Geometry calculator

What this tool does

It computes the measurements you usually need from the dimensions you have, and draws the shape with those dimensions written on it:

• 2D shapes: circle, ellipse, circular sector, rectangle, parallelogram, trapezoid and regular polygon: area, perimeter or circumference, diagonals, heights, apothem and angles.
• 3D solids: cube, box, sphere, cylinder, cone, square pyramid, torus and frustum of a cone: volume, total and lateral surface area, slant heights and diagonals.
• Triangle solver: from three sides (SSS), two sides and the angle between (SAS), two angles and a side (ASA or AAS), two sides and an angle opposite one (SSA), or base and height. It returns every side and angle, the area, perimeter, circumradius, inradius and the three heights, names the triangle (scalene, isosceles, equilateral; acute, right, obtuse) and draws it to scale.

Pick the input unit (mm, cm, m, in or ft) and the result unit separately: areas convert with the square of the factor and volumes with its cube, using exact definitions (1 in = 25.4 mm, 1 ft = 0.3048 m). Angles can be entered in degrees or radians. Precision shows up to ten significant digits automatically, or a fixed 3 to 6 significant figures, which is how a measured answer should usually be given.

The formulas behind it

• Circle: area πr², circumference 2πr. Sector of angle θ (radians): area θr²/2, arc rθ, chord 2r sin(θ/2).
• Ellipse: area πab. Its perimeter has no closed formula; the tool uses Ramanujan's second approximation, accurate to about 1 part in 100,000 even for a 10:1 ellipse, and marks it with ≈.
• Regular polygon with n sides of length s: apothem s / (2 tan(π/n)), area perimeter × apothem / 2, interior angle (n − 2) × 180° / n.
• Cylinder: volume πr²h, surface 2πr(r + h). Cone: slant √(r² + h²), volume πr²h/3, surface πr(r + slant).
• Sphere: volume 4πr³/3, surface 4πr². Square pyramid: volume a²h/3. Frustum: volume πh(r₁² + r₁r₂ + r₂²)/3. Torus: volume 2π²Rr², surface 4π²Rr.

Worked example: a cone with radius 3 cm and height 4 cm has a 5 cm slant, a volume of 12π ≈ 37.70 cm³ and a total surface of 24π ≈ 75.40 cm². In mm that is 37,699 mm³: a cubic centimetre is 1,000 cubic millimetres, not 10.

Solving triangles, and the SSA trap

With three sides the triangle is fixed: the angles follow from the law of cosines, and the area from Heron's formula (computed in a numerically stable form, so very thin triangles stay accurate). SAS, ASA and AAS also fix a single triangle.

SSA is different. Given sides a and b and the angle A opposite a, the law of sines gives sin B = b · sin A / a, and a sine has two angles between 0° and 180°: B and 180° − B. So there can be two triangles, one, or none. With a = 8, b = 10 and A = 40°: sin B = 0.8035, so B is 53.46° or 126.54°, and both leave room for the third angle; the tool shows both triangles, with c = 12.42 and c = 2.90. With a = 5 instead, sin B would be 1.29, which is impossible: side a is too short to reach the base. That is why SSA is not a congruence rule in school geometry.

Tips and limits

• The 2D drawings keep the shape's proportions; the 3D drawings are schematic and not to scale.
• Results are floating-point (about 15 significant digits). Choose a number of significant figures that matches how precisely the inputs were measured: a box measured to the nearest centimetre does not have a volume known to the cubic millimetre.
• The circumradius R and inradius r are useful for fitting: R is the smallest circle through the corners, r the largest circle inside.
• For unit conversions beyond lengths, the unit converter covers mass, volume and more. For the formulas in practice, read geometry formulas you actually use.