Finding the measurement of the third side of a triangle when you know the measurement of the other two sides only works if you have a right triangle or the measurement of at least one other angle. Without this information you do not have enough data in order to find out the length of the third side. A right triangle has a built in third angle, as one of the angles has to be 90 degrees.
Right Triangle Using Pythagorean Theorem
Draw the triangle on your paper labeling the two sides adjacent to the right angle, or legs, “a” and “b.” Label the hypotenuse, or third side “c.”
Set up your equation so that
a^2 + b^2 = c^2
This is the Pythagorean Theorem used for solving for the unknown side.
Fill in the lengths you know in the equation. The hypotenuse is always the longest side in a right triangle. This is a great way to check your work, because if either of the legs is longer than the hypotenuse, you know you have made an error.
Solve for the unknown side. If you are solving for the hypotenuse, you fill in the “a” and “b,” square both numbers and then add the numbers together. Use your calculator to get the square root of the resulting sum to reach your answer. If you are solving for one of the legs of, you need to move the other leg to the same side as the “c” by subtracting. This leaves the remaining leg alone, allowing you to solve for it. This means you square the “c” number and the known leg. Subtract the squared leg value from the squared c value. Get the square root of the resulting number and you have your answer for the unknown leg.
Using the Law of Sines
Set up the triangle so that the side opposite the angle is matched with the angle. Label the side opposite angle A as a, the side across from angle B as b and the side opposite angle C as c.
Write the equation out to read
\frac{a}{\sin A}= \frac{b}{\sin B} = \frac{c}{\sin C}
This gives you the basics for solving for your unknown side.
Take the angle you know and use the calculator to determine the sine of that angle. Most scientific calculators have you enter the angle number and then hit the button labeled “sin.” Write down the value.
Divide the length of the side associated with the angle by the value of the sin of that angle. This gives you a number typically written as an approximation, as the decimal places go off indefinitely. Call this new number X for the purpose of this example.
Take the value of the other known side and divide it by X. This new number equals the sine of the new angle.
Enter the number in the calculator and hit the “sin-1” to get the angle in degrees. You can now solve for the angle of the unknown side.
Add the two known angles together and subtract the total from 180. All angles inside a triangle must add up to 180 degrees.
Calculate the sine of the new angle by entering it in the calculator and hitting the “sin” button. Multiply the answer by X and this gives you the length of the unknown side.
For an example using the Pythagorean Theorem as well as a new method, solving using the Law of Cosines, watch the video below:
Tip: Law of Sines can be worked as stated or by inverting all of the information so that the sine of the angle is divided by the length of the side.
Warning: Draw the problem to see what you are multiplying and dividing in order to ensure you understand how the problem is working. Remember, you must do the same thing to both sides of the equation in order to keep the sides equal.
Video transcript
- [Instructor] The triangle shown below has an area of 75 square units. Find the missing side. So pause the video and see if you can find the length of this missing side. Alright, now let's work through this together. They give us the area, they give us this side right over here, this 11. They give us this length 10, which, if we rotate this triangle you can view it as an altitude. And in fact let me do that. Let me rotate this triangle, because then I think it might jump out at you how we can tackle this. So let me copy and let me paste it. So if I move it here, but I'm gonna rotate it. So if I rotate, that is our rotated triangle and now it might be a little bit clearer what we're talking about, this length x that we want to figure out, this is our base. And they give us our height and they give us our area. And we know how base, height and area relate for a triangle. We know that area is equal to 1/2 times the base times the height, and they tell us that our area is 75 units squared. So this is 75 is equal to 1/2. What is our base? Our base is the variable x. So let's just write that down. 1/2 times x and then what is our height? Well, our height is actually the 10. If x is the length of our base, then the height of our triangle is gonna be 10, we actually don't even need to use this 11. They're putting that there just to distract you. So, this is going to be our height, times 10. So 75 is equal to 1/2 times x times 10, or, let me just rewrite it this way. We can say 75 is equal to 1/2 times 10 is equal to five times x is equal to five, let me do the x in that same color, is equal to five times x. So what is x going to be? There's a couple of ways you could think about it. You could say five times what is equal to 75? And you might be able to figure that out. You might say, OK, five times 10 is 50, and then let's see, I need another 25, so put another five there, so it's really five times 15, or you could do it a little bit more systematically. You can divide both sides by what you're multiplying by x. So if you divide this side by five, five times x divided by five, well, you're just going to have an x left over. But these two things were equal, so you can't just do it to one side, you have to do it to both sides. So you have to divide both sides by five. And what's 75 divided by five? Well that is 15. So you get x is equal to 15. And you can verify that. If x is equal to 15, base times height times 1/2. Well, it's 15 times 10 times 1/2, or 15 times five which is going to be 75 square units.