# What is Pythagorean Theorem Formula

The Pythagorean theorem is sometimes referred to as Pythagoras theorem. It is a fundamental concept in Euclidean geometry. It provides the relation between three sides of a right-angled triangle.

## The Pythagorean Theorem Statement

The Pythagoras theorem says that the square of the hypotenuse is equal to the sum of the squares of the perpendicular and the base.

The Pythagorean Theorem formula is given as:

Where H is the length of the hypotenuse, P is the length of the perpendicular, and B is the length of the base.

## Pythagorean Theorem Explanation

In a right-angled triangle, the side opposite to the right angle is the largest and is called the hypotenuse. While the other two sides are called the base and perpendicular. The side opposite to the angle you’re interested in is called the perpendicular and the other one becomes the base.

According to the Pythagorean Theorem formula, if two of the sides are known the third can be calculated using the above-mentioned relation.

## A Brief History of the Pythagorean Theorem

The Pythagorean Theorem is named after Pythagoras who was a Greek mathematician. Although evidence indicates that the ancient mathematicians of Babylon, Mesopotamia, China, and India might have used the theorem, Pythagoras is credited to be the first one to provide mathematical proof for it. The others also provided proofs but for some special cases of the theorem.

## More about Pythagorean Theorem formula

We now have several proofs of the theorem; in fact, Pythagorean Theorem has the most proofs than any other mathematical theorem. It has geometric as well as algebraic proofs dating thousands of years back.

Moreover, the theorem can be generalized to provide relations for higher-dimensional spaces and even non-Euclidean spaces. In fact, it can be generalized for objects that are not right-angled triangles to n-dimensional objects that are not even triangles.

### An Interesting Fact

The Pythagoras theorem is not only used by mathematicians, it is also used by businessmen and other people outside the mathematical world. The theorem is taken as the symbol of mathematical perplexity, mystique, and intellectual power. It is a popular reference in plays, songs, stamps, cartoons, and literature.

## Proof by Rearrangement

As discussed above, the Pythagoras theorem is thought to date back to even before Pythagoras. However, he is most probably the first one to prove it. This is called the proof by rearrangement.

By William B. Faulk – Own work, CC BY-SA 4.0

Observe the two larger squares shown in the figure. Each square has four triangles. The difference between them is that the triangles are arranged in a different manner. This means that the white portion of the larger squares must have the same area. If you equate the two areas, you will end up with the Pythagorean Theorem formula shown in figure.

Here, c is taken as the length of the hypotenuse, a is the length of the perpendicular, and b is the length of the base.

The area of the smaller square in the first figure is:

Area1=cxc=c2

The area of the smaller squares in the other figure is:

Area2=a2

and

Area3=b2

If both figures are same and only differ in the manner in which their triangles are arranged, then these areas can be equated as:

Area1=Area2+Area3

c2=a2+b2

This proof is called the Pythagorean proof; however, several other proofs are also available. This particular proof is also described in the writings of Proclus, another Greek philosopher of a later time.

## Other Proofs of the Pythagorean Theorem Formula

As we discussed before that the Pythagorean Theorem Formula is given by:

a2+ b2= c2

Here, c is the length of the hypotenuse, a is the length of the perpendicular, and b is the length of the base.

Let’s suppose you know the length of the base and perpendicular, you can calculate the length of hypotenuse by using the formula:

c = a2+ b2

If you are given the length of the hypotenuse and one side, either a or b, the other side can be found out by using the following formula:

a = c2- b2

or

b = c2- a2

So, we can say that if any of the two sides of a right-angled triangle are given, the third side can be determined using the Pythagorean theorem formula. However, the length of the hypotenuse will always be greater than the other two sides and always remain less than their sum.

## Other Formulas Derived from the Pythagoras Theorem

### Law of cosines:

The Pythagoras theorem forms the base of several other theorems and formulas. One of these is the law of cosines.

Using the law of cosines, you can determine any side of a triangle if the other two sides and the angle between them is given. The law of cosines is given by the formula:

c2= a2+ b2- 2abcos γ

Here a, b, and c are the sides of the triangle. γ is the angle between a and b, and is opposite to the side c.

If the angle γ is equal to 90 degrees, then its cosine will be zero. The law of cosines is therefore reduced to the Pythagoras theorem formula.

c2= a2+ b2

### Distance Formula

The distance formula states that:

d=∆x2+∆y2

Here

x = x1 – x2 and y = y1 – y2

If you square on both sides, you end up with the Pythagoras Theorem:

d2 = x2 + y2

### Example 1:

If a right triangle has a 2m base and 3m perpendicular, what is the length of the hypotenuse?

