The Pythagoras Theorem states that the area of square of Hypotenuse side of all right-angled triangle is equals to the sum of areas of square of two other sides of triangle.
It means, \( a^2 + b^2 = c^2 \)
Pythagorean Theorem Calculator
What is Pythagorean Theorem?
Pythagorean Theorem is another name for Pythagoras theorem. This theorem was derived 2000 years ago by a Greek mathematician Thales and his student Anaximander. They contributed their great efforts to the field of mathematics.
This theorem tells the relation in between the three sides of a right-angle triangle. The Pythagorean formula can be defined as:
“In A Right-Angle Triangle, The Sum Of The Squares Of The Two Sides Of A Right-Angle Triangle Is Equal To The Square Of The Hypotenuse”
You must be thinking what is the hypotenuse? Here is the answer!
In a right angle triangle (a triangle having 90°) has three sides;
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Base
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Perpendicular
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Hypotenuse
The base is the bottom line of a triangle, perpendicular is the straight line on which 90° angle is made and the side opposite to 90° is known as hypotenuse.
In this figure side ‘a’ is perpendicular, side ‘b’ is base and the side ‘c’ is the hypotenuse.
Pythagoras Formula
The Pythagoras formula is as follows;
\( (HYPOTENUS)^2 = (PERPENDICULAR)^2 + (BASE)^2 \)
This formula can be used for the above-mentioned figure as;
\( C² = A² + B² \)
Pythagoras Theorem Proof
There are many algebraic proofs of this theorem. However, we have presented a very simple derivation of this formula. Let’s have a look at the derivation explained below.
Consider the above diagram. In the figure, two different alignments of the right-angle triangle are displayed. One square having a right angle triangle is greater than the other square box as shown I and ii in the figure.
Proof 1:
Solving the first square box labelled as I in the figure. The four duplicates of a similar triangle are orchestrated around a square with sides’ c. The outcomes in the development of a bigger square with sides of length b + a, and area of \( (b + a)^2 \). The amount of the territory of these four triangles and the more modest square should approach the region of the bigger square which can be written as;
\( (b + a)^2 = c^2 + \dfrac{2ab}{2} \)
\( = c^2 + 2ab \)
Which gives us;
\( c^2 = (b + a)^2 – 2ab \)
\( = b^2 + 2ab + a^2 – 2ab \)
\( = a^2 + b^2 \)
Hence, \( c^2 = a^2 + b^2 \)
Proof 2:
Another proof of Pythagoras theorem is taken from the second squarer mentioned as ii in the figure.
The four duplicates of a similar triangle are organized with the end goal that they structure an encased square with sides of length \( b – a \), and region \( (b – a)^2 \). The four triangles with area ab/2 likewise structure a bigger square with sides of length c. The region of the bigger square should then rise to the amount of the zones of the four triangles and the more modest square with the end goal that:
\( =(b – a)^2 + \dfrac{2ab}{2} \)
\( =(b – a)^2 + 2ab \)
\( = b^2 – 2ab + a^2 + 2ab \)
\( =a^2 + b^2 \)
Since the bigger square has sides c and the area c2, the above can be modified as:
\( c^2 = a^2 + b^2 \)
This equation is known as Pythagoras formula.
There are many other ways but the above two derivations are the simplest to understand and widely used.
Applications of Pythagorean Theorem
Construction
Pythagorean Theorem helps you to calculate the diagonal length. This is used in construction, architecture and woodworking. For example, if you are an architecture and want to build the roof of the house than you can use Pythagoras Formula to find the slope of the roof. You can use the information to place the correct beam for supporting the roof.
Structure of Buildings
Another application of Pythagoras theorem is for making the houses and buildings square. You can calculate the sides by using right-angle triangle so that the builders can construct perfect square buildings.
Navigation
The values derived from the Pythagoras formula is also used for navigation purpose. This principle is used for finding pathways in air and sea. For example, it can help the pilot to locate the correct path to reach its destination by using the height of the plan and the distance from the airport to the destination.
