What is a "Theorem"? And Who is Pythagoras?
In math, a theorem is essentially a statement that has been proven to be true, from other known facts or accepted truths. As for Pythagoras, he was a brilliant scholar who lived long ago in ancient Greece. And he proved it. Consequently, this theorem is commonly referred to as the Pythagorean Theorem.
Historians aren't sure if Pythagoras himself proved the theorem. It's possible that his students or followers did, but most people give him the credit.
The most important thing you need to know is that the Pythagorean Theorem describes an important geometric relationship between the three sides of a right triangle. We will talk about what this relationship means in a moment. But first, you need to learn a few basic ideas before you can really understand the Pythagorean Theorem or use it to solve problems.
What Prerequisites Do You Need?
First, to understand the Pythagorean Theorem, you need to be familiar with angles and triangles, and you also need a basic understanding of powers and square roots. Therefore, if these topics are entirely new to you, please be sure to read our articles covering these subjects first. Second, although the Pythagorean Theorem falls under the realm of geometry, applying it in practice requires a grasp of basic algebra. Specifically, you need to understand variables and how to solve basic algebraic equations involving powers. We cover these topics in our "Algebra Basics" section.
What Does the Pythagorean Theorem Explain?
Now, let's take a look at what the Pythagorean Theorem actually states. The theorem can be phrased in various ways, but our preferred formulation is this: for any right triangle with two legs labeled "a" and "b," and a hypotenuse labeled "c," the relationship holds that a² + b² = c². As this definition implies, the Pythagorean Theorem does not apply to *all* triangles. It applies exclusively to right triangles.
What Is a Right Triangle?
As you may already know, a right triangle always contains one right angle. The sides that form that 90° The two sides that form the right angle are called the legs. In formulas, they are typically denoted as *a* and *b*. You need to identify which angle is the right angle. This helps you recognize what the hypotenuse is. The hypotenuse is the longest side of a right-angled triangle, and it is always located "opposite" the right angle. We designate it as *c*.
What Exactly Does This Formula Express?
Well, now that we have familiarized ourselves with the various components of the Pythagorean theorem, let's consider what this equation (*a² + b² = c²*) is actually telling us. It states that if you "square" the lengths of the two legs (i.e., *a* and *b*) and then add these two "squared values" together, the result will equal the value of the hypotenuse when it is "squared" (i.e., *c* squared). Does that still sound a bit abstract? In that case, let's immediately look at the most classic example.
3, 4, 5 Triangles
Let's look at a special example of a right-angled triangle that will help us better understand the Pythagorean theorem. This right-angled triangle is known as the "3-4-5 triangle," because the ratio of the lengths of its three sides is 3, 4, and 5.
By "ratio of lengths" here, we mean that the specific unit of measurement for the length does not matter. The side lengths can be expressed in any unit (inches, meters, miles, etc.). Therefore, as long as the ratio of the three side lengths remains 3, 4, and 5, the triangle can be of any size.
Now, let's verify whether it satisfies the Pythagorean theorem. Starting with the side that is 3 units long (let's call it "side *a*"), if we square this side, 3², which equals 9. Next, let's look at the side that is 4 units long("side *b*"), 4², which equals 16.
Finally, let's address the hypotenuse ("side *c*"), the longest side. It is 5 units long. 5² = 25. If we add "a²" and "b²" together, they indeed equal "c²," because 9 + 16 = 25. It truly satisfies the Pythagorean theorem.
The Most Common Uses
Given Two Sides, Find the Third Side
The Pythagorean theorem is a useful tool that helps you use known information to calculate unknown information. Specifically, if you have a right-angled triangle but only know the lengths of two of its sides, the theorem tells you how to calculate the length of the third, unknown side.
Example 1: Given the two legs, find the Hypotenuse
For instance, suppose we have a right-angled triangle where one side is 2 centimeters long and another side is 3 centimeters long, but we do not know how long the hypotenuse is. No problem! The Pythagorean theorem describes the relationship between the three sides of any right-angled triangle, so we can calculate it. We know that "a² + b² =" ...c².” So, let's substitute the known values into this equation and solve for the unknown variable.
Once again, it does not matter which of the two legs is designated as “a” or “b.” So, we will label them as such, and then, within the Pythagorean theorem equation, we will replace “a” with 2 and “b” with 3. This gives us an algebraic equation with only one unknown variable: “c.” We now have “2² + 3² = c²,” and we need to solve for “c.”
Step 1: 2² + 3² = c²
Step 2: 4 + 9 = c²
Step 3: c² = 4 + 9 = 13
Step 4: c = √13
If you require a precise numerical value, you can use a calculator to find the decimal approximation. However, in mathematics, unless a square root can be easily simplified, it is typically left in its radical form. Therefore, the side lengths of this right triangle are 2 cm, 3 cm, and √13 cm.
Example 2: Given the hypotenuse and one leg, find the other leg
Let's look at another example. For this right triangle, we know the length of the hypotenuse (6 meters) and the length of one of the legs (4 meters), but the length of the other leg is unknown. Therefore, we will use the Pythagorean theorem to determine this unknown length.
As usual, we will designate the hypotenuse as side “c.” We will label the known leg as side “a” and the unknown leg as side “b.” We can then substitute the known values into the Pythagorean theorem to solve for the unknown variable. Replacing “c” with 6 and “a” with 4 yields the equation “4² + b² = 6²”; we now need to simplify this equation and solve for “b.”
Step 1: 4² + b² = 6²
Step 2: 16 + b² = 36
Step 3: b² = 36 - 16 = 20
Step 4: b = √20 = 2√5
If you are unsure how to simplify √20... How do you simplify this to 2√5? You can simply leave the answer as √20 meters. We won't delve deeply into simplifying radicals here. If you would like to learn how to perform such simplifications, you can study the step-by-step solutions provided in the Simplify Calculator.
Determine if It Is a Right Triangle
That covers how to use the Pythagorean theorem to find the length of an unknown side in a right triangle. However, I would also like to mention another way the Pythagorean theorem can be used. You can also use the theorem to "check" a triangle to see if it is, in fact, a right triangle.
For example, suppose someone shows you this triangle and asks: "Is this a right triangle?" Well, it certainly looks like one, but judging by eye, it is difficult to determine whether that angle is *exactly* 90 degrees. Perhaps it is very close, say 89 degrees. Don't worry, as long as you know the lengths of the triangle's three sides, the Pythagorean theorem can provide a definitive answer.
If we know the lengths of the three sides (labeled a, b, and c), we can substitute those values into the formula to verify whether the equation holds. For instance, if a triangle has two shorter sides measuring 3 centimeters each, and a longest side measuring 4 centimeters, we can substitute these values for "a," "b," and "c," and then simplify the expression to see what the result is.
3² + 3² ?= 4²
3² + 3² =18 = 16. It doesn't look quite right! There is no doubt that this triangle is not a right triangle.
Final Summary
Now you understand what the Pythagorean theorem is and how to apply it. You can use it to find the length of an unknown side in any right-angled triangle, or to verify whether a given triangle meets the criteria for being a right-angled triangle. Remember that you cannot learn mathematics simply by reading. You have to practice all the time.
