This right triangle calculator finds every remaining side and angle of a right-angled triangle from any two known measurements: two sides, or one side and one acute angle. Enter what is known, and the tool applies the Pythagorean theorem or the trigonometric ratios, whichever fits, and shows the substituted steps.
A right triangle always has one 90-degree angle, which means only two further measurements are needed to fix the whole shape, one fewer than the three measurements a general triangle requires. The same page works as a right angle triangle calculator, right angled triangle calculator, or general triangle calculator when the figure is right-angled.
Hypotenuse calculator: find the hypotenuse from two legs
As a hypotenuse calculator, this right triangle calculator uses the Pythagorean theorem to relate the two legs (the sides meeting at the right angle) to the hypotenuse (the side opposite the right angle, always the longest side).
c = √(a² + b²)Worked example: legs a = 3 and b = 4, the smallest whole-number right triangle.
1. Square each leg. 3² = 9, 4² = 16.
2. Add the squares. 9 + 16 = 25.
3. Take the square root. √25 = 5, the hypotenuse.
Find a missing leg from the hypotenuse and one leg
Rearranging the same theorem solves for a leg when the hypotenuse and the other leg are known.
b = √(c² − a²)Worked example: hypotenuse c = 13, leg a = 5.
1. Square both values. 13² = 169, 5² = 25.
2. Subtract. 169 − 25 = 144.
3. Take the square root. √144 = 12, the missing leg.
Triangle angle calculator: solve with one side and one angle
When one acute angle and one side are known, this triangle angle calculator uses the three primary trigonometric ratios to finish the right angle triangle.
sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacentWorked example: an acute angle of 30° and an adjacent leg of 10.
1. Choose the ratio that connects the known and unknown sides. tan(30°) = opposite / 10.
2. Solve for the opposite side. opposite = 10 × tan(30°) ≈ 10 × 0.5774 ≈ 5.77.
3. Find the hypotenuse. hypotenuse = adjacent / cos(30°) = 10 / 0.8660 ≈ 11.55.
4. Find the second acute angle. 90 − 30 = 60°, since the two acute angles in a right triangle always sum to 90 degrees.
Right angle calculator note
A right angle calculator step for this figure is fixed: one corner is 90°. The acute angles are what remain to find. The tool does not invent a different right-angle measure; it uses that 90° corner and solves the rest of the triangle.
Calculate triangle area
To calculate triangle area for a right triangle, take half the product of the two legs, since each leg doubles as a base and the height for the other.
A = ½ × a × bWorked example: legs 3 and 4 from the earlier example.
½ × 3 × 4 = 6 square units.
If only a leg and the hypotenuse are known, the missing leg is found first with the Pythagorean theorem, then this formula applies to the two legs.
Verify a right triangle configuration is possible
Not every combination of two given values produces a valid right triangle. If an entered hypotenuse is shorter than a given leg, the configuration is impossible, since the hypotenuse is always the longest side. The solver flags this rather than returning a negative number under a square root.
Use SOH-CAH-TOA to remember the ratios
SOH-CAH-TOA is the standard memory aid for the three trigonometric ratios: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, and Tangent is Opposite over Adjacent. Picking the correct ratio for a given problem comes down to identifying which two sides (or side and hypotenuse) are involved, relative to the known angle.
Work through a second Pythagorean triple
Legs of 5 and 12, another common whole-number right triangle, show the same process with different numbers. 1. Square each leg. 5² = 25, 12² = 144. 2. Add the squares. 25 + 144 = 169. 3. Take the square root. √169 = 13, the hypotenuse.
4. Find the area. ½ × 5 × 12 = 30 square units.
This 5-12-13 triangle, like the earlier 3-4-5 example, is a Pythagorean triple: a set of three whole numbers that satisfy a² + b² = c² exactly, with no rounding needed at any step.
Recognize other common Pythagorean triples
Beyond 3-4-5 and 5-12-13, a handful of other whole-number triples come up often enough to be worth recognizing on sight: 8-15-17 and 7-24-25 both satisfy a² + b² = c² exactly. Any whole-number multiple of a known triple also works, so 6-8-10 is simply the 3-4-5 triple doubled.
Spotting one of these patterns in a problem can save the square-root step entirely, since the hypotenuse or missing leg is already known from the pattern.
Frequently asked questions
How do you find the hypotenuse of a right triangle?
Use the Pythagorean theorem: c = √(a² + b²). For legs 3 and 4, the hypotenuse is √(9+16) = √25 = 5. That is the core hypotenuse calculator step on this page.
How do you find a missing leg of a right triangle?
Rearrange the Pythagorean theorem: b = √(c² − a²). For a hypotenuse of 13 and a leg of 5, the missing leg is √(169−25) = √144 = 12.
How do you solve a right triangle with one angle and one side?
Use the trigonometric ratios sin, cos, or tan, choosing whichever relates the known side to the side you want to find, given the known angle. That is the triangle angle calculator path for a right angled triangle.
What is SOH-CAH-TOA?
A memory aid for the trigonometric ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Do the two acute angles in a right triangle always add up to 90 degrees?
Yes. Since all three angles sum to 180° and one is fixed at 90°, the remaining two, both acute, always sum to exactly 90°.
How do you find the area of a right triangle?
Multiply the two legs and divide by 2: A = ½ × a × b. For legs 3 and 4, the area is 6 square units.
What if the given hypotenuse is shorter than a given leg?
That combination is impossible, since the hypotenuse is always the longest side of a right triangle. The solver will flag this rather than attempt an invalid calculation.
Summary
A right triangle is solved from any two known measurements using the Pythagorean theorem, c² = a² + b², for two sides, or the trigonometric ratios sine, cosine and tangent when one side and one acute angle are known. Whether you call it a right triangle calculator, right angle triangle calculator, or right angled triangle calculator, the methods on this page are the same.
Legs of 3 and 4 give a hypotenuse of 5 and an area of 6 square units, while a hypotenuse of 13 and a leg of 5 give a missing leg of 12.
The two acute angles always sum to exactly 90 degrees.