Surveying
If you want to find the steepness of a mountain or a hill you can simply use this Pythagoras theorem. A surveyor uses this formula to calculate the length. He uses the stick’s height and the horizontal distance to find the length. Thus, Pythagorean Theorem is widely used to find the depth of the mountain.
Why Do You Use Pythagorean Formula?
The Pythagoras theorem is used to find one of the sides of a triangle. This formula can be used when you are given two sides of a triangle. You can use the values of two sides and can find the unknown side of a right-angle triangle.
We have gathered some basic examples for you which will tell you how to use this formula. Lets’ have a look at a few of the examples.
Basic Example
In this figure three sides are given in which the perpendicular side’s length is 7, the base side’s length is 8. In this figure, the length of the hypotenuse is unknown which needs to be figured out using the Pythagoras equation.
\( a^2 = b^2 + c^2 \)
Enter these values from the figure into the formula;
\( C^2= (7)^2 + (8)^2 \)
\( C^2= 49 + 64 \)
\( C^2 = 113 \)
Taking square of the formula;
\( C= \sqrt{113} \)
\( C= 10.630146 \)
This is how you can get the length of any side.
Similarly, you can also calculate the Alpha (α) and Beta (β) angles for side a and b respectively.
Solution for Angle (α):
\( <α = arcsin (\dfrac{a}{c}) \)
\( <α= arcsin (0.66) \)
\( <α=0.718830rad \) or \( 41.185925° \)
Solution for Angle (β):
\( <β= arcsin (\dfrac{b}{c}) \)
\( <β= arcsin (0.75) \)
\( <β=0.851966rad \) or \(48.814075° \)
You can also calculate the area of right angle triangle by using the values of a, b and c
The formula of area is as follows;
\( Area= \dfrac{a × b}{2} \)
\( Area = \dfrac{7 × 8}{2} \)
\(Area = 28 \)
The perimeter of the given triangle can be calculated as;
\(Perimeter= a+b+c\)
\( Perimeter= 7 + 8+ 10.630142 \)
\(Perimeter=25.630146\)
The solution for h is as follows;
\(h = \dfrac{a × b}{c} \)
\( h = \dfrac{7 × 8}{10.360146} \)
\(h = 5.268037\)
Further Examples:
In Pythagoras theorem the value of a and b always positive because it is the length of perpendicular and base of a triangle. The value of c is the length of hypotenuse. The Pythagoras Theorem is used to calculate the length of unknown side.
\( c^2= a^2 + b^2 \)
Example:
\( a = 3 \)
\( b = 4 \)
\( c^2 = a^2 + b^2 \)
\( c^2 = 3^2 + 4^2 \)
\( c^2 = 9 + 16 \)
\( c^2 = 25 \)
\( c = 5 \)
Area: To calculate area the value of a and b are multiply and then divide by 2.
Area = A*B/2
\( Area = \frac{a * b}{2} \)
\( Area = \frac{3 * 4}{2} \)
\( Area = 6 \)
Perimeter: The sum of base, perpendicular and hypotenuse are known as perimeter.
\( Perimeter = a + b + c \)
\( Perimeter = 3 + 4 + 5 \)
\( Perimeter = 12 \)
To calculate h: The product of A and B then divide by c is known as h.
\( H = \frac{a * b}{5} \)
\( H = \frac{3 * 4}{5} \)
\( H = \frac{12}{5} \)
\( H = 2.40 \)
To calculate ∠α: The sine inverse of the ration of a and c is known as sign alpha.
\( ∠α = sin-1(a/c) \)
\( ∠α = sin-1(3/5) \)
\( ∠α = sin-1(0.60) \)
\( ∠α = 36.869° \)
To calculate ∠β: The sine inverse of the ration of b and c is known as sign beta.
\( ∠ β = sin-1(b/c) \)
\( ∠ β = sin-1(4/5) \)
\( ∠ β = sin-1(0.80) \)
\( ∠ β = 53.130° \)