Triangle Calculator
Solve a triangle from three sides or two sides and the included angle — get every angle, the area, perimeter, heights, and the triangle's classification.
كيفية استخدام هذه الحاسبة
- 1Choose what you know: three sides, two sides with the angle between them, or the two legs of a right triangle.
- 2Enter the measurements in any consistent unit.
- 3Read the area, all three angles, and the classification. The height table gives the altitude to each side.
طريقة الحساب
Triangle formulas
Heron: s = (a + b + c) ÷ 2, A = √(s(s−a)(s−b)(s−c)) Law of cosines: c² = a² + b² − 2ab·cos(C) Angle: cos(A) = (b² + c² − a²) ÷ 2bc Pythagoras: c² = a² + b² (right triangles only) Height: hₐ = 2A ÷ a
The law of cosines is the general form of Pythagoras: when angle C is 90° its cosine is zero and the correction term vanishes, leaving c² = a² + b². Rearranged, it gives every angle from the three sides alone. Heron's formula then gives the area without needing any angle or height, which is why three tape-measure readings are enough to solve a triangle completely.
مثال محلول
A 3-4-5 triangle has a semi-perimeter of 6, so Heron's formula gives √(6 × 3 × 2 × 1) = 6 square units. The law of cosines confirms the angles as 36.87°, 53.13°, and exactly 90° — the classic right triangle.
Triangle Calculator: الدليل الكامل
Heron's formula: area from three sides
Most people learn area as ½ × base × height, which is fine when you have a height — but on a real site you almost never do. Measuring a perpendicular height requires establishing a right angle first, while measuring three side lengths requires nothing but a tape.
Heron's formula solves this. Take half the perimeter, subtract each side from it in turn, multiply all four numbers together, and take the square root. It is exact, it needs no angles, and it is why surveyors have triangulated land for two thousand years.
The triangle inequality
Three lengths only form a triangle if each one is shorter than the sum of the other two. Sides of 2, 3, and 9 cannot close — the short pair laid end to end still falls short of the long one.
The calculator checks this before solving, because a violation produces a negative value inside Heron's square root and a meaningless answer. If you get this error on real measurements, one of the three is almost certainly a mis-read.
Checking a right angle on site
The 3-4-5 rule is the fastest way to square a corner without a framing square. Measure 3 units along one edge and 4 along the other; if the diagonal between those marks is exactly 5, the corner is a true 90°. Any multiple works — 6-8-10 and 12-16-20 are common on larger layouts, and bigger triangles are more forgiving of small measurement errors.
This is Pythagoras used in reverse. The law of cosines generalises it: if the diagonal comes out longer than 5 the corner is obtuse, and if it comes out shorter the corner is acute. Enter your three measurements above and the calculator reports the exact angle.
Inradius, circumradius, and the special centres
The inradius is the radius of the largest circle that fits inside the triangle, touching all three sides; it equals the area divided by the semi-perimeter. The circumradius is the radius of the circle passing through all three vertices, and equals abc ÷ 4A.
For a right triangle, the circumradius is always exactly half the hypotenuse — the right angle sits on the circle and the hypotenuse is a diameter. That relationship, Thales' theorem, is a quick way to find the centre of a circle from any right angle inscribed in it.
الأسئلة الشائعة
How do I find the area of a triangle without the height?
Use Heron's formula with the three side lengths, which is what this calculator does. Take the semi-perimeter s = (a+b+c)/2, then the area is √(s(s−a)(s−b)(s−c)). No angle or perpendicular height is needed.
How do I find the hypotenuse?
For a right triangle, square both legs, add them, and take the square root: c = √(a² + b²). Choose the "two legs of a right triangle" option above and the hypotenuse is returned as side c, along with both acute angles.
Why do the three angles always add to 180 degrees?
Draw a line through one vertex parallel to the opposite side. The two outer angles it creates equal the other two angles of the triangle (alternate angles), and together with the vertex angle they form a straight line — which is 180°.
What do SSS and SAS mean?
They describe which measurements you have. SSS is side-side-side, three lengths. SAS is side-angle-side, two lengths with the angle between them. Both determine a triangle uniquely, which is why either is enough to solve it completely.