How to use it
- Pick the type: right triangle, SSS, SAS or ASA.
- Enter the values you know.
- Read all three sides and angles.
- Check the area, perimeter, heights and triangle type.
Four ways to solve a triangle
- Right triangle (Pythagorean): enter any two of leg a, leg b and hypotenuse c.
- SSS: all three sides.
- SAS: two sides and the angle between them.
- ASA: one side and the two angles at its ends.
From these, the calculator finds every side and angle, the area, the perimeter, the three heights, the radius of the circumscribed and inscribed circles, and whether the triangle is acute, right or obtuse, and scalene, isosceles or equilateral. Everything is calculated in your browser.
The Pythagorean theorem
In a right triangle, a² + b² = c², where c is the hypotenuse, the side opposite the right angle.
- Legs 6 and 8: c = √(36 + 64) = √100 = 10
- Hypotenuse 13, leg 5: other leg = √(169 − 25) = √144 = 12
Ladder example: a 13-foot ladder with its base 5 feet from the wall reaches 12 feet up.
Common whole-number right triangles (Pythagorean triples): 3-4-5, 5-12-13, 8-15-17 and 7-24-25, plus any multiple such as 6-8-10.
Laws of cosines and sines
For triangles without a right angle:
- Law of cosines: c² = a² + b² − 2ab × cos(C). Used for SAS and to find angles from SSS.
- Law of sines: a / sin(A) = b / sin(B) = c / sin(C). Used for ASA.
SAS example: a = 5, b = 7, C = 60°. c² = 25 + 49 − 2 × 5 × 7 × 0.5 = 39, so c ≈ 6.245.
The three angles always add up to 180°. For the 3-4-5 triangle they are about 36.87°, 53.13° and 90°.
Area, heights and circles
The calculator gives the area using whichever formula fits (half base times height, or Heron's formula from three sides). It also shows:
- Heights to each side: h = 2 × area / side
- Circumradius R, the radius of the circle through all three corners
- Inradius r, the radius of the largest circle that fits inside
For the 3-4-5 triangle: area 6, perimeter 12, R = 2.5 and r = 1.
Squaring a corner with 3-4-5
Builders use the Pythagorean theorem to lay out square corners for decks, patios and foundations. Measure 3 ft along one side and 4 ft along the other; if the diagonal between those marks is exactly 5 ft, the corner is a right angle. For bigger layouts, use 6-8-10 or 9-12-15 for more accuracy.
For a rectangle, compare the two diagonals: if they are equal, the corners are square. A 12 × 16 ft deck should have 20 ft diagonals.
Common mistakes
- Using the hypotenuse as a leg. The hypotenuse is always the longest side, opposite the right angle.
- Entering three values in right-triangle mode. Enter exactly two and leave the third blank.
- Impossible triangles. Sides 2, 3 and 6 cannot form a triangle because 2 + 3 < 6; the calculator says so.
- Degree mode. Angles here are in degrees. For radians or trig values, use the trig calculator.
- Two possible triangles. Knowing two sides and an angle that is not between them (SSA) can fit two different triangles, which is why this calculator asks for SAS or ASA instead.
- Angles that add up to 180° or more. In ASA mode, the two angles you enter must total less than 180°, because the third angle takes up the rest.
Frequently asked questions
How do I find the hypotenuse?
Square both legs, add them and take the square root: c = √(a² + b²). Legs 6 and 8 give 10.
Can I solve a triangle that is not a right triangle?
Yes. Use SSS, SAS or ASA mode; the calculator applies the laws of cosines and sines.
Why does it say the triangle does not exist?
The values break a rule: any two sides must add up to more than the third, and the angles must total 180°.
What is a 3-4-5 triangle?
A right triangle with sides 3, 4 and 5. Builders use it to check square corners.
Does the calculator give the area?
Yes, along with the perimeter, heights and the inscribed and circumscribed circle radii.
Last updated: November 12, 2026