Right Triangle Calculator

Right Triangle Calculator 

Right triangle calculator is an all-at-one place tool that finds almost every basic unknown quantity of a right triangle. You can find any angle or side using this tool.

What is a right triangle? 

A triangle that has one angle at 90 degrees is called a right-angled triangle. The side that is opposite the right angle (90 degrees) is called hypotenuse and the other two are the adjacent and the opposite sides. 

The three sides of a right triangle are usually referred to by variables commonly c for hypotenuse and a and b for the other sides. 

One of the properties of a right triangle is the Pythagorean theorem, which states that in any right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This can be written as:

a² + b² = c²

where:

• a and b are the lengths of the two sides that form the right angle, and
• c is the length of the hypotenuse.

How to find the sides or angles of the right triangle?

Given different pieces of information, there are various methods to find the sides and angles of a right triangle.

To find the sides from sides:

The main method to find the sides of a triangle is using the Pythagorean theorem. Input the known sides and change the formula to separate the unknown value on one side. 

Example: 

A right triangle has the following data

Side b = 4cm

Side a = 3cm 

Find the side c.

Solution:

Since side c is the hypotenuse, no change in the Pythagorean theorem is required. Just enter the values and solve. 

a² + b² = c²

(3)² + (4)² = c²

9 + 16 = c²

25 = c²

5 = c

The length of the hypotenuse is 5

To find the angles from angles:

The total sum of the angles of a triangle is always 180 degrees. Since a right triangle always has a 90 degrees angle, the sum of the other two angles is also 90. I.e. 

Hypotenuse (90) + Other two angles (90) = Total angle (180) 

Example: 

A right triangle has an angle A of 65 degrees. Find angle B. 

Solution:

We know the measurement of the right angle i.e. 90 degrees. 

So to find angle B,

90 + 65 + B = 180

155 + B = 180 

B = 180 - 155

B = 25

Or you can simply subtract the value of angle A or B whichever is provided from 90 to find the unknown angle.

To find a side and an angle:

There are many possibilities for this case. But all of the questions can be solved using trigonometric functions. In trigonometry 

    a / sin(a) = b / sin(b) = c / sin(c)

Also 

Sin (a) = a / c

Cos (a) = b / c

Tan (a) = a / b

Cot (a) = b / a

Sin (b) = b / c

Cos (b) = a / c

Tan (b) = b / a

Cot (b) = a / b

These rules can be used to find any length or angle if provided any two quantities. 

According to the given data, look for the right formula. Say that you have two angles (A, B) and a side (a) and you have to find the side(b). Use the formula 

a/ sin (a) = b / sin (b)

b  = (a)(sin (b)) / sin (a)

This formula can also be used when two sides and one angle is provided to find the second angle.

Example 1:

A Right triangle has an angle of 30 degrees and an adjacent side (b) of length 5. Find the opposite side (a) and the hypotenuse (c).

1. To find a (the opposite side), Use the tangent function, tan(θ) = a/b. So a = btan(30) = 5tan(30) = 2.5.
2. To find c (the hypotenuse), Use the cosine function, cos(θ) = b/c. So c = b/cos(30) = 5/cos(30) = approximately 5.77.
3. The other acute angle in the triangle would be 90 - 30 = 60 degrees (since the sum of the two acute angles in a right triangle is 90 degrees).

Example 2:

A Right triangle has a hypotenuse (c) of length 10 and an adjacent side (b) of length 8. Find the opposite side (a) and the angle (θ).

1. To find a (the opposite side), Use the Pythagorean theorem: a² = c² - b² = 10² - 8² = 100 - 64 = 36. Taking the square root of both sides gives a = √36 = 6.
2. To find θ (the angle), Use the cosine function, cos(θ) = b/c. So θ = arccos(b/c) = arccos(8/10) = arccos(0.8) = approximately 36.87 degrees.
3. The other acute angle in the triangle would be 90 - 36.87 = 53.13 degrees.