Pythagorean Theorem Calculator
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Pythagorean Theorem Calculator
This calculator solves the Pythagorean Theorem equation for sides a, b, or the hypotenuse c. The hypotenuse is the side of the triangle opposite the right angle.
For right triangles only, enter any two values to find the third. View the solution with steps using the Pythagorean Theorem formula.
Using the Pythagorean Theorem formula for right triangles, you can find the length of the third side if you know the length of any two other sides. Read below to see solution formulas derived from the Pythagorean Theorem:
Solve for the Length of the Hypotenuse ( c )
The length of the hypotenuse is the square root of the sum of the squares of sides ( a ) and ( b ).
[ c = sqrt{a^2 + b^2} ]
Solve for the Length of Side ( a )
The length of side ( a ) is the square root of the squared hypotenuse minus the square of side ( b ).
[ a = sqrt{c^2 – b^2} ]
Solve for the Length of Side ( b )
The length of side ( b ) is the square root of the squared hypotenuse minus the square of side ( a ).
[ b = sqrt{c^2 – a^2} ]
How to Find the Length of the Hypotenuse Using the Pythagorean Theorem
To determine the length of the hypotenuse in a right triangle, use the Pythagorean Theorem, a fundamental principle in geometry.
The Formula:
The theorem is represented by the equation:
[ c = sqrt{a^2 + b^2} ]
Here, ( c ) is the hypotenuse, and ( a ) and ( b ) are the lengths of the other two sides.
Steps to Solve:
- Square the Side Lengths:
Calculate the squares of the two shorter sides: ( a^2 ) and ( b^2 ). - Sum the Squares:
Add the results: ( a^2 + b^2 ). - Take the Square Root:
Find the square root of the sum to get the length of the hypotenuse ( c ).
Example:
Imagine a triangle where ( a = 3 ) and ( b = 4 ).
- Square these values:
( 3^2 = 9 )
( 4^2 = 16 ) - Add the squares:
( 9 + 16 = 25 ) - Calculate the square root:
( sqrt{25} = 5 )
Thus, the hypotenuse ( c ) is 5.
Tips for Calculation:
- Use a calculator for precise computations, especially with more complex numbers.
- Verify your inputs to avoid simple arithmetic errors.
This process allows for quick and accurate determination of the hypotenuse, streamlining your geometry tasks.
How to Solve for the Length of Side ( b ) Using the Pythagorean Theorem
To find the length of side ( b ) in a right triangle using the Pythagorean Theorem, follow these steps:
Understand the Formula:
The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse ( c ) is equal to the sum of the squares of the other two sides ( a ) and ( b ):
[ c^2 = a^2 + b^2 ]
Rearrange the Formula:
To solve for side ( b ), isolate ( b ) on one side:
[ b^2 = c^2 – a^2 ]
Calculate the Difference:
Subtract the square of side ( a ) from the square of the hypotenuse ( c ):
[ b^2 = c^2 – a^2 ]
Compute the Square Root:
Take the square root of the result to find the length of side ( b ):
[ b = sqrt{c^2 – a^2} ]
Example
Suppose the hypotenuse ( c ) measures 10 units and side ( a ) measures 6 units. Here’s how to calculate side ( b ):
- Calculate ( c^2 ):
( 10^2 = 100 ) - Calculate ( a^2 ):
( 6^2 = 36 ) - Find ( b^2 ):
( 100 – 36 = 64 ) - Compute the square root:
( b = sqrt{64} = 8 ) units
Using this method ensures you’re correctly applying the Pythagorean Theorem to determine the unknown side of a right triangle.
How to Solve for the Length of Side ( a ) Using the Pythagorean Theorem
To find the length of side ( a ) in a right triangle, use the Pythagorean Theorem. This theorem states that in a right-angled triangle, the square of the hypotenuse ( c ) is equal to the sum of the squares of the other two sides ( a ) and ( b ):
[ c^2 = a^2 + b^2 ]
Steps to Follow:
- Identify the Known Values:
- Length of the hypotenuse ( c ).
- Length of side ( b ).
- Rearrange the Formula:
Solve for ( a ):[ a^2 = c^2 – b^2 ]
- Calculate Side ( a ):
Subtract the square of side ( b ) from the square of the hypotenuse ( c ):[ a^2 = c^2 – b^2 ]
- Take the Square Root:
Find ( a ) by taking the square root of both sides:[ a = sqrt{c^2 – b^2} ]
Example:
Suppose the hypotenuse ( c ) is 10 units, and side ( b ) is 6 units.
