How To Find Angles Of A Triangle With Sides

Right Triangle Side and Bending Calculator

Created by Hanna Pamuła , PhD candidate

Reviewed by

Bogna Szyk and Jack Bowater

Last updated:

Nov 12, 2021

Table of contents:

How to notice the sides of a right triangle
How to find the angle of a right triangle
How do yous solve a right angle triangle with simply one side?
How to find 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. Choose ii given values, type them into the calculator and the remaining unknowns will exist adamant in a blink of an eye! If yous are wondering how to find the missing side of a right triangle, go on scrolling and you'll find the formulas behind our calculator.

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 lot can use different relationships or laws to find the missing side:

Given two sides

If you know 2 other sides of the correct triangle, it's the easiest option; all you demand to do is apply the Pythagorean theorem:

a² + b² = c²

if leg a is the missing side, so transform the equation to the grade when a is on one side, and have 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

Utilize 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 one leg

Find the missing leg using trigonometric functions:

a = b * tan(α)
b = a * tan(β)

Given area and one leg

Equally we remember from basic triangle area formula, we tin can calculate the area by multiplying triangle height and base and dividing the result by 2. 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, so the equation gets simplified to:

area = a * b / 2

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

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

How to find the angle of a right triangle

If you know ane bending apart from the correct angle, calculation of the tertiary one is a piece of cake:

Given β: α = xc - β

Given α: β = xc - α

However, if but 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) (changed sine)
cos(α) = b / c then α = arccos(b / c) (inverse cosine)
tan(α) = a / b and then α = arctan(a / b) (changed tangent)
cot(α) = b / a so α = arccot(b / a) (inverse cotangent)

and for β

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

How practise yous solve a right angle triangle with simply one side?

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

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

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

Assume we desire to find the missing side given area and 1 side. Select the proper option from a drop-downward list. Information technology'south the third one.
Blazon in the given values. For instance, an area of a right triangle is equal to 28 in² and b = 9 in.
Our right triangle side and bending estimator displays missing sides and angles! Now we know that:

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

Now, let's check how does finding angles of a correct triangle piece of work:

Refresh the calculator. Pick the option you need. Assume that we have two sides and we want to notice all angles. The default pick is the right one.
Enter the side lengths. Our correct triangle has a hypotenuse equal to 13 in and a leg a = five in.
Missing side and angles appear. In our instance, b = 12 in, α = 67.38° and β = 22.62°.

FAQ

How many lines of symmetry does a right triangle take?

If a correct triangle is isosceles (i.e., its 2 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 correct triangle cannot have all 3 sides equal, as all 3 angles cannot as well exist equal, as i has to be ninety° by definition. A correct triangle can, however, have its two non-hypotenuse sides be equal in length. This would likewise 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 similar if all their angles are the aforementioned 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