution methods.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Right Triangle Calculator
Calculator
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Formula: c = sqrt(a^2 + b^2), Area = (1/2)ab
Worked example — Hypotenuse = 5, Angles: 36.87 and 53.13 deg, Area = 6, Perimeter = 12, Inradius = 1, Circumradius = 2.5
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 Angle A = arctan(3/4) = 36.8699 degrees Angle B = arctan(4/3) = 53.1301 degrees Area = (1/2)(3)(4) = 6 Perimeter = 3 + 4 + 5 = 12 Altitude to hypotenuse = (3)(4)/5 = 2.4 Inradius = (3+4-5)/2 = 1 Circumradius = 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. Other leg b = sqrt(13^2 - 5^2) = sqrt(169 - 25) = sqrt(144) = 12 feet Angle with ground = arctan(12/5) = 67.38 degrees Area = (1/2)(5)(12) = 30 sq ft This 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.
How do you calculate the hypotenuse from two legs?
The hypotenuse is calculated using the Pythagorean theorem: c = sqrt(a^2 + b^2), where a and b are the two legs. Simply square each leg, add the results, and take the square root. For legs of 3 and 4: c = sqrt(9 + 16) = sqrt(25) = 5. For legs of 5 and 12: c = sqrt(25 + 144) = sqrt(169) = 13. The hypotenuse is always longer than either leg but shorter than the sum of both legs (triangle inequality). When both legs are equal (isosceles right triangle), the hypotenuse equals the leg length times sqrt(2). This calculation is used millions of times daily in computer graphics, GPS systems, physics simulations, and engineering applications worldwide.
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.
What is the altitude to the hypotenuse and why is it important?
The altitude to the hypotenuse is the perpendicular line from the right angle vertex to the hypotenuse, with length h = ab/c. This altitude creates two smaller right triangles that are both similar to the original triangle and to each other. This similarity leads to the geometric mean relationships: each leg is the geometric mean of the hypotenuse and its projection onto the hypotenuse (a^2 = c times p_a, b^2 = c times p_b), and the altitude is the geometric mean of the two projections (h^2 = p_a times p_b). These relationships provide alternative proofs of the Pythagorean theorem and are used in geometric constructions. The altitude to the hypotenuse also represents the shortest distance from the right angle vertex to the hypotenuse.
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.
How do right triangles apply to distance calculations?
Right triangles are the basis of virtually all distance calculations in mathematics and science. The Euclidean distance between two points (x1,y1) and (x2,y2) is the hypotenuse of a right triangle with legs (x2-x1) and (y2-y1), giving d = sqrt((x2-x1)^2 + (y2-y1)^2). This extends to 3D: d = sqrt(dx^2 + dy^2 + dz^2). GPS receivers calculate your position using multiple right triangles formed between satellites and your location. Computer screens measure resolution diagonally as the hypotenuse of width and height. Pilots calculate ground distance using right triangles formed by altitude and slant range. Even walking diagonally across a rectangular field involves right triangle calculations to determine the distance saved compared to walking along two sides.
What is the median to the hypotenuse?
The median to the hypotenuse in a right triangle is the line segment from the right angle vertex to the midpoint of the hypotenuse, and it has a remarkable property: its length always equals exactly half the hypotenuse. This means the median to the hypotenuse equals the circumradius, because the midpoint of the hypotenuse is the circumcenter. For a 3-4-5 triangle, the median to the hypotenuse is 2.5. This property creates an isosceles triangle between the right angle vertex and the two endpoints of the hypotenuse when the median is drawn, since the median and two halves of the hypotenuse all have the same length. This property is unique to right triangles and provides a quick way to verify that a triangle is a right triangle.
How do you solve a right triangle completely?
Solving a right triangle means finding all three sides and all three angles. Since one angle is always 90 degrees, you need to find the other two angles and any unknown sides. If you know two sides, use the Pythagorean theorem for the third side and inverse trigonometric functions for the angles. If you know one side and one acute angle, the other angle is 90 minus the known angle, and the sides are found using sine, cosine, or tangent. For example, given leg a = 7 and angle A = 35 degrees: angle B = 55 degrees, leg b = a / tan(A) = 7 / tan(35) = 9.997, hypotenuse c = a / sin(A) = 7 / sin(35) = 12.204. Always verify your solution satisfies the Pythagorean theorem as a check on accuracy.
What are special right triangles?
Special right triangles have fixed side ratios that make calculations exact without a calculator. In a 45-45-90 triangle (an isosceles right triangle), the two legs are equal and the hypotenuse is leg × √2. If each leg is 1, the hypotenuse is √2 ≈ 1.414. In a 30-60-90 triangle, the sides are in ratio 1 : √3 : 2, where the shortest side is opposite the 30° angle and the hypotenuse is twice the shortest side. These triangles appear constantly in engineering, architecture, and physics — for instance, a roof pitch of 45° forms a 45-45-90 triangle, and an equilateral triangle bisected diagonally creates two 30-60-90 triangles. Knowing these ratios eliminates the need for trigonometric tables in common scenarios.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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