Geometry problems become much easier when you start with the information you already know instead of searching for a different calculator for every formula. A circle, rectangle, triangle or solid is a system of related measurements: once enough of them are known, many others can be deduced.

The Bethemesh Geometry Calculator follows that principle. Select a shape, indicate the dimensions you know, enter their values and the calculator derives everything that can be determined without inventing missing information.

Circle: radius, diameter, circumference and area

For a circle, the main quantities are directly related. If the radius is (r), the diameter is (2r), the circumference is (2\pi r), and the area is (\pi r^2).

For example, a circle with a radius of 5 cm has a diameter of 10 cm, a circumference of about 31.42 cm and an area of about 78.54 cm².

Calculate a circle from its radius.

If the value you know is the diameter instead, there is no need to rearrange the formulas yourself. Open the circle calculator from the diameter.

Square

A square is completely determined by one side. For side (a), its perimeter is (4a), its area is (a^2), and its diagonal is (a\sqrt{2}).

A 6 cm square therefore