Right Triangle Calculator (With Steps) — Find the Hypotenuse, Legs, or Area of a Right Triangle (2024)

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Please provide values for any three of the six fields below. At least one of those values must be a side length.

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Please provide your input and click the calculate button

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About the Right Triangle Calculator

This right triangle calculator lets you calculate the length of the hypotenuse or a leg or the area of a right triangle. For each case, you may choose from different combinations of values to input.

Also, the calculator will give you not just the answer, but also a step by step solution. So you can use it as a great tool to learn about right triangles.

Usage Guide

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i. Valid Inputs

The calculator needs exactly two inputs, at least one of which must be a side (a leg or the hypotenuse).

All inputs can be in any of the three number formats listed below.

Note — The input for each of the two angles must be between $\hspace{0.2em} 0 \hspace{0.2em}$ 0 and $\hspace{0.2em} 90 \hspace{0.2em}$ 90.

ii. Example

If you would like to see an example of the calculator's working, just click the "example" button.

iii. Solutions

As mentioned earlier, the calculator won't just tell you the answer but also the steps you can follow to do the calculation yourself. The "show/hide solution" button would be available to you after the calculator has processed your input.

iv. Share

We would love to see you share our calculators with your family, friends, or anyone else who might find it useful.

By checking the "include calculation" checkbox, you can share your calculation as well.

Right Triangles

A right triangle (or right-angled triangle) is a triangle in which one of the three internal angles is a right angle ( $\hspace{0.2em} 90 \degree \hspace{0.2em}$ 90°).

The longest side in a right triangle (known as the hypotenuse) is the side opposite the right angle.

Pythagorean Theorem

Now one feature of right triangles that makes them so useful and important is that they obey the pythagorean theorem.

And according to the Pythagorean theorem, the square of the hypotenuse is equal to the sum of the squares of the other two sides (called legs). So, for the triangle above –

$BC^2 + AC^2 = {\color{Red} AB} ^2$ BC2+AC2=AB2

Example

The two legs of a right triangle measure $\hspace{0.2em} 5 \text{ cm} \hspace{0.2em}$ 5cm and $\hspace{0.2em} 12 \text{ cm} \hspace{0.2em}$ 12cm in length. Find the length of the hypotenuse.

Solution

If $\hspace{0.2em} x \hspace{0.2em}$ x denotes the length of the hypotenuse, according the pythagorean theorem —

$\begin{align*} x^2 \hspace{0.25em} &= \hspace{0.25em} 5^2 + 12^2 \\[1em] &= \hspace{0.25em} 169 \end{align*}$ x2=52+122=169

Taking the square root on both sides, we have

$\hspace{0.25em} = \hspace{0.25em} 13$ =13

So the hypotenuse has a length of $\hspace{0.2em} 13 \text{ cm} \hspace{0.2em}$ 13cm.

Trigonometric Ratios

Another concept that makes right triangles great for the study of triangles is that of trigonometric ratios. Here's a brief explanation.

We'll start with the figure below.

Now, when we talk about trigonometric ratios, those ratios are with respect to a reference angle. And that reference angle can be any of the two acute angles in a right triangle.

Also, as the figure shows, we have special names for the two legs. The one opposite to the angle is termed "opposite" and the one adjacent to it is called "adjacent".

Trigonometric ratios are ratios between the side lengths of a right triangle. And the value of a trigonometric ratio depends on the reference angle alone.

Here's a table listing the six trigonometric ratios.

Ratio	Formula
Sine	$\sin \theta \hspace{0.25em} = \hspace{0.25em} \frac{\text{opposite}}{\text{hypotenuse}}$ sinθ=hypotenuseopposite
Cosine	$\cos \theta \hspace{0.25em} = \hspace{0.25em} \frac{\text{adjacent}}{\text{hypotenuse}}$ cosθ=hypotenuseadjacent
Tangent	$\tan \theta \hspace{0.25em} = \hspace{0.25em} \frac{\text{opposite}}{\text{adjacent}}$ tanθ=adjacentopposite
Cosecant	$\csc \theta \hspace{0.25em} = \hspace{0.25em} \frac{\text{hypotenuse}}{\text{opposite}}$ cscθ=oppositehypotenuse
Secant	$\sec \theta \hspace{0.25em} = \hspace{0.25em} \frac{\text{hypotenuse}}{\text{adjacent}}$ secθ=adjacenthypotenuse
Cotangent	$\cot \theta \hspace{0.25em} = \hspace{0.25em} \frac{\text{adjacent}}{\text{opposite}}$ cotθ=oppositeadjacent

As mentioned earlier, values for trigonometric ratios depend only on the reference angle $\hspace{0.2em} (\theta) \hspace{0.2em}$ (θ). This is crucial, as you'll see in the second example below.

