Geometry calculator

Pick a shape, type its measurements and get its area, perimeter or volume at once, each result with its formula and the numbers substituted.

What you can do

Plane figures: square, rectangle, triangle, equilateral triangle, circle, ellipse, rhombus, parallelogram, trapezoid, regular n-sided polygon, circular sector and annulus.
Solids: cube, cuboid, regular prisms with any number of sides and prisms with any base, cylinder, cone, conical frustum, regular pyramid, sphere, hemisphere, spherical cap, ellipsoid and torus.
Volume, lateral and total area, diagonals, slant heights, apothems and angles.
Start from another value: the side of a square of known area, the radius of a sphere of known volume…
Solve triangles from three values (SSS, SAS, ASA, AAS, SSA or two sides and the area): sides, angles, area, perimeter, heights, medians, inradius and circumradius and the type of triangle, with a drawing to scale and both triangles of the ambiguous case.

Guide: areas, volumes and solving triangles

This tool computes the measurements of plane figures and solids, and solves triangles from three given values. It also works backward: if you know the area or the volume, it solves for the missing measurement. This guide collects the main formulas, explains the laws of sines and cosines including the ambiguous case, and works examples.

Most used formulas

Circle: A = πr², perimeter = 2πr.
Triangle: A = ½·base·height, or A = ½·a·b·sin C with two sides and the angle between them.
Trapezoid: A = (B + b)·h/2.
Cylinder: V = πr²h; lateral area = 2πrh; total area = 2πr(r + h).
Cone: V = πr²h/3; lateral area = πr·s, with the slant height s = √(r² + h²).
Sphere: V = 4πr³/3; area = 4πr².

Example 1: a cylinder, both ways

A cylinder with radius 3 and height 10: V = π·3²·10 = 90π ≈ 282.74; lateral area = 2π·3·10 = 60π ≈ 188.50; area of one base = 9π ≈ 28.27; total area = 2π·3·13 = 78π ≈ 245.04.

Backward: if the volume must be 500 with the same radius, h = V/(πr²) = 500/(9π) ≈ 17.68. And a circle of area 50 has r = √(50/π) ≈ 3.989. The tool solves for any measurement this way from the area, perimeter or volume; if data are missing it says which, and if there are several solutions it shows them all.

Solving a triangle

A triangle is determined by three values, at least one of them a side. With the usual notation (side a opposite angle A, and so on):

Law of sines: a / sin A = b / sin B = c / sin C

Law of cosines: a² = b² + c² − 2bc · cos A

SSS (three sides) and SAS (two sides and the included angle): start with the law of cosines.
ASA and AAS (one side and two angles): the third angle comes from the 180° sum and the sides from the law of sines.
SSA (two sides and a non-included angle): this is the ambiguous case, which can have zero, one or two solutions.

Example 2: SAS

b = 7, c = 9 and A = 40°.

a² = 49 + 81 − 2·7·9·cos 40° = 130 − 126·0.7660 ≈ 33.48, so a ≈ 5.786.
cos B = (a² + c² − b²)/(2ac) gives B ≈ 51.05°.
C = 180° − 40° − 51.05° ≈ 88.95°.
Area = ½·b·c·sin A = ½·7·9·sin 40° ≈ 20.25.

Example 3: the ambiguous case

a = 7, b = 9 and A = 40°. By the law of sines, sin B = 9·sin 40°/7 ≈ 0.8264. Two angles have that sine, B ≈ 55.73° and B ≈ 124.27°, and both fit in a triangle together with A = 40°:

With B ≈ 55.73°: C ≈ 84.27° and c ≈ 10.84.
With B ≈ 124.27°: C ≈ 15.73° and c ≈ 2.95.

Many students give only the first solution. The tool shows both, and in the earlier examples as well, every value matches what it computes.

How to use the calculator

Choose plane figures, solids or triangles, and then the figure.
Type the measurements you know. You can also enter the area, perimeter or volume as data and leave blank the measurement you want.
Read the results with their formulas and the steps: the formula, the substitution of the values and the result.

Common mistakes

Degrees and radians: sin 40 in radians is not sin 40°.
Forgetting the second solution in the SSA case.
Using the cone's slant height where the height belongs, or the other way around.
Mixing units: if r is in cm and h in m, the volume comes out wrong.

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