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Mathematics & calculation

Geometry Calculator: Area, Perimeter and Volume

Geometry solver for 19 flat shapes and solids: square, rectangle, circle, triangles, trapezoid, rhombus, ellipse, sector, regular polygon, cube, cuboid, sphere, cylinder, cone, pyramid and torus. Enter what you know and the tool derives every quantity it can, with the formulas explained.

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How does this tool work?

This geometry calculator solves the main flat shapes and common solids from only the dimensions you know. Pick a shape, enter for example a radius, two sides and an angle, or three dimensions, and the tool works out every quantity that can be derived: area, perimeter, diagonals, angles, apothem, lateral or total surface area and volume. It replaces a dozen separate calculators for revision, homework or job-site estimates.

The calculator works as a solver: for each shape, several combinations of inputs are offered where possible. A circle can be solved from its radius, diameter or circumference; a triangle from three sides, a base and a height, or two sides and the included angle; a right triangle from its two legs or from the hypotenuse and one leg.

The shapes covered are, in 2D: square, rectangle, circle, general triangle, right triangle, trapezoid, parallelogram, rhombus, ellipse, annulus, circular sector and regular polygon; in 3D: cube, cuboid (rectangular box), sphere, cylinder, cone, square pyramid and torus.

Each field has its own unit: lengths are converted to a common unit before the calculation, then the results are expressed in the unit of the first field, squared for areas and cubed for volumes. Angles can be entered in degrees, radians or gradians.

Only quantities that can genuinely be derived are shown. A base and a height are enough for the area of a triangle, but not for its perimeter, so the tool does not make one up. Inconsistent data, such as three sides that break the triangle inequality or a tube radius larger than the major radius of a torus, produce no result.

Only one result is an approximation: the perimeter of an ellipse, which has no simple exact formula. The tool uses Ramanujan's approximation, which is very accurate for everyday ellipses.

How to use this tool

  1. Choose a shape

    Select a 2D shape or a 3D solid from the list.

  2. Say what you know

    When several methods exist, choose the combination of dimensions you have.

  3. Enter the dimensions

    Type the values with their units; every result that can be calculated appears immediately.

Geometry formulas used

A selection of the relationships the solver applies, for flat shapes and solids.

Trapezoid

A = (B + b) × h ÷ 2

The average of the two bases times the height.

Rhombus

A = D × d ÷ 2

The product of the diagonals divided by two.

Regular polygon

A = P × a ÷ 2, with a = c ÷ (2 tan(π ÷ n))

Half the product of the perimeter and the apothem a.

Ellipse (Ramanujan)

P ≈ π(3(a + b) − √((3a + b)(a + 3b)))

Approximation of the perimeter; the exact area is πab.

Cylinder

V = π × r² × h

The base area times the height.

Cone and pyramid

V = base area × h ÷ 3

One third of the cylinder or prism with the same base and height.

Sphere

V = 4/3 × π × r³, S = 4πr²

Volume and surface area of a ball of radius r.

A, P
area and perimeter of a flat shape
V, S
volume and surface area of a solid
r, h
radius and height
B, b, D, d
long and short base, long and short diagonal
n, c, a
number of sides, side length and apothem of a regular polygon

Example: a ball with a 6 cm radius has a volume of 288π ≈ 904.8 cm³.

Use cases

Revision and homework

Check an area, perimeter or volume from middle school to high school, and verify intermediate values: diagonal, slant height, apothem, angles.

Volumes and capacity

Calculate the volume of a tank, a planter, a silo, a raised bed or a hole in m³ or ft³, then convert it to liters (1 m³ = 1,000 L).

Surfaces to cover

Estimate the lateral area of a cylinder to paint, the surface of a conical roof or an annulus to pave.

Plans and fabrication

Size a regular polygon, a sector or an ellipse for cutting, 3D printing or woodworking.

Examples

Volume of a cylindrical tank

V = π × 0.5² × 1.2 ≈ 0.942 m³, about 942 liters (roughly 249 US gallons).

Input

cylinder, r = 0.5 m, h = 1.2 m

Result

V ≈ 0.942 m³

Sphere

V = 4/3 × π × r³ and S = 4πr²; the volume is about 4.19 liters.

Input

r = 10 cm

Result

V ≈ 4,188.8 cm³, S ≈ 1,256.6 cm²

Cone

The slant height is √(3² + 4²) = 5 cm; V = π × 9 × 4 ÷ 3 ≈ 37.70 cm³ and the total surface area is π × 3 × (3 + 5) ≈ 75.40 cm².

Input

r = 3 cm, h = 4 cm

Result

slant height 5 cm, V ≈ 37.70 cm³

Trapezoid

(10 + 6) × 4 ÷ 2 = 32 m²; the midline is 8 m.

Input

bases 10 and 6 m, height 4 m

Result

A = 32 m²

Regular hexagon

Apothem ≈ 1.732 cm, perimeter 12 cm, A = 12 × 1.732 ÷ 2 ≈ 10.39 cm².

Input

6 sides of 2 cm

Result

A ≈ 10.39 cm²

Circular sector

The arc length is r × θ (θ in radians) and the area r² × θ ÷ 2.

Input

r = 5 cm, angle 60°

Result

arc ≈ 5.24 cm, A ≈ 13.09 cm²

Tips and best practices

  • Mix units if you need to: each field is converted before the calculation, and the result follows the unit of the first field.
  • To convert a volume to liters: 1 m³ = 1,000 L and 1 dm³ = 1 L; 1,000 cm³ is also 1 L. One US gallon is about 3.785 L.
  • For a composite shape, solve each part separately, then add or subtract the areas and volumes.
  • If an expected result is missing, it cannot be derived from the data: choose another combination of dimensions.
  • Angles for a sector or a triangle can be entered in degrees, radians or gradians.

Sources and references

Frequently asked questions

Which shapes are supported?

Twelve flat shapes (square, rectangle, circle, general and right triangle, trapezoid, parallelogram, rhombus, ellipse, annulus, sector, regular polygon) and seven solids (cube, cuboid, sphere, cylinder, cone, square pyramid, torus).

Why are some results not shown?

A shape is not always fully determined by the data you enter. With a base and a height, the area of a triangle is known but not its sides, so not its perimeter either. The solver only shows what can be calculated with certainty.

How do you calculate the volume of a cylinder?

V = π × r² × h: the area of the circular base times the height. With r = 10 cm and h = 30 cm, V ≈ 9,425 cm³, or about 9.4 liters.

What is the formula for the volume of a sphere and of a cone?

Sphere: V = 4/3 × π × r³. Cone: V = π × r² × h ÷ 3, one third of the cylinder with the same base and height. The same goes for a pyramid compared with a prism.

Is the perimeter of the ellipse exact?

No, it is an approximation: there is no simple exact formula. The tool uses Ramanujan's, whose error is negligible for everyday use.

Can I use several units in the same calculation?

Yes. Each dimension has its own selector (mm, cm, m, km, inches, feet, yards) and the conversion happens before the calculation.

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