Understanding the Right Triangle
A right triangle is any triangle containing one interior angle of exactly 90°. That single constraint is what makes the shape so powerful: the fixed right angle locks the relationship between the sides and the angles, and those locked relationships are the foundation that all of trigonometry is built on.
The side sitting directly across from the right angle is the hypotenuse, and it is always the longest of the three. Convention labels the sides a, b, and c, with c reserved for the hypotenuse. Each angle takes the name of the side it faces, so the 90° corner sits opposite c. This calculator uses the Greek letters α (alpha) and β (beta) for the two unknown acute angles: α faces side a, β faces side b. The letter h marks the altitude, the perpendicular segment dropped from the right-angle vertex onto the hypotenuse. That altitude carves the triangle into two smaller triangles, both similar to each other and to the triangle you started with.
When all three side lengths turn out to be whole numbers, the shape is called a Pythagorean triangle and the trio of lengths is a Pythagorean triple, such as 3-4-5 or 5-12-13.
The usual mistakes are simple ones: labeling a leg as c, forgetting the square root at the end, mixing units between the two sides, or leaving the angle unit on degrees while typing a radian value. Area and perimeter behave the same as in any triangle, though the perpendicular legs make the area especially clean, since one leg already serves as the height for the other.
How to Calculate a Right Triangle (step by step)
Every output comes from resolving the two legs first, then deriving everything else. The worked examples below use the calculator’s defaults: a = 3 and c = 5, with b, α, and β left at 0.
Step One: Resolve the two legs from whatever you entered
The calculator’s first job is to pin down a and b. Which route it takes depends on the pair of values you supplied.
two sides given → missing leg = sqrt(c^2 - known leg^2)
one leg + one angle → other leg = leg / tan(angle) or leg × tan(angle)
hypotenuse + angle → leg = c × sin(angle) or c × cos(angle)
Given a = 3 and c = 5:
b = sqrt(5^2 - 3^2) = sqrt(25 - 9) = sqrt(16) = 4
Step Two: Confirm the hypotenuse with the Pythagorean theorem
With both legs known, the hypotenuse is the square root of the sum of their squares. If you supplied c yourself, this step simply returns it.
c = sqrt(a^2 + b^2)
sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5
Step Three: Find the acute angles with inverse tangent
Each acute angle is the arctangent of the leg opposite it divided by the leg beside it. Arctangent returns radians, so degree mode multiplies the result by 180/π.
α = arctan(a / b) β = arctan(b / a)
degrees = radians × 180 / π
α = arctan(3 / 4) = 0.6435 rad = 36.8699°
β = arctan(4 / 3) = 0.9273 rad = 53.1301°
36.8699° + 53.1301° = 90° ← always check this
Step Four: Find the altitude to the hypotenuse
Because the legs are perpendicular, the area can be written two ways: half of a × b, or half of c × h. Setting those equal and solving for h gives a shortcut.
(1/2) × c × h = (1/2) × a × b → h = (a × b) / c
(3 × 4) / 5 = 12 / 5 = 2.4
Step Five: Find the area
One leg is the base, the other is the height. No extra measurement needed.
A = (a × b) / 2
(3 × 4) / 2 = 6 square units
Step Six: Find the perimeter
Add the three sides together.
P = a + b + c
3 + 4 + 5 = 12 units
Working in Degrees or Radians
Degrees are the everyday unit: a full turn is 360°, and both acute angles in a right triangle land somewhere between 0° and 90°. Radians measure the same angles by arc length instead, with a full turn equal to 2π, which is why calculus and most programming languages default to them.
radians = degrees × π / 180
degrees = radians × 180 / π
36.8699° × π / 180 = 0.6435 rad
π / 3 ≈ 1.0472 rad = 60°
π / 4 ≈ 0.7854 rad = 45°
π / 6 ≈ 0.5236 rad = 30°
Switch the unit selector before you type an angle. In radian mode, enter the decimal equivalent (1.0472 rather than pi/3), and every angle the calculator returns will come back in radians as well.
Special Right Triangles Worth Memorizing
Two right triangles appear often enough that knowing their side ratios saves real time on exams and job sites alike.
The 30°-60°-90° Triangle
Its name lists its three angles, and its sides always hold the ratio 1 : √3 : 2, running from the side opposite 30° up to the hypotenuse. Since the ratio never changes, one known length unlocks the other two.
side opposite 30° : side opposite 60° : hypotenuse = 1 : √3 : 2
If the 60° side = 5:
short leg = 5 / √3 ≈ 2.8868
hypotenuse = (5 × 2) / √3 = 10 / √3 ≈ 5.7735
This is the standard tool for evaluating trigonometric functions at multiples of π/6.
The 45°-45°-90° Triangle
Also called the isosceles right triangle, because its two legs are equal. The sides follow the ratio 1 : 1 : √2, so again a single known length settles the rest.
leg : leg : hypotenuse = 1 : 1 : √2
If the hypotenuse c = 5:
each leg = 5 / √2 ≈ 3.5355
Use this one for trigonometric functions at multiples of π/4.