Right Triangle Calculator: Solve Sides & Angles

Solve a Right Triangle from Two Measurements

Right Triangle Formula

Formula

\[\begin{aligned}c&=\sqrt{a^2+b^2}, & A&=\frac{ab}{2}, & P&=a+b+c\\ \alpha&=\arctan\!\left(\frac{a}{b}\right), & \beta&=90^\circ-\alpha\\ h&=\frac{ab}{c}, & r&=\frac{ab}{P}, & R&=\frac{c}{2}\end{aligned}\]

Where:

\(a\) and \(b\) — the perpendicular legs, opposite angles \(\alpha\) and \(\beta\), respectively
\(c\) — the hypotenuse, opposite the right angle
\(A\) — area; \(P\) — perimeter
\(h\) — height from the right-angle vertex to the hypotenuse
\(r\) — inradius; \(R\) — circumradius

The Pythagorean theorem and trigonometry determine the sides and acute angles. Once the triangle is solved, its sides also determine its area, perimeter, height, and circle radii.

Right Triangle – Calculation Example

Example: the default values are Leg a = 3 and Leg b = 4. Find the remaining measurements.

\[c=\sqrt{3^2+4^2}=5,\qquad \alpha=\arctan\!\left(\frac{3}{4}\right)\approx36.8699^\circ,\qquad \beta\approx53.1301^\circ\]

\[A=\frac{3\cdot4}{2}=6,\qquad P=3+4+5=12\]

\[h=\frac{3\cdot4}{5}=2.4,\qquad r=\frac{3\cdot4}{12}=1,\qquad R=\frac{5}{2}=2.5\]

Result: the sides are 3, 4, and 5 units; the acute angles are approximately 36.8699° and 53.1301°. The area is 6 square units and the perimeter is 12 units.

How to use the calculator

Enter two known measurements, including at least one side. The side fields are Leg a, Leg b, and Hypotenuse c. Leg a is opposite Angle α; Leg b is opposite Angle β.
For a side-and-angle calculation, enter a side and either Angle α or Angle β in degrees. Alternatively, pair a side with Area or Perimeter. Enter all side lengths in the same unit and area in its corresponding square unit.
Clear the default values in Leg a and Leg b if they are not among your known measurements. Leave fields you do not want to use empty.
Read Solved Triangle for the sides, angles in degrees and radians, area, perimeter, height to hypotenuse, inradius, and circumradius. Check Measurements used to see which inputs determined the result.

What the solved triangle tells you

A right triangle has one 90° angle. Its two legs meet at that angle, and the hypotenuse is the side opposite it. The acute angles α and β add to 90°. Knowing two suitable measurements, at least one of them a side, usually fixes both the triangle’s shape and its size. Two angles alone fix the shape but cannot determine side lengths.

The hypotenuse can be found from two legs using \(a^2+b^2=c^2\). If a leg and an acute angle are known, trigonometric ratios relate them to the other sides. For example, \(\tan\alpha=a/b\) and \(\sin\alpha=a/c\). For a focused calculation using these relationships, see the Pythagorean Theorem Calculator or Trigonometry Calculator.

The area is half the product of the legs because they serve as a perpendicular base and height. The reported Height to hypotenuse is a different height: it runs from the right-angle vertex to the hypotenuse. The inradius is the radius of a circle touching all three sides; the circumradius is the radius of a circle passing through all three vertices. In a right triangle, the circumradius always equals half the hypotenuse. If you only need area from a different set of triangle measurements, use the Triangle Area Calculator.

Input rules and practical limits

Enter positive, finite measurements. Each acute angle must be greater than 0° and less than 90°, and the hypotenuse must be longer than either known leg. A side paired with area or perimeter must also describe a possible right triangle; the calculator reports an error when it does not.

The calculator uses only two entered values. It takes the first two available sides in the order Leg a, Leg b, Hypotenuse c. If only one side is entered, it pairs that side with the first available value in this order: Angle α, Angle β, Area, Perimeter. Other entries are ignored rather than checked for consistency, so clear unwanted fields and consult Measurements used.

A hypotenuse paired with an area or perimeter can allow two labelings of the same shape: swapping legs a and b also swaps angles α and β. The result and scaled drawing show the labeling with \(a\leq b\); a note under Measurements used identifies when the swapped labeling is also possible. The drawing is scaled to show proportions, not to provide a physical measurement at screen size.

Frequently Asked Questions

How do I find a missing leg if I know the hypotenuse and the other leg?

Subtract the square of the known leg from the square of the hypotenuse, then take the square root. For a hypotenuse of 13 and a leg of 5, the missing leg is \(\sqrt{13^2-5^2}=12\). If the known leg is as long as or longer than the hypotenuse, those measurements cannot form a right triangle.

How do I solve a right triangle by hand from one side and one angle?

Choose a trigonometric ratio based on where the known side sits relative to the angle. For example, if the hypotenuse is 10 and an acute angle is 30°, the opposite leg is \(10\sin 30^\circ=5\), and the adjacent leg is \(10\cos 30^\circ\approx8.66\). Set your calculator to degree mode when using an angle given in degrees.

How can I check whether three measured sides form a right triangle?

Identify the longest side, then compare its square with the sum of the squares of the other two sides. Lengths 6, 8, and 10 pass because \(6^2+8^2=10^2\). Real-world measurements may differ slightly because of rounding or measurement error, so an approximate match is not proof of an exact right angle.

What is special about a 45-45-90 right triangle?

Its two legs are equal, so each acute angle is 45° and the hypotenuse is \(\sqrt{2}\) times either leg. For legs of length 7, the hypotenuse is \(7\sqrt{2}\approx9.90\). This is also the largest-area right triangle possible for a fixed hypotenuse; making one leg longer than the other reduces its area.

Can I use centimeters for one side and meters for another?

Convert all lengths to one unit before calculating. For example, 50 cm and 1 m should be entered as 50 cm and 100 cm, or as 0.5 m and 1 m. Area uses the square of that unit: \(10{,}000\text{ cm}^2=1\text{ m}^2\). Angles are entered in degrees; to convert radians to degrees, multiply by \(180/\pi\).

Why might a triangle with a very small angle give an unexpected result?

A very small acute angle produces a thin triangle, so one leg may be tiny compared with the other. Small rounding changes to the angle or a side can then noticeably change the tiny leg. Use the most precise measurements you have, and avoid rounding intermediate calculations. An angle of exactly 0° or 90° does not produce a nondegenerate right triangle.

Is there a limit to the area if I know only the hypotenuse?

Yes. For a hypotenuse of length \(c\), the area cannot exceed \(c^2/4\). Equality occurs when the legs are equal. For example, a hypotenuse of 10 permits an area of at most 25 square units. A smaller positive area determines the leg lengths, although either leg can be assigned the label \(a\).