Example

In $\hspace{0.2em} \triangle ABC \hspace{0.2em}$ △ABC, $\hspace{0.2em} \angle C = 90 \degree \hspace{0.2em}$ ∠C=90°, $\hspace{0.2em} \angle B = 40 \degree \hspace{0.2em}$ ∠B=40°, and $\hspace{0.2em} AC = 5 \text{ in} \hspace{0.2em}$ AC=5in. Find the lengths of $\hspace{0.2em} AB \hspace{0.2em}$ AB and $\hspace{0.2em} BC \hspace{0.2em}$ BC.

Solution

Let's start with a rough labeled sketch of the triangle.

Next, we know

$\sin \theta \hspace{0.25em} = \hspace{0.25em} \frac{\text{opposite}}{\text{hypotenuse}}$ sinθ=hypotenuseopposite

So, for the given triangle

$\begin{align*} \sin B \hspace{0.25em} &= \hspace{0.25em} \frac{AC}{AB} \\[1.5em] \sin 40 \degree \hspace{0.1em} &= \hspace{0.25em} \frac{5}{AB} \end{align*}$ sinBsin40°=ABAC=AB5

Now, because the value of a trigonometric ratio depends only on the angle, $\hspace{0.2em} \sin 40 \degree \hspace{0.2em}$ sin40° would be a constant. As a calculator would tell you, $\hspace{0.2em} \sin 40 \degree \hspace{0.1em} \approx \hspace{0.25em} 0.59 \hspace{0.2em}$ sin40°≈0.59.

Substituting the value of $\hspace{0.2em} \sin 40 \degree \hspace{0.2em}$ sin40° into the equation above, we have

$\begin{align*} 0.59 \hspace{0.25em} &\approx \hspace{0.25em} \frac{5}{BC} \\[1.5em] BC \hspace{0.25em} &\approx \hspace{0.25em} 8.47 \end{align*}$ 0.59BC≈BC5≈8.47

Similarly,

$\begin{align*} \tan \theta \hspace{0.25em} &= \hspace{0.25em} \frac{\text{opposite}}{\text{adjacent}} \\[1.5em] \tan B \hspace{0.25em} &= \hspace{0.25em} \frac{AC}{BC} \\[1.5em] \tan 40 \degree \hspace{0.1em} &= \hspace{0.25em} \frac{5}{BC} \end{align*}$ tanθtanBtan40°=adjacentopposite=BCAC=BC5

Substituting $\hspace{0.2em} \tan 40 \degree \hspace{0.1em} \approx \hspace{0.25em} 0.73 \hspace{0.2em}$ tan40°≈0.73,

$\begin{align*} 0.73 \hspace{0.25em} &\approx \hspace{0.25em} \frac{5}{BC} \\[1.5em] BC \hspace{0.25em} &\approx \hspace{0.25em} 6.85 \end{align*}$ 0.73BC≈BC5≈6.85

Right Triangle Calculator (With Steps) — Find the Hypotenuse, Legs, or Area of a Right Triangle (2024)

FAQs

How do you find the hypotenuse of a right triangle with the area? ›

Hypotenuse =root(a^2+b^2). Area is given. Then ab=2*area. Frame all combinations and get hypotenuse.

See Details ›

Is 14 48 50 a right triangle? ›

Hence, it is a right triangle.

How to find the legs of a right triangle with only the hypotenuse 30-60-90? ›

When the hypotenuse of a 30-60-90 triangle is given, divide that length by 2 to get the shorter side. Multiply the shorter side by the square root of 3 to get the longer side.

Know More ›

What is the length of an unknown leg in the right triangle? ›

How do you find the length of an unknown leg in a right triangle? Use the Pythagorean theorem to write an equation substituting the known leg and hypotenuse, and then solve for the unknown leg.

Explore More ›

How to find the missing length of a triangle? ›

The Pythagorean theorem states that a² + b² = c² in a right triangle where c is the longest side. You can use this equation to figure out the length of one side if you have the lengths of the other two. The figure shows two right triangles that are each missing one side's measure.

How do you find the third length of a triangle? ›

When given the lengths of two sides of a right triangle, we find the length of the third side of the triangle using the Pythagorean Theorem. To do this, We plug the known side lengths into the Pythagorean equation, a2 + b2 = c2, appropriately, and then we solve for the remaining variable.