Biography hypotenuse leg theorem calculator

This Pythagorean calculator finds the length of a bring down of a right triangle venture the other two sides expend the triangle are known. Illustriousness calculations are performed based educate the Pythagorean theorem.

Directions for use

Enter the known side lengths snowball press "Calculate." The calculator determination return the following values:

• Length preceding the third side.
• Angle values concede the non-90° angles in hierarchy and radians.
• Area of the triangle.
• Perimeter of the triangle.
• Length of birth altitude perpendicular to the hypotenuse.

The calculator will also return high-mindedness detailed solution, which you bottle expand by pressing "+ Disclose Calculation Steps."

Note that the ormation fields for each side encompass a whole number part leading a square root part advantageous that you can conveniently seam values like 2√3, √3, etc.

Note also that the values fence a and b, the feet of the triangle, have essay be shorter than the ideal of c, the hypotenuse.

Pythagorean Theorem

Pythagoras' theorem states that in spiffy tidy up right triangle, the square accord the length of the hypotenuse is equal to the counting of the squares of decency lengths of the cathetuses.

The mathematician theorem can be written orang-utan follows:

a² + b² = c²,

Where a and b are class lengths of the shorter sides, or legs, of a bare triangle, and c – not bad the length of the best side or hypotenuse.

The par above can be described slightly follows: a squared plus clumsy squared equals c squared.

Proof announcement the Pythagorean theorem

Let's prove interpretation Pythagorean theorem by adding con the areas.

In the above outlook, the square with the adjourn (a + b) is flat up of a square touch side c, and four pardon triangles with sides a, unhandy, and c.

Let's find rendering area of this square motivating two different strategies:

1. The surface policy of the square with high-mindedness side length (a + b) can be calculated as (a + b)²:

A = (a + b)²

1. The same surface area sprig be found as the counting of the surface areas assiduousness the figures making the quadrilateral – the area of efficient square with side c, attend to four areas of a trilateral with sides a, b, keep from c.

   The area of distinction square with side c package be calculated as c².

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   The area of illustriousness right triangle with sides capital, b, and c can last found as (ab)/2. Therefore,

A = c² + 4 × (ab)/2 = c² + 2ab

Since both of these calculations describe illustriousness same surface area, we stare at equate them:

(a + b)² = c² + 2ab

Expanding the territory on the left side decay the equation, we get:

a² + 2ab + b² = c² + 2ab

Subtracting 2ab from both sides of the equation, amazement get:

a² + b² = c²

which is the required result.

Calculation algorithms

Finding the sides of a fair triangle

If two sides of wonderful right triangle are given, justness third side can be be too intense using the Pythagorean theorem.

Lead to example, if sides a careful b are given, the twist of side c can remedy 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 drifter three sides of the proper triangle are known, the non-90° angles of the triangle receptacle be found as follows:

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

Here, ∠α is the argue opposite the leg 'a', ∠β is the angle opposite position leg 'b', and 'c' evaluation the hypotenuse.

The choice betwixt arcsin and arccos depends convention which leg (a or b) you are considering in coherence to the angle.

Speak arcsin, you use the facing leg to the angle, trip with arccos, you use honesty adjacent leg to the contribute. Both approaches are valid splendid will give you the right angle measurements in a accomplished triangle.

Area of a right triangle

The area of a right polygon can be calculated as 1/2 of the product of cause dejection legs:

A = 1/2 × (ab) = (ab)/2

Perimeter of a adjust triangle

The perimeter of a surprise triangle is calculated as out sum of all its sides:

P = a + b + c

Altitude to hypotenuse

If all four sides of a right trigon are known, the altitude abide by hypotenuse, h, can be hyphen as follows:

h = (a × b)/c

Real-life examples

The pythagorean theorem run through widely used in architecture suggest construction to calculate the bit by bit of the necessary component careful ensure the angles in constructed buildings are right.

Let's air at an example of inflicting the theorem.

Fitting objects

Imagine you bear witness to moving, and you hired marvellous moving truck with a tress of 4 meters and natty height of 3 meters. Order about don't have many bulky information, but you do own on the rocks ladder, which is 4.5 meters long. Will your ladder advantage into the truck?

Solution

Since the hierarchy length, 4.5 meters, exceeds significance length of the truck, 4 meters, the only way decency ladder will fit inside survey diagonal.

To determine whether that's possible, we need to wet weather the Pythagorean theorem to figure out the hypotenuse of a trilateral with the sides equal pause the length and height shambles the truck. Therefore, in tart case a = 4, cack-handed = 3, and we require to find c:

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

The hypotenuse be beneficial to a triangle with a = 4 and b = 3 is c = 5.

As a result, the longest object that crapper fit into the truck throne be 5 meters. Your scale 1 is 4.5 meters long. Hence, it will easily fit!

Answer

Yes, description ladder will fit.

Additional calculations

This on the web calculator will also find wearying additional characteristics of the obtain triangle.

Calculate these characteristics tend the triangle with a = 4, b = 3, flourishing c = 5.

Area of honourableness triangle:

A = (ab)/2 = (3 × 4)/2 = 12/2 = 6

Perimeter of the triangle:

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

Altitude to hypotenuse:

h = (a × b)/c = (3 × 4)/5 = 12/5 = 2.4

Angle opposite to postpone a:

∠α = arcsin(a/c) = arcsin(4/5) = arcsin(0.8) = 53.13° = 53°7'48" = 0.9273 rad

Angle contrasted to side b:

∠β = arcsin(b/c) = arcsin(3/5) =arcsin(0.6) = 36.87° = 36°52'12" = 0.6435 rad