Hypotenuse Calculator is a free online tool that helps to find the hypotenuse of a right-angled triangle. The hypotenuse is the longest side of a right triangle. Furthermore, it is the side opposite the right angle.
What is a Hypotenuse Calculator?
Hypotenuse Calculator helps to calculate the hypotenuse of a right triangle with a given base and height. The Pythagoras Theorem is used to calculate the hypotenuse of a right triangle. To use the hypotenuse calculator, enter the values in the given input boxes.
Hypotenuse Calculator
NOTE: Enter values upto 3 digits only.
How to Use the Hypotenuse Calculator?
Follow the steps given below to find the hypotenuse of a right triangle using the hypotenuse calculator:
- Step 1: Go to Cuemath's online Hypotenuse Calculator.
- Step 2: Enter the base and height of the triangle in the respective input boxes.
- Step 3: Click on the "Calculate" button to find the length of the hypotenuse.
- Step 4: Click on the "Reset" button to clear the fields and enter new values.
How Does Hypotenuse Calculator Work?
A triangle in which one angle measures 90 degrees and the remaining two angles are acute is known as a right-angled triangle or right triangle. The hypotenuse of a right triangle can be determined by the Pythagoras Theorem. This theorem gives the fundamental relationship between the three sides of a right triangle. It states that the sum of squares of the height and the base of a right triangle will be equal to the square of the hypotenuse. Moreover, the three sides of a right triangle are also known as Pythagorean triples. The formula for the Pythagoras theorem is given by:
Hypotenuse2 = (Base2 + Height2)
The steps to find the hypotenuse of a right triangle are given below:
- Step 1: Square the base and the height; Base2, Height2
- Step 2: Add the two values obtained in step 1; Base2 + Height2
- Step 3: Take the square root of the value from step 2 to get the length of the hypotenuse; Hypotenuse = √(Base2 + Height2)
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Solved Examples on Hypotenuse Calculator
Example 1:
Find the hypotenuse of a right-angled triangle if its base is 3 units and height is 4 units. Verify it using the online hypotenuse calculator.
Solution:
Hypotenuse = √(Base2 + Height2)
= √(32 + 42)
= √25
= 5 units.
Example 2:
Find the hypotenuse of a right-angled triangle if its base is 2.5 units and height is 3.2 units. Verify it using the online hypotenuse calculator.
Solution:
Hypotenuse = √(Base2 + Height2)
= √(2.52 + 3.22)
= √16.49
= 4.061 units.
Now, you can try the hypotenuse calculator to find the hypotenuse of the triangles with the following dimensions:
- Base = 7.2 units, Height = 15.6 units.
- Base = 23 units, Height = 47 units.
☛ Related Articles:
- Hypotenuse
- Right Angled Triangle
- Pythagoras Theorem
☛ Math Calculators:
Pythagoras
Over 2000 years ago there was an amazing discovery about triangles:
When a triangle has a right angle (90°) ...
... and squares are made on each of the three sides, ...
geometry/images/pyth2.js
... then the biggest square has the exact same area as the other two squares put together!
It is called "Pythagoras' Theorem" and can be written in one short equation:
a2 + b2 = c2
Note:
- c is the longest side of the triangle
- a and b are the other two sides
Definition
The longest side of the triangle is called the "hypotenuse", so the formal definition is:
In a right angled triangle:
the square of the hypotenuse is equal to
the sum of the squares of the other two sides.
Sure ... ?
Let's see if it really works using an example.
Example: A "3, 4, 5" triangle has a right angle in it.
Let's check if the areas are the same: 32 + 42 = 52 Calculating this becomes: 9 + 16 = 25 It works ... like Magic! |
Why Is This Useful?
If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!)
How Do I Use it?
Write it down as an equation:
a2 + b2 = c2 |
Then we use algebra to find any missing value, as in these examples:
Example: Solve this triangle
Start with:a2 + b2 = c2
Put in what we know:52 + 122 = c2
Calculate squares:25 + 144 = c2
25+144=169:169 = c2
Swap sides:c2 = 169
Square root of both sides:c = √169
Calculate:c = 13
Read Builder's Mathematics to see practical uses for this.
Also read about Squares and Square Roots to find out why √169 = 13
Example: Solve this triangle.
Start with:a2 + b2 = c2
Put in what we know:92 + b2 = 152
Calculate squares:81 + b2 = 225
Take 81 from both sides: 81 − 81 + b2 = 225 − 81
Calculate: b2 = 144
Square root of both sides:b = √144
Calculate:b = 12
Example: What is the diagonal distance across a square of size 1?
Start with:a2 + b2 = c2
Put in what we know:12 + 12 = c2
Calculate squares:1 + 1 = c2
1+1=2: 2 = c2
Swap sides: c2 = 2
Square root of both sides:c = √2
Which is about:c = 1.4142...
It works the other way around, too: when the three sides of a triangle make a2 + b2 = c2, then the triangle is right angled.
Example: Does this triangle have a Right Angle?
Does a2 + b2 = c2 ?
- a2 + b2 = 102 + 242 = 100 + 576 = 676
- c2 = 262 = 676
They are equal, so ...
Yes, it does have a Right Angle!
Example: Does an 8, 15, 16 triangle have a Right Angle?
Does 82 + 152 = 162 ?
- 82 + 152 = 64 + 225 = 289,
- but 162 = 256
So, NO, it does not have a Right Angle
Example: Does this triangle have a Right Angle?
Does a2 + b2 = c2 ?
Does (√3)2 + (√5)2 = (√8)2 ?
Does 3 + 5 = 8 ?
Yes, it does!
So this is a right-angled triangle
And You Can Prove The Theorem Yourself !
Get paper pen and scissors, then using the following animation as a guide:
- Draw a right angled triangle on the paper, leaving plenty of space.
- Draw a square along the hypotenuse (the longest side)
- Draw the same sized square on the other side of the hypotenuse
- Draw lines as shown on the animation, like this:
- Cut out the shapes
- Arrange them so that you can prove that the big square has the same area as the two squares on the other sides
Another, Amazingly Simple, Proof
Here is one of the oldest proofs that the square on the long side has the same area as the other squares.
Watch the animation, and pay attention when the triangles start sliding around.
You may want to watch the animation a few times to understand what is happening.
The purple triangle is the important one.
We also have a proof by adding up the areas.
Historical Note: while we call it Pythagoras' Theorem, it was also known by Indian, Greek, Chinese and Babylonian mathematicians well before he lived.
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