This triangle calculator lets you solve a triangle. It calculates the missing measurements of a triangle if you know any one side and any two from the remaing five mesurements.

The calculator will give you not just the answers, but also a step-by-step solution.

Here's a quick overview of what it means to solve a triangle and a few related concepts to help you make sense of the solutions provided by the triangle calculator.

For those interested, we have a more comprehensive tutorial on solving triangles.

## Solving a Triangle

There are six values describing the six parts of a triangle — three sides and three angles. Now, if we know one side and any two of the other five values, we can use that information to find the remaining three.

Finding the unknown measurements of a triangles from what is known is referred to as solving triangles.

## Important Concepts

Let's look at a few of the important concepts that help us solve triangles.

### Angle Sum Property

The sum of the three internal angles of a triangle is $\hspace{0.2em} 180 \degree$.

$A + B + C = 180 \degree$

### Sine Rule

The sine rule states that the ratio of side length to the sine of opposite angle is the same for all sides in a triangle.

$\frac{a}{\sin A} \, = \, \frac{b}{\sin B} \, = \, \frac{c}{\sin C}$

### Cosine Rule

The cosine rule gives the relationship between the side lengths of a triangle and the cosine of any of its angles. It says —

$a^2 = b^2 + c^2 - 2bc \, \cos A$

Re-framing the formula for other sides, we have

$b^2 = a^2 + c^2 - 2ac \, \cos B$

$c^2 = a^2 + b^2 - 2ab \, \cos C$

For cases where we need to find angles using the cosine rule, the three formulas can be rearranged as —

$\begin{align*} \cos A \, = \, \frac{b^2 + c^2 - a^2}{2bc} \\[1.5em] \cos B \, = \, \frac{a^2 + c^2 - b^2}{2ac} \\[1.5em] \cos C \, = \, \frac{a^2 + b^2 - c^2}{2ab} \end{align*}$

## Problems

When it comes to solving triangles, there are five different types of problems depending on which three of the triangle's measurements we know.

- $\hspace{0.2em} SSS \hspace{0.2em}$ — all three sides are known
- $\hspace{0.2em} SAS \hspace{0.2em}$ — two sides and the included angle
- $\hspace{0.2em} SSA \hspace{0.2em}$ — two sides and a non-included angle
- $\hspace{0.2em} ASA \hspace{0.2em}$ — two angles and the included side
- $\hspace{0.2em} AAS \hspace{0.2em}$ — two angles and the non-included side

While every problem can be solved using the fundamentals discussed earlier and a basic knowledge of triangles, each type has a sequence of steps that you can use to solve problems of that type.

Let me show you what I mean using an example.

Example

The lengths of the three sides of a triangle are $\hspace{0.2em} 6 \hspace{0.2em}$, $\hspace{0.2em} 7 \hspace{0.2em}$, and $\hspace{0.2em} 8 \hspace{0.2em}$. Solve the triangle.

Solution

The question gives us the three sides of the triangle. So the problem is of type $\hspace{0.2em} SSS \hspace{0.2em}$. Solving the triangle would mean calculating its three angles.

Step 0. We start by drawing a rough sketch of the triangle and labeling the information given in the question. It’s not necessary but often makes things easier and helps avoid silly mistakes.

5

Step 1. Use the Cosine Rule to find the largest angle

When we know all the side lengths, we can use the Cosine Rule to find any of the angles.

It's best to find the largest angle first — the angle opposite to the longest side.

That's because if there is an obtuse angle $\hspace{0.2em} (>90 \degree) \hspace{0.2em}$in the triangle, it has to be this angle. So in the next step, we don't need to worry about the obtuse solutions when taking sine inverse.

Here the largest angle would be $\hspace{0.2em} C \hspace{0.2em}$. So using the formula for $\hspace{0.2em} \cos C \hspace{0.2em}$, we have

$\begin{align*} \cos C \, &= \, \frac{c^2 - b^2 - a^2}{-2ab} \\[1.3em] &= \, \frac{8^2 - 7^2 - 6^2}{-2 \cdot 6 \cdot 7} \\[1.3em] &\approx \, 0.25 \end{align*}$

Taking cos inverse on both sides.

$\begin{align*} C \, &\approx \, \cos ^{-1} (0.25) \\[1em] &\approx \, 75.52 \degree \end{align*}$

Step 2. Use the Sine Rule for one of the remaining angles

Now that we know the three sides and one angle, we can use the Sine Rule to find any of the remaining two angles. Let's calculate $\hspace{0.2em} A \hspace{0.2em}$.

According to the sine rule

$\frac{a}{\sin A} \, = \, \frac{c}{\sin C}$

Substituting the known values and solving for B, we have

$\begin{align*} \frac{6}{\sin A} \, &\approx \, \frac{8}{\sin 75.52 \degree} \\[1.75em] \sin A &\approx 0.7262 \\[1.75em] A &\approx 46.57 \degree \end{align*}$

Step 3. Use the Angle Sum Property to find the third angle

So,

$\begin{align*} A + B + C \, &= \, 180 \degree \\[1em] 46.57 \degree + B + 75.52 \degree\, &= \, 180 \degree \\[1em] B \, &= \, 57.91 \degree \end{align*}$

And we have solved the triangle.