Enter any two known values into the right triangle calculator above, and it returns the missing sides, angles, area, and perimeter. Whether you have one side and one angle, two sides, or just need a quick Pythagorean theorem check, this tool handles the math in seconds.
What This Right Triangle Calculator Tool Does
This calculator solves a right triangle from the minimum information needed: any two known values. That means you can enter:
- Two side lengths
- One side and one non-right angle
- The hypotenuse and one leg
From those inputs, the tool calculates every remaining unknown. You get the missing side lengths, both acute angles, the area, and the perimeter.
It works for homework, construction layouts, DIY projects, and quick geometry checks. Just enter what you know, and the calculator fills in the rest.
What Is a Right Triangle?
A right triangle is a triangle that contains exactly one 90° angle. The two remaining angles are always acute, meaning each is less than 90°. Because all three interior angles must add up to 180°, those two acute angles always sum to exactly 90°.
The Right Angle and Sides of a Right Triangle
The right angle is the defining feature. It is the 90° angle, typically marked with a small square in diagrams.
The two sides that form the right angle are called legs (often labeled a and b). The side across from the right angle is the hypotenuse (labeled c). Every right triangle has exactly this structure: two legs and one hypotenuse.
Which side is the hypotenuse? It is always the side opposite the 90° angle. It is also always the longest side of the triangle.
Hypotenuse, Side Length, and How They Relate
The hypotenuse is the longest side because it sits opposite the largest angle. No leg can ever be longer than the hypotenuse in a valid right triangle.
The three side lengths are connected by a strict relationship: the Pythagorean theorem. If you know any two sides, you can calculate the third. If you know one side and one acute angle, trigonometry gives you the rest.
This relationship is why a right triangle calculator needs only two inputs to solve everything.
How to Solve a Right Triangle With One Side and One Angle
When you know one side length and one acute angle, you have enough information to solve the entire triangle. The right angle is already known (90°), so the second acute angle is simply 90° minus the one you entered.
Finding a Side Length From One Known Value
Use sine, cosine, or tangent depending on which side you know and which side you need.
- Know the hypotenuse, need a leg? Multiply the hypotenuse by the sine or cosine of the known angle.
- Know a leg, need the hypotenuse? Divide the leg by the sine or cosine of the appropriate angle.
- Know one leg, need the other leg? Use the tangent ratio, or find the hypotenuse first and then use the Pythagorean theorem.
How do you solve a right angle triangle with only one side? You cannot, not without at least one angle (besides the 90° angle). One side alone leaves too many unknowns. You need one side plus one acute angle, or two sides.
Finding an Angle When Two Sides Are Known
When you know two side lengths, inverse trigonometric functions give you the angles.
- tan⁻¹(opposite / adjacent) returns the angle directly.
- sin⁻¹(opposite / hypotenuse) works when the hypotenuse is one of your known sides.
- cos⁻¹(adjacent / hypotenuse) is the alternate option.
Once you find one acute angle, subtract it from 90° to get the other.
Do I use degrees or radians? This calculator uses degrees by default. If your coursework or application requires radians, convert by multiplying degrees by π/180.
The Pythagorean Equation and Formula
The Pythagorean theorem is the foundational equation for right triangles. It states:
a² + b² = c²
Here, a and b are the two legs, and c is the hypotenuse. This relationship holds true for every right triangle, regardless of size.
Using the Pythagorean Theorem to Calculate Side Length
To find a missing side:
- Missing the hypotenuse? Add the squares of both legs, then take the square root. c = √(a² + b²)
- Missing a leg? Subtract the square of the known leg from the square of the hypotenuse, then take the square root. a = √(c² − b²)
How to calculate the missing side of a right triangle? If you have both other sides, plug them into the formula above. If you have one side and one angle, use trigonometry instead.
Why does entering "c and a" sometimes fail? If the value you entered for a is larger than c, the equation produces a negative number under the square root. That is not a valid right triangle. The hypotenuse must always be the longest side.
Pythagorean Triples: Common Right Triangle Examples
A Pythagorean triple is a set of three positive integers that satisfy a² + b² = c². These are exact, whole-number right triangles with no rounding.
Common examples:
- 3, 4, 5 (the most well-known triple)
- 5, 12, 13
- 8, 15, 17
- 7, 24, 25
Any multiple of a triple also works. For example, 6, 8, 10 is just the 3, 4, 5 triple scaled by 2.
What is the 3 4 5 right triangle rule? If a triangle has sides in a 3:4:5 ratio, it is guaranteed to contain a right angle. Builders use this rule to verify square corners on job sites. Measure 3 feet along one wall, 4 feet along the other, and the diagonal should be exactly 5 feet.
