5
units
6
sq units
12
units
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°
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°
2.4
units
5
units
6
sq units
12
units
—
°
—
°
2.4
units
The Right Triangle Calculator computes all properties of a right triangle from its two legs. A right triangle contains one 90-degree angle, making it the cornerstone of trigonometry, construction, and countless practical applications. Simply enter the two leg lengths, and this tool instantly returns the hypotenuse, area, perimeter, both acute angles, and the altitude to the hypotenuse.
The right triangle holds a special place in mathematics. The Pythagorean theorem, arguably the most famous theorem in all of mathematics, states that in a right triangle the square of the hypotenuse equals the sum of the squares of the two legs. Written as $$c^2 = a^2 + b^2$$, this relationship has been known for over 4,000 years. Clay tablets from ancient Babylon (c. 1800 BCE) list Pythagorean triples, and the theorem appears in the mathematical traditions of India, China, and Greece independently.
Right triangles are indispensable in practical work. Carpenters and builders use the 3-4-5 rule to verify that corners are square. Surveyors measure horizontal distances and elevation changes that form right triangles. Navigators decompose courses into north-south and east-west components using right-triangle trigonometry. Physicists resolve forces and velocities into perpendicular components. In computer graphics, distance calculations between points rely on the Pythagorean theorem applied in two or three dimensions.
The two acute angles in a right triangle are complementary: they always sum to 90 degrees. Each acute angle can be found using the arctangent function. The angle opposite a given leg equals the arctangent of that leg divided by the other leg. These relationships form the basis of trigonometric ratios — sine, cosine, and tangent — which extend the utility of right triangles far beyond simple geometry.
The altitude from the right-angle vertex to the hypotenuse creates two smaller triangles that are each similar to the original. This property, known as the geometric mean relation, yields the altitude formula $$h = \frac{ab}{c}$$, where $$h$$ is the altitude, $$a$$ and $$b$$ are the legs, and $$c$$ is the hypotenuse. The altitude is useful in structural engineering for calculating the shortest distance from the right-angle vertex to the hypotenuse.
This calculator handles any valid pair of positive leg lengths. Whether you are working in millimeters for a precision machining task or in kilometers for a geographic distance problem, enter the two legs and receive all derived quantities instantly. Results are presented with four decimal places for engineering-grade accuracy.
Given two legs $$a$$ and $$b$$ of a right triangle, all other properties are derived as follows.
Hypotenuse (Pythagorean Theorem):
$$c = \sqrt{a^2 + b^2}$$
Area: The two legs serve as base and height since they are perpendicular:
$$A = \frac{a \cdot b}{2}$$
Perimeter:
$$P = a + b + c = a + b + \sqrt{a^2 + b^2}$$
Acute Angles: Using the arctangent function:
$$\alpha = \arctan\left(\frac{a}{b}\right) \cdot \frac{180}{\pi}$$
$$\beta = \arctan\left(\frac{b}{a}\right) \cdot \frac{180}{\pi}$$
Note that $$\alpha + \beta = 90°$$ always holds.
Altitude to Hypotenuse: The perpendicular distance from the right-angle vertex to the hypotenuse:
$$h = \frac{a \cdot b}{c}$$
This formula follows from equating two expressions for the triangle's area: $$\frac{ab}{2} = \frac{ch}{2}$$.
The Hypotenuse is the longest side of the right triangle, lying opposite the 90° angle. It represents the direct distance between the endpoints of the two legs.
The Area is the enclosed surface measured in square units. Since the legs are perpendicular, one serves as the base and the other as the height.
The Perimeter is the total boundary length, useful for fencing, framing, or trim calculations.
Angle A and Angle B are the two acute angles. They are complementary (sum to 90°). The larger acute angle is always opposite the longer leg.
The Altitude to Hypotenuse is the shortest distance from the right-angle vertex to the hypotenuse. It is useful in engineering for clearance calculations and in geometry for proving similarity relationships.
Inputs
Results
Hypotenuse c = sqrt(9 + 16) = sqrt(25) = 5. Area = 3*4/2 = 6. Perimeter = 3+4+5 = 12. Angle A = atan(3/4) = 36.87°. Angle B = atan(4/3) = 53.13°. Check: 36.87 + 53.13 = 90°. Altitude h = 3*4/5 = 2.4.
Inputs
Results
Hypotenuse c = sqrt(25 + 144) = sqrt(169) = 13 (another Pythagorean triple). Area = 5*12/2 = 30. Perimeter = 5+12+13 = 30. Angle A = atan(5/12) = 22.62°. Altitude h = 5*12/13 = 4.6154.
You can find the missing leg using the rearranged Pythagorean theorem: if you know hypotenuse c and leg a, then b = sqrt(c² - a²). Calculate the missing leg first, then enter both legs into this calculator.
A Pythagorean triple is a set of three positive integers (a, b, c) such that a² + b² = c². Common examples include (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). Any multiple of a Pythagorean triple is also a Pythagorean triple.
The altitude creates two smaller right triangles that are similar to the original. It represents the shortest distance from the right-angle vertex to the hypotenuse, which is important in engineering clearance calculations and in proving geometric theorems about similar triangles.
Yes, for 3D distances, apply the Pythagorean theorem twice: first find the horizontal distance d = sqrt(x² + y²), then the 3D distance = sqrt(d² + z²) = sqrt(x² + y² + z²). This is equivalent to using three perpendicular legs.
The two acute angles in a right triangle are always complementary, meaning they sum to exactly 90°. If one acute angle is 30°, the other is 60°. This complementary relationship is fundamental to trigonometric identities like sin(θ) = cos(90° - θ).
All six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) are originally defined as ratios of sides in a right triangle. For example, sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent. These ratios extend to the unit circle for all angles.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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