How to Find the Angles of a Right Triangle if You Know the Sides

Right Triangle Side and Angle Calculator

Created by Hanna Pamuła , PhD candidate

Reviewed by

Bogna Szyk and Jack Bowater

Final updated:

Nov 12, 2021

Table of contents:

• How to detect the sides of a right triangle
• How to discover the angle of a right triangle
• How do you solve a correct angle triangle with just one side?
• How to notice the missing side of a right triangle? How to find the angle? Example
• FAQ

Finding out the missing side or angle couldn't be easier than with our great tool - right triangle side and angle calculator. Cull ii given values, blazon them into the calculator and the remaining unknowns will exist determined in a glimmer of an eye! If you lot are wondering how to observe the missing side of a correct triangle, proceed scrolling and you lot'll notice the formulas behind our figurer.

How to observe the sides of a correct triangle

There are a few methods of obtaining right triangle side lengths. Depending on what is given, you can use different relationships or laws to find the missing side:

1. Given 2 sides

If you know two other sides of the right triangle, it'south the easiest option; all yous demand to exercise is use the Pythagorean theorem:

a² + b² = c²

• if leg a is the missing side, then transform the equation to the form when a is on one side, and take a square root:

  a = √(c² - b²)

• if leg b is unknown, and so

  b = √(c² - a²)

• for hypotenuse c missing, the formula is

  c = √(a² + b²)

2. Given angle and hypotenuse

Apply the law of sines or trigonometry to find the right triangle side lengths:

• a = c * sin(α) or a = c * cos(β)

• b = c * sin(β) or b = c * cos(α)

3. Given angle and 1 leg

Find the missing leg using trigonometric functions:

• a = b * tan(α)

• b = a * tan(β)

4. Given area and i leg

As nosotros remember from basic triangle expanse formula, nosotros can calculate the expanse by multiplying triangle pinnacle and base of operations and dividing the result by 2. A right triangle is a special case of a scalene triangle, in which one leg is the summit when the 2nd leg is the base, and so the equation gets simplified to:

area = a * b / ii

For example, if we know only the correct triangle expanse and the length of the leg a, we can derive the equation for other sides:

• b = 2 * area / a
• c = √(a² + (2 * expanse / a)²)

How to find the angle of a right triangle

If you know one angle autonomously from the correct angle, calculation of the 3rd one is a piece of cake:

Given β: α = 90 - β

Given α: β = xc - α

Withal, if only two sides of a triangle are given, finding the angles of a correct triangle requires applying some basic trigonometric functions:

for α

• sin(α) = a / c then α = arcsin(a / c) (inverse sine)
• cos(α) = b / c and so α = arccos(b / c) (inverse cosine)
• tan(α) = a / b so α = arctan(a / b) (inverse tangent)
• cot(α) = b / a and so α = arccot(b / a) (inverse cotangent)

and for β

• sin(β) = b / c so β = arcsin(b / c) (inverse sine)
• cos(β) = a / c so β = arccos(a / c) (inverse cosine)
• tan(β) = b / a so β = arctan(b / a) (inverse tangent)
• cot(β) = a / b and so β = arccot(a / b) (changed cotangent)

How do yous solve a right angle triangle with merely one side?

To solve a triangle with one side, you likewise demand 1 of the not-right angled angles. If not, it is incommunicable:

1. If you accept the hypotenuse, multiply it by sin(θ) to become the length of the side reverse to the angle.
2. Alternatively, multiply the hypotenuse past cos(θ) to get the side adjacent to the angle.
3. If you have the non-hypotenuse side adjacent to the angle, divide it by cos(θ) to become the length of the hypotenuse.
4. Alternatively, multiply this length by tan(θ) to go the length of the side opposite to the bending.
5. If y'all have an angle and the side opposite to it, you tin can divide the side length past sin(θ) to get the hypotenuse.
6. Alternatively, divide the length by tan(θ) to get the length of the side adjacent to the angle.

How to find the missing side of a right triangle? How to find the angle? Case

Let's show how to discover the sides of a right triangle with this tool:

1. Assume we want to find the missing side given area and i side. Select the proper option from a drib-downward list. Information technology'southward the third ane.
2. Type in the given values. For example, an area of a right triangle is equal to 28 in² and b = 9 in.
3. Our right triangle side and bending calculator displays missing sides and angles! Now nosotros know that:

• a = 6.222 in
• c = x.941 in
• α = 34.66°
• β = 55.34°

Now, let'due south check how does finding angles of a right triangle work:

1. Refresh the calculator. Pick the option you lot need. Presume that we have 2 sides and we desire to notice all angles. The default choice is the right one.
2. Enter the side lengths. Our correct triangle has a hypotenuse equal to 13 in and a leg a = 5 in.
3. Missing side and angles appear. In our example, b = 12 in, α = 67.38° and β = 22.62°.

FAQ

How many lines of symmetry does a right triangle accept?

If a right triangle is isosceles (i.east., its ii not-hypotenuse sides are the aforementioned length) it has one line of symmetry. Otherwise, the triangle will have no lines of symmetry.

Can a right angled triangle accept equal sides?

No, a right triangle cannot have all 3 sides equal, as all three angles cannot also be equal, as one has to exist xc° by definition. A right triangle can, still, accept its two non-hypotenuse sides be equal in length. This would also mean the two other angles are equal to 45°.

Are all right triangles similar?

Non all right angled triangles are similar, although some can be. They are like if all their angles are the same length, or if the ratio of 2 of their sides is the same.

Hanna Pamuła , PhD candidate

30 sixty xc triangle 45 45 90 triangle Surface area of a right triangle … 13 more

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Source: https://www.omnicalculator.com/math/right-triangle-side-angle

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