How to Calculate Using the Pythagorean Theorem
The Pythagorean Theorem is a fundamental principle in geometry that relates the lengths of the sides in a right triangle. It's an essential tool for solving problems involving right triangles and has numerous applications in mathematics, science, and engineering.
For a right triangle with legs a and b, and hypotenuse c:
\( a^2 + b^2 = c^2 \)
Where:
- a and b are the lengths of the legs (the sides adjacent to the right angle)
- c is the length of the hypotenuse (the side opposite the right angle)
Calculation Steps
- Identify the known values among the side lengths (a, b, or c).
- Determine which side length you need to calculate.
- Apply the Pythagorean Theorem formula based on the known and unknown values.
- Substitute the known values into the chosen equation.
- Solve for the unknown side length.
- If needed, use the results to calculate the area or perimeter of the triangle.
Example Calculation
Let's solve a right triangle with leg a = 3 units and leg b = 4 units.
- Use the Pythagorean Theorem: \( a^2 + b^2 = c^2 \)
- Substitute the known values: \( 3^2 + 4^2 = c^2 \)
- Calculate: \( 9 + 16 = c^2 \)
- Simplify: \( 25 = c^2 \)
- Take the square root of both sides: \( c = \sqrt{25} = 5 \)
- Calculate the area: Area = \( \frac{1}{2} \times a \times b = \frac{1}{2} \times 3 \times 4 = 6 \) square units
- Calculate the perimeter: Perimeter = a + b + c = 3 + 4 + 5 = 12 units
Visual Representation
This diagram illustrates the right triangle from our example, showing how the Pythagorean Theorem relates the sides of the triangle.