- Calculate ( c^2 ):
( 10^2 = 100 ) - Calculate ( b^2 ):
( 6^2 = 36 ) - Find ( a^2 ):
( 100 – 36 = 64 ) - Compute the square root:
( a = sqrt{64} = 8 ) units
By following these steps, you can easily determine the length of side ( a ) using the Pythagorean Theorem.
Understanding the Pythagorean Theorem
The Pythagorean Theorem is a fundamental principle in geometry, specifically dealing with right triangles. It provides a way to determine the relationship between the sides of a right triangle.
Formula Breakdown
At its core, the theorem asserts that the square of the length of the hypotenuse ( c ) equals the sum of the squares of the other two sides ( a ) and ( b ). This can be expressed with the formula:
[ a^2 + b^2 = c^2 ]
In this formula:
- a and b represent the lengths of the triangle’s legs.
- c stands for the hypotenuse.
Practical Application
By knowing the lengths of any two sides of a right triangle, this theorem allows you to calculate the missing side. It’s a powerful tool in various fields including architecture, engineering, and computer graphics.
Key Points
- Right Triangle Focus: This theorem is applicable only to right triangles.
- Essential in Math: It is a cornerstone in trigonometry and is often a foundational concept taught in geometry classes worldwide.
- Real-World Relevance: Many real-life problems involve calculating distances or dimensions, where the Pythagorean Theorem can be directly applied.
The elegance and utility of the Pythagorean Theorem extend well beyond academic exercises, cementing its role in both theoretical and applied mathematics.
Understanding Pythagorean Triples
Pythagorean triples are fascinating sets of three whole numbers that neatly fit into the Pythagorean Theorem—a cornerstone of geometry. This theorem states that, for a right-angled triangle, the square of the length of the hypotenuse ( c ) is equal to the sum of the squares of the lengths of the other two sides ( a ) and ( b ):
[ a^2 + b^2 = c^2 ]
The Simplest Example
The most basic Pythagorean triple is composed of the numbers 3, 4, and 5. Here’s how it works:
- Side ( a ): 3
- Side ( b ): 4
- Hypotenuse ( c ): 5
To see the magic:
[ 3^2 + 4^2 = 9 + 16 = 25 ] [ c^2 = 5^2 = 25 ]
Both expressions equal 25, confirming that these numbers are indeed a Pythagorean triple.
Exploring More Triples
Many such triples exist beyond 3, 4, 5, such as:
- 5, 12, 13:
[ 5^2 + 12^2 = 25 + 144 = 169 ] [ 13^2 = 169 ]
- 8, 15, 17:
[ 8^2 + 15^2 = 64 + 225 = 289 ] [ 17^2 = 289 ]
Generating Pythagorean Triples
These triples aren’t just rare occurrences. They can be systematically generated using formulas, such as:
Euclid’s Formula: For any two positive integers ( m ) and ( n ) where ( m > n ):
[ a = m^2 – n^2, quad b = 2mn, quad c = m^2 + n^2 ]
This ensures you can discover infinitely many Pythagorean triples by cleverly picking values for ( m ) and ( n ).
Conclusion
Understanding these triples deepens one’s appreciation for the inherent beauty and order of mathematics. Whether you’re solving problems, constructing right-angled triangles, or simply exploring number relationships, Pythagorean triples offer an endless world of discovery.
What is a Pythagorean Triple?
A Pythagorean triple consists of three positive integers that satisfy the equation derived from the Pythagorean Theorem:
[ a^2 + b^2 = c^2 ]
In this context:
- a and b represent the legs of a right triangle.
- c is the hypotenuse.
A Classic Example: ( (3, 4, 5) )
Let’s break it down:
- Square each of the two smaller numbers:
[ 3^2 = 9 ] [ 4^2 = 16 ]
- Add these results:
[ 9 + 16 = 25 ]
- Verify with the hypotenuse:
[ 5^2 = 25 ]
Because both sides of the equation equal 25, ( (3, 4, 5) ) forms a valid Pythagorean triple. This simple example highlights how these integers harmoniously create right-angled triangles and follow the rules set out by the Pythagorean Theorem.
Note: For precise calculations, especially with more complex numbers, it’s recommended to use a calculator with square root functionality, such as those from Texas Instruments or Casio.
You can access the Pythagorean Theorem Calculator here.