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Pythagorean theorem calculator finds the unknown side length of a right triangle. It also calculates angles, area, perimeter, and altitude to hypotenuse.
a = 3 area A = 6
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This Pythagorean calculator finds the length of a side of a right triangle if the other two sides of the triangle are known. The calculations are performed based on the Pythagorean theorem.
Enter the known side lengths and press "Calculate." The calculator will return the following values:
The calculator will also return the detailed solution, which you can expand by pressing "+ Show Calculation Steps."
Note that the input fields for each side include a whole number part and a square root part so that you can conveniently enter values like 2√3, √3, etc.
Note also that the values of a and b, the legs of the triangle, have to be shorter than the value of c, the hypotenuse.
Pythagoras' theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the cathetuses.
The pythagorean theorem can be written as follows:
a² + b² = c²,
Where a and b are the lengths of the shorter sides, or legs, of a right triangle, and c – is the length of the longest side or hypotenuse. The equation above can be described as follows: a squared plus b squared equals c squared.
Let's prove the Pythagorean theorem by adding up the areas.
In the above image, the square with the side (a + b) is made up of a square with side c, and four right triangles with sides a, b, and c. Let's find the area of this square using two different strategies:
A = (a + b)²
A = c² + 4 × (ab)/2 = c² + 2ab
Since both of these calculations describe the same surface area, we can equate them:
(a + b)² = c² + 2ab
Expanding the square on the left side of the equation, we get:
a² + 2ab + b² = c² + 2ab
Subtracting 2ab from both sides of the equation, we get:
a² + b² = c²
which is the required result.
Finding the sides of a right triangle
If two sides of a right triangle are given, the third side can be found using the Pythagorean theorem. For example, if sides a and b are given, the length of side c can be found as follows:
If all three sides of the right triangle are known, the non-90° angles of the triangle can be found as follows:
∠α = arcsin(a/c) ∠β = arcsin(b/c)
The area of a right triangle can be calculated as 1/2 of the product of its legs:
A = 1/2 × (ab) = (ab)/2
The perimeter of a right triangle is calculated as a sum of all its sides:
P = a + b + c
If all three sides of a right triangle are known, the altitude to hypotenuse, h, can be found as follows:
h = (a × b)/c
The pythagorean theorem is widely used in architecture and construction to calculate the lengths of the necessary component and ensure the angles in constructed buildings are right. Let's look at an example of applying the theorem.
Imagine you are moving, and you hired a moving truck with a length of 4 meters and a height of 3 meters. You don't have many bulky items, but you do own a ladder, which is 4.5 meters long. Will your ladder fit into the truck?
Since the ladder length, 4.5 meters, exceeds the length of the truck, 4 meters, the only way the ladder will fit inside is diagonal. To determine whether that's possible, we need to use the Pythagorean theorem to calculate the hypotenuse of a triangle with the sides equal to the length and height of the truck. Therefore, in our case a = 4, b = 3, and we need to find c:
The hypotenuse of a triangle with a = 4 and b = 3 is c = 5. Therefore, the longest object that can fit into the truck can be 5 meters. Your ladder is 4.5 meters long. Therefore, it will easily fit!
Yes, the ladder will fit.
This online calculator will also find some additional characteristics of the given triangle. Calculate these characteristics for the triangle with a = 4, b = 3, and c = 5.
Area of the triangle:
A = (ab)/2 = (3 × 4)/2 = 12/2 = 6
Perimeter of the triangle:
P = a + b + c = 3 + 4 + 5 = 12
Altitude to hypotenuse:
h = (a × b)/c = (3 × 4)/5 = 12/5 = 2.4
Angle opposite to side a:
∠α = arcsin(a/c) = arcsin(4/5) = arcsin(0.8) = 53.13° = 53°7'48" = 0.9273 rad
Angle opposite to side b:
∠β = arcsin(b/c) = arcsin(3/5) =arcsin(0.6) = 36.87° = 36°52'12" = 0.6435 rad