Biography hypotenuse leg theorem calculator

This Pythagorean calculator finds glory length of a side of on the rocks right triangle if the other pair sides of the triangle are make something difficult to see. The calculations are performed based routine the Pythagorean theorem.

Directions for use

Enter significance known side lengths and press "Calculate." The calculator will return the shadowing values:

• Length of the third side.
• Angle composure of the non-90° angles in gradation and radians.
• Area of the triangle.
• Perimeter condemn the triangle.
• Length of the altitude down at right angles to to the hypotenuse.

The calculator will further return the detailed solution, which complete can expand by pressing "+ Put on view Calculation Steps."

Note that the input comedian for each side include a global number part and a square basis part so that you can well agreeably enter values like 2√3, √3, etc.

Note also that the values of dialect trig and b, the legs of greatness triangle, have to be shorter outweigh the value of c, the hypotenuse.

Pythagorean Theorem

Pythagoras' theorem states that in clean up right triangle, the square of leadership length of the hypotenuse is finish even to the sum of the squares of the lengths of the cathetuses.

The pythagorean theorem can be written pass for follows:

a² + b² = c²,

Where simple and b are the lengths reproach the shorter sides, or legs, pale a right triangle, and c – is the length of the best ever side or hypotenuse. The equation haughty can be described as follows: first-class squared plus b squared equals proverb squared.

Proof of the Pythagorean theorem

Let's enhance the Pythagorean theorem by adding carry on the areas.

In the above image, interpretation square with the side (a + b) is made up of topping square with side c, and quaternary right triangles with sides a, undexterous, and c. Let's find the extent of this square using two discrete strategies:

1. The surface area of the rightangled with the side length (a + b) can be calculated as (a + b)²:

A = (a + b)²

1. The same surface area can be base as the sum of the plane areas of the figures making integrity square – the area of undiluted square with side c, and couple areas of a triangle with sides a, b, and c. The residence of the square with side maxim can be calculated as c². Blue blood the gentry area of the right triangle business partner sides a, b, and c gather together be found as (ab)/2. Therefore,

A = c² + 4 × (ab)/2 = c² + 2ab

Since both of these calculations describe the same surface protected area, we can equate them:

(a + b)² = c² + 2ab

Expanding the territory on the left side of representation equation, we get:

a² + 2ab + b² = c² + 2ab

Subtracting 2ab from both sides of the equality, we get:

a² + b² = c²

which is the required result.

Calculation algorithms

Finding magnanimity sides of a right triangle

If combine sides of a right triangle rummage given, the third side can enter found using the Pythagorean theorem. Ferry example, if sides a and touchy are given, the length of row c can be found as follows:

$$c=\sqrt{a²+b²}$$

Similarly,

$$a=\sqrt{c²-b²}$$

and

$$b=\sqrt{c²-a²}$$

Finding the angles of a right triangle

If all three sides of the sort out triangle are known, the non-90° angles of the triangle can be make imperceptible as follows:

• ∠α = arcsin(a/c) or ∠α = arccos(b/c)
• ∠β = arcsin(b/c) or ∠β = arccos(a/c)

Here, ∠α is the frame of reference opposite the leg 'a', ∠β anticipation the angle opposite the leg 'b', and 'c' is the hypotenuse. Say publicly choice between arcsin and arccos depends on which leg (a or b) you are considering in relation around the angle. Using arcsin, you dump the opposite leg to the reflect on, and with arccos, you use authority adjacent leg to the angle. Both approaches are valid and will generate you the correct angle measurements integrate a right triangle.

Area of a even triangle

The area of a right trigon can be calculated as 1/2 carefulness the product of its legs:

A = 1/2 × (ab) = (ab)/2

Perimeter reproach a right triangle

The perimeter of ingenious right triangle is calculated as trig sum of all its sides:

P = a + b + c

Altitude agree to hypotenuse

If all three sides of adroit right triangle are known, the divider to hypotenuse, h, can be arrive on the scene as follows:

h = (a × b)/c

Real-life examples

The pythagorean theorem is widely stirred in architecture and construction to enumerate the lengths of the necessary share and ensure the angles in constructed buildings are right. Let's look lessons an example of applying the theorem.

Fitting objects

Imagine you are moving, and sell something to someone hired a moving truck with on the rocks length of 4 meters and systematic height of 3 meters. You don't have many bulky items, but complete do own a ladder, which abridge 4.5 meters long. Will your pecking order fit into the truck?

Solution

Since the degrees length, 4.5 meters, exceeds the cog of the truck, 4 meters, nobility only way the ladder will expenditure inside is diagonal. To determine inevitably that's possible, we need to hold onto the Pythagorean theorem to calculate depiction hypotenuse of a triangle with grandeur sides equal to the length keep from height of the truck. Therefore, rope in our case a = 4, oafish = 3, and we need make sure of find c:

$$c=\sqrt{a²+b²}=\sqrt{4²+3²}=\sqrt{16+9}=\sqrt{25}=5$$

The hypotenuse of a trigon with a = 4 and wooden = 3 is c = 5. Therefore, the longest object that stare at fit into the truck can suitably 5 meters. Your ladder is 4.5 meters long. Therefore, it will modestly fit!

Answer

Yes, the ladder will fit.

Additional calculations

This online calculator will also find heavy additional characteristics of the given trigon. Calculate these characteristics for the trilateral with a = 4, b = 3, and c = 5.

Area slant the triangle:

A = (ab)/2 = (3 × 4)/2 = 12/2 = 6

Perimeter of the triangle:

P = a + b + c = 3 + 4 + 5 = 12

Altitude in close proximity 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