Trigonometry Formulas for Right Triangles
Sine, Cosine, and Tangent for Finding Side and Angle Values
Trigonometric functions define the ratio between sides of a right triangle relative to a specific angle (call it θ):
- sin(θ) = opposite side / hypotenuse
- cos(θ) = adjacent side / hypotenuse
- tan(θ) = opposite side / adjacent side
"Opposite" and "adjacent" depend on which angle you are referencing. The side directly across from angle θ is the opposite. The side touching angle θ (that is not the hypotenuse) is the adjacent.
These ratios let you find any unknown side when you know one side and one angle. They also let you find any angle when you know two sides, using the inverse functions (sin⁻¹, cos⁻¹, tan⁻¹).
The calculator above applies these formulas automatically based on your inputs.
How to Calculate the Area and Perimeter of a Right Triangle
Area of a right triangle uses a straightforward formula:
Area = (1/2) × a × b
The two legs serve as the base and height because they meet at the 90° angle. No need for a separate altitude calculation.
Perimeter is the sum of all three sides:
Perimeter = a + b + c
If you only know two sides, find the third side first using the Pythagorean theorem, then add all three.
Can a right angled triangle have equal sides? Not all three. If all three sides were equal, every angle would be 60°, and there would be no 90° angle. However, a right triangle can be isosceles, meaning the two legs are equal. That is the 45-45-90 triangle.
Special Right Triangles: 45-45-90 and 30-60-90 Examples
Two right triangles appear so frequently in math, construction, and design that their ratios are worth memorizing.
45-45-90 triangle
- Angles: 45°, 45°, 90°
- Side ratio: 1 : 1 : √2
- Both legs are equal (isosceles). The hypotenuse is the leg length times √2.
- Example: legs of 5 each produce a hypotenuse of 5√2 ≈ 7.07.
30-60-90 triangle
- Angles: 30°, 60°, 90°
- Side ratio: 1 : √3 : 2
- The side opposite 30° is the shortest. The side opposite 60° is that value times √3. The hypotenuse is twice the shortest side.
- Example: shortest side of 4 gives sides of 4, 4√3 ≈ 6.93, and 8.
These ratios let you solve special right triangles without a calculator if you recognize the angle pattern.
How many lines of symmetry does a right triangle have? Most right triangles have zero lines of symmetry. The only exception is the 45-45-90 (isosceles right triangle), which has exactly one line of symmetry running from the right angle vertex to the midpoint of the hypotenuse.
Right Triangle Calculator Example With Solution
Example: Finding the Length of One Side
Given: Side a = 3, side b = 4. Find the hypotenuse c.
- Apply the Pythagorean theorem: c² = a² + b²
- c² = 3² + 4² = 9 + 16 = 25
- c = √25 = 5
This is the classic 3, 4, 5 Pythagorean triple. The remaining angles are approximately 36.87° and 53.13°.
Area = (1/2) × 3 × 4 = 6 square units Perimeter = 3 + 4 + 5 = 12 units
Example: Solve for an Angle Value
Given: Side a = 7 (opposite angle A), hypotenuse c = 13. Find angle A.
- Use inverse sine: A = sin⁻¹(a / c)
- A = sin⁻¹(7 / 13) = sin⁻¹(0.5385)
- A ≈ 32.58°
The other acute angle B = 90° − 32.58° = 57.42°
Missing leg b = √(c² − a²) = √(169 − 49) = √120 ≈ 10.95
Are all right triangles similar? No. Two right triangles are similar only when they share the same acute angle measures. The right angle alone is not enough to guarantee similarity.
Right Triangle Calculator Formula and Equation Reference
Quick reference for every formula this calculator uses:
Pythagorean theorem
- c = √(a² + b²)
- a = √(c² − b²)
- b = √(c² − a²)
Trigonometric ratios (given angle θ)
- sin(θ) = opposite / hypotenuse
- cos(θ) = adjacent / hypotenuse
- tan(θ) = opposite / adjacent
Inverse trig (given two sides)
- θ = sin⁻¹(opposite / hypotenuse)
- θ = cos⁻¹(adjacent / hypotenuse)
- θ = tan⁻¹(opposite / adjacent)
Area
- Area = (1/2) × a × b
Perimeter
- Perimeter = a + b + c
Angle sum
- Angle A + Angle B = 90°
When This Triangle Calculator Gives an Estimate, Not an Exact Solution
This calculator uses standard trigonometric functions and the Pythagorean theorem. For most inputs, the results are mathematically exact or rounded to a reasonable number of decimal places.
However, keep a few things in mind:
- Decimal rounding can introduce tiny differences from exact values. A result like 6.9282 is a rounded form of 4√3.
- Pythagorean triples (like 3, 4, 5) produce exact integer answers. Most real-world measurements do not.
- This tool is for math, education, and planning. It does not replace professional surveying, engineering calculations, or structural analysis where certified precision is required.
If your application demands exact tolerances, use these results as a starting point and verify with the appropriate professional tools or guidance.