Solution:

Using the Pythagoras Theorem, we know that

c2=a2+b2

So,

c=a2+b2

The length of the hypotenuse then comes out to be,

c=22+32=13m

### Example 2:

Find the length of the base of a right triangle if the length of the hypotenuse is 55m and that of the perpendicular is 10m.

Solution:

Using the Pythagoras Theorem, we know that

c2=a2+b2

Separating the length of the base, we get

b2=c2-a2

b=c2-a2

Using the values given in the question, the base is found to be

b=552-102=54.083m

### Example 3:

Consider a right triangle abc. If the base is 30m wide and the hypotenuse is measured to be 50m, find out the length of the perpendicular of the triangle.

Solution:

Using the Pythagoras theorem, we know that

c2=a2+b2

Now, simple separate the length of the perpendicular to get

a2=c2-b2

And

a=c2-b2

Using the values given in the question, we find the perpendicular as,

a=502-302

a=900

a=30m

### Example 4:

If the lengths of the sides of a triangle ABC are given below, determine whether it is a right-angled triangle or not.

AB=45, BC=55, CA=75

Solution:

For any right-angled triangle, the Pythagoras theorem should be satisfied. We know that in a right-angled triangle the largest of the three sides is the hypotenuse. Therefore, in this case, we will consider CA=75 as the hypotenuse. Furthermore, let AB=45 be the base and BC=55 be the perpendicular.

Now, the Pythagoras Theorem states that,

c2=a2+b2

Taking the left hand-side of the equation:

c2=752

c2=5625

Now, taking the right hand-side of the equation,

a2+b2=452+552

a2+b2=2025+3025

a2+b2=5050

Since, left hand-side and right hand-side of the equations are not equal; therefore, the given triangle is not a right-angled triangle.

### Example 5:

Repeat example 4 with the triangle PQR. Where:

PQ=28, QR=53, and PR=45.

Solution:

For any right-angled triangle the Pythagoras theorem should be saitsfied. We know that in a right-angled triangle the largest of the three sides is the hypotenuse. Therefore, in this case, we will consider QR=53 as the hypotenuse. Furthermore, let PQ=28 be the base and PR=45 be the perpendicular.

Now, the Pythagoras Theorem states that,

c2=a2+b2

Taking the left hand-side of the equation:

c2=532

c2=2809

Now, taking the right hand-side of the equation,

a2+b2=282+452

a2+b2=784+2025

a2+b2=2809

Since, left hand-side and right hand-side of the equations are equal; therefore, the given triangle is a right-angled triangle.

# Our Service Charter

1. ### Excellent Quality / 100% Plagiarism-Free

We employ a number of measures to ensure top quality essays. The papers go through a system of quality control prior to delivery. We run plagiarism checks on each paper to ensure that they will be 100% plagiarism-free. So, only clean copies hit customers’ emails. We also never resell the papers completed by our writers. So, once it is checked using a plagiarism checker, the paper will be unique. Speaking of the academic writing standards, we will stick to the assignment brief given by the customer and assign the perfect writer. By saying “the perfect writer” we mean the one having an academic degree in the customer’s study field and positive feedback from other customers.
2. ### Free Revisions

We keep the quality bar of all papers high. But in case you need some extra brilliance to the paper, here’s what to do. First of all, you can choose a top writer. It means that we will assign an expert with a degree in your subject. And secondly, you can rely on our editing services. Our editors will revise your papers, checking whether or not they comply with high standards of academic writing. In addition, editing entails adjusting content if it’s off the topic, adding more sources, refining the language style, and making sure the referencing style is followed.
3. ### Confidentiality / 100% No Disclosure

We make sure that clients’ personal data remains confidential and is not exploited for any purposes beyond those related to our services. We only ask you to provide us with the information that is required to produce the paper according to your writing needs. Please note that the payment info is protected as well. Feel free to refer to the support team for more information about our payment methods. The fact that you used our service is kept secret due to the advanced security standards. So, you can be sure that no one will find out that you got a paper from our writing service.
4. ### Money Back Guarantee

If the writer doesn’t address all the questions on your assignment brief or the delivered paper appears to be off the topic, you can ask for a refund. Or, if it is applicable, you can opt in for free revision within 14-30 days, depending on your paper’s length. The revision or refund request should be sent within 14 days after delivery. The customer gets 100% money-back in case they haven't downloaded the paper. All approved refunds will be returned to the customer’s credit card or Bonus Balance in a form of store credit. Take a note that we will send an extra compensation if the customers goes with a store credit.

We have a support team working 24/7 ready to give your issue concerning the order their immediate attention. If you have any questions about the ordering process, communication with the writer, payment options, feel free to join live chat. Be sure to get a fast response. They can also give you the exact price quote, taking into account the timing, desired academic level of the paper, and the number of pages.

Excellent Quality
Zero Plagiarism
Expert Writers

or

Instant Quote