How To Find An Angle With 2 Sides

Correct Triangle Side and Angle Figurer

Created by Hanna Pamuła , PhD candidate

Reviewed by

Bogna Szyk and Jack Bowater

Last updated:

Nov 12, 2021

Table of contents:

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

Finding out the missing side or bending couldn't be easier than with our cracking tool - right triangle side and bending calculator. Cull ii given values, type them into the computer and the remaining unknowns volition be determined in a glimmer of an eye! If you lot are wondering how to find the missing side of a right triangle, go along scrolling and you'll find the formulas behind our computer.

How to find the sides of a right triangle

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

Given ii sides

If you know two other sides of the right triangle, information technology's the easiest option; all you lot need to exercise is apply the Pythagorean theorem:

a² + b² = c²

if leg a is the missing side, and so transform the equation to the class when a is on ane side, and take a square root:

a = √(c² - b²)
if leg b is unknown, then

b = √(c² - a²)
for hypotenuse c missing, the formula is

c = √(a² + b²)

Given bending and hypotenuse

Right triangle with law of sines formulas. a over sin(α) equals b over sin(β) equals c, because sin(90°) = 1

Utilize the police 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)

Given angle and one leg

Find the missing leg using trigonometric functions:

a = b * tan(α)
b = a * tan(β)

Given area and ane leg

As nosotros remember from basic triangle area formula, we tin calculate the area by multiplying triangle top and base of operations and dividing the event by two. A right triangle is a special case of a scalene triangle, in which one leg is the height when the second leg is the base of operations, then the equation gets simplified to:

expanse = a * b / two

For instance, if we know only the right triangle area and the length of the leg a, we can derive the equation for other sides:

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

How to notice the angle of a correct triangle

If you know one angle apart from the right angle, calculation of the third one is a like shooting fish in a barrel:

Given β: α = 90 - β

Given α: β = 90 - α

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

for α

sin(α) = a / c so 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) (changed 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 so β = arccot(a / b) (inverse cotangent)

How do you solve a right angle triangle with only one side?

To solve a triangle with ane side, yous also need one of the non-right angled angles. If not, it is impossible:

If you take the hypotenuse, multiply it by sin(θ) to get the length of the side opposite to the angle.
Alternatively, multiply the hypotenuse by cos(θ) to become the side adjacent to the bending.
If you have the non-hypotenuse side next to the angle, divide it past cos(θ) to go the length of the hypotenuse.
Alternatively, multiply this length by tan(θ) to get the length of the side opposite to the angle.
If you accept an angle and the side opposite to it, you tin dissever the side length by sin(θ) to get the hypotenuse.
Alternatively, dissever the length past tan(θ) to get the length of the side adjacent to the angle.

How to observe the missing side of a right triangle? How to find the bending? Instance

Let'due south show how to find the sides of a right triangle with this tool:

Assume we want to find the missing side given area and one side. Select the proper option from a drop-down list. It's the third one.
Type in the given values. For case, an surface area of a right triangle is equal to 28 in² and b = 9 in.
Our right triangle side and angle calculator displays missing sides and angles! Now nosotros know that:

a = 6.222 in
c = 10.941 in
α = 34.66°
β = 55.34°

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

Refresh the reckoner. Pick the pick you demand. Assume that nosotros have two sides and nosotros want to find all angles. The default option is the right i.
Enter the side lengths. Our correct triangle has a hypotenuse equal to thirteen in and a leg a = v in.
Missing side and angles announced. In our instance, b = 12 in, α = 67.38° and β = 22.62°.

FAQ

How many lines of symmetry does a right triangle have?

If a right triangle is isosceles (i.due east., its two non-hypotenuse sides are the same length) it has i 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 3 angles cannot as well be equal, as one has to be 90° past definition. A right triangle tin can, however, accept its two non-hypotenuse sides be equal in length. This would also mean the ii other angles are equal to 45°.

Are all correct triangles like?

Not all right angled triangles are similar, although some can exist. They are similar if all their angles are the same length, or if the ratio of 2 of their sides is the aforementioned.

Hanna Pamuła , PhD candidate

Right triangle with sides a,b,c and angles α and β