How To Find The Last Side Of A Triangle

Right Triangle Side and Angle Calculator

Created by Hanna Pamuła , PhD candidate

Reviewed by

Bogna Szyk and Jack Bowater

Concluding updated:

November 12, 2021

Table of contents:

How to discover the sides of a right triangle
How to find the angle of a right triangle
How do you lot solve a correct angle triangle with simply 1 side?
How to find the missing side of a correct triangle? How to discover the bending? Instance
FAQ

Finding out the missing side or bending couldn't exist easier than with our great tool - right triangle side and bending computer. Choose ii given values, type them into the calculator and the remaining unknowns volition exist adamant in a blink of an eye! If you are wondering how to find the missing side of a right triangle, continue scrolling and you'll find the formulas behind our computer.

How to discover the sides of a right triangle

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

Given two sides

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

a² + b² = c²

if leg a is the missing side, then transform the equation to the course when a is on 1 side, and accept 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 angle and hypotenuse

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

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)

Given bending and ane leg

Find the missing leg using trigonometric functions:

a = b * tan(α)
b = a * tan(β)

Given area and one leg

Every bit we remember from bones triangle area formula, we can calculate the area by multiplying triangle peak and base of operations and dividing the result by two. A right triangle is a special case of a scalene triangle, in which one leg is the height when the 2nd leg is the base of operations, so the equation gets simplified to:

expanse = a * b / 2

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

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

How to find the bending of a correct triangle

If you know 1 angle autonomously from the right bending, calculation of the tertiary i is a like shooting fish in a barrel:

Given β: α = 90 - β

Given α: β = 90 - α

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

for α

sin(α) = a / c so α = arcsin(a / c) (inverse sine)
cos(α) = b / c then α = arccos(b / c) (changed 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 then β = arccot(a / b) (inverse cotangent)

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

To solve a triangle with i side, y'all also need 1 of the not-right angled angles. If not, it is impossible:

If you have the hypotenuse, multiply it past sin(θ) to get the length of the side opposite to the angle.
Alternatively, multiply the hypotenuse past cos(θ) to get the side next to the angle.
If you have the non-hypotenuse side adjacent to the angle, carve up it by cos(θ) to get the length of the hypotenuse.
Alternatively, multiply this length past tan(θ) to go the length of the side contrary to the bending.
If you have an bending and the side opposite to it, you can divide the side length by sin(θ) to get the hypotenuse.
Alternatively, divide the length by tan(θ) to get the length of the side side by side to the angle.

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

Permit'southward show how to discover the sides of a right triangle with this tool:

Presume we want to find the missing side given area and one side. Select the proper option from a drop-down list. It's the tertiary one.
Blazon in the given values. For instance, an surface area of a correct triangle is equal to 28 in² and b = 9 in.
Our right triangle side and bending reckoner displays missing sides and angles! At present nosotros know that:

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

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

Refresh the calculator. Pick the option you need. Assume that we accept 2 sides and nosotros want to find all angles. The default option is the right 1.
Enter the side lengths. Our right triangle has a hypotenuse equal to thirteen in and a leg a = 5 in.
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 correct triangle have?

If a correct triangle is isosceles (i.e., its two non-hypotenuse sides are the same length) information technology has one line of symmetry. Otherwise, the triangle will have no lines of symmetry.

Tin a correct angled triangle have equal sides?

No, a right triangle cannot take all iii sides equal, equally all three angles cannot also exist equal, as 1 has to be 90° by definition. A correct triangle can, however, have its two non-hypotenuse sides be equal in length. This would also hateful the two other angles are equal to 45°.

Are all right triangles similar?

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

Hanna Pamuła , PhD candidate

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