How to find the missing side of a right triangle

Use the Pythagorean Theorem for Right Triangles...

Question:

If I'm given a right triangle and two of its sides, how can I find the length of the third side? Can I do this if it's not a right triangle?

Answer

Finding the missing side of a right triangle is a pretty simple matter if two sides are known. One of the more famous mathematical formulas is \(a^2+b^2=c^2\), which is known as the Pythagorean Theorem. The theorem states that the hypotenuse of a right triangle can be easily calculated from the lengths of the sides. The hypotenuse is the longest side of a right triangle.

If you're given the lengths of the two sides it is easy to find the hypotenuse. Just square the sides, add them, and then take the square root. Here's an example:

Since we are given that the two legs of the triangle are 3 and 4, plug those into the Pythagorean equation and solve for the hypotenuse:

$$ a^2+b^2=c^2 $$ $$ 3^2+4^2=c^2 $$ $$ 25 = c^2 $$ $$ c = \sqrt{25} $$ $$ c = 5 $$

If you are given the hypotenuse and one of the legs, it's going to be slightly more complicated, but only because you have to do some algebra first. Suppose you know that one leg is 5 and the hypotenuse (longest side) is 13. Plug those into the appropriate places in the Pythagorean equation:

$$ a^2+b^2=c^2 $$ $$ 5^2+b^2=13^2 $$ $$ 25+b^2=169 $$ $$ b^2=144 $$ $$ b = 12 $$

As you can see, it is pretty simple to use the Pythagorean Theorem to find the missing side length of a right triangle. But -- what if it's not a right triangle? If you change that angle in the triangle there can obviously be any number of possibilities for the hypotenuse! Thus, you need more information to solve the problem. You can try using the Law of Sines or the Law of Cosines to determine side lengths in other triangles.

Try the "Triangle Calculator" below:

Find a Side when we know another Side and Angle

We can find an unknown side in a right-angled triangle when we know:

• one length, and
• one angle (apart from the right angle, that is).

Example: Depth to the Seabed

The ship is anchored on the seabed.

We know:

• the cable length (30 m), and
• the angle the cable makes with the seabed

So we should be able to find the depth!

But How?

The answer is to use Sine, Cosine or Tangent!

But Which One?

Which one of Sine, Cosine or Tangent to use?

To find out which, first we give names to the sides:

• Adjacent is adjacent (next to) to the angle,
  
• Opposite is opposite the angle,
  
• and the longest side is the Hypotenuse.

Now, for the side we already know and the side we are trying to find, we use the first letters of their names and the phrase "SOHCAHTOA" to decide which function:

Like this:

Example: Depth to the Seabed (Continued)

Find the names of the two sides we are working on:

• the side we know is the Hypotenuse
• the side we want to find is Opposite the angle (check for yourself that "d" is opposite the angle 39°)

Now use the first letters of those two sides (Opposite and Hypotenuse) and the phrase "SOHCAHTOA" which gives us "SOHcahtoa", which tells us we need to use Sine:

Sine: sin(θ) = Opposite / Hypotenuse

Now put in the values we know:

sin(39°) = d / 30

And solve that equation!

But how do we calculate sin(39°) ... ?

sin(39°) = 0.6293...

So now we have:

0.6293... = d / 30

Now we rearrange it a little bit, and solve:

Start with:0.6293... = d / 30

Swap sides:d / 30 = 0.6293...

Multiply both sides by 30:d = 0.6293... x 30

Calculate:d = 18.88 to 2 decimal places

The depth the anchor ring lies beneath the hole is 18.88 m

Step By Step

These are the four steps to follow:

• Step 1 Find the names of the two sides we are using, one we are trying to find and one we already know, out of Opposite, Adjacent and Hypotenuse.

• Step 2 Use SOHCAHTOA to decide which one of Sine, Cosine or Tangent to use in this question.

• Step 3 For Sine write down Opposite/Hypotenuse, for Cosine write down Adjacent/Hypotenuse or for Tangent write down Opposite/Adjacent. One of the values is the unknown length.

• Step 4 Solve using your calculator and your skills with Algebra.

Examples

Let’s look at a few more examples:

Example: find the height of the plane.

We know the distance to the plane is 1000
And the angle is 60°

What is the plane's height?

Careful! The 60° angle is at the top, so the "h" side is Adjacent to the angle!

• Step 1 The two sides we are using are Adjacent (h) and Hypotenuse (1000).

• Step 2 SOHCAHTOA tells us to use Cosine.

• Step 3 Put our values into the Cosine equation:

  cos 60° = Adjacent / Hypotenuse
  = h / 1000

• Step 4 Solve:

Start with:cos 60° = h/1000

Swap:h/1000 = cos 60°

Calculate cos 60°:h/1000 = 0.5

Multiply both sides by 1000:h = 0.5 x 1000

 h = 500

The height of the plane = 500 meters

Example: Find the length of the side y:

• Step 1 The two sides we are using are Opposite (y)
  and Adjacent (7).

• Step 2 SOHCAHTOA tells us to use Tangent.

• Step 3 Put our values into the tangent function:

  tan 53° = Opposite/Adjacent 
  = y/7

• Step 4 Solve:

Start with:tan 53° = y/7

Swap:y/7 = tan 53°

Multiply both sides by 7:y = 7 tan 53°

Calculate:y = 7 x 1.32704...

 y = 9.29 (to 2 decimal places)

Side y = 9.29

Example: Radio Mast

There is a mast that is 70 meters high.

A wire goes to the top of the mast at an angle of 68°.

How long is the wire?

• Step 1 The two sides we are using are Opposite (70) and Hypotenuse (w).

• Step 2 SOHCAHTOA tells us to use Sine.

• Step 3 Write down:

  sin 68° = 70/w

• Step 4 Solve:

The unknown length is on the bottom (the denominator) of the fraction!

So we need to follow a slightly different approach when solving:

Start with:sin 68° = 70/w

Multiply both sides by w:w × (sin 68°) = 70

Divide both sides by "sin 68°":w = 70 / (sin 68°)

Calculate:w = 70 / 0.9271...

 w = 75.5 m (to 1 place)

The length of the wire = 75.5 m

How do you find a missing third side of a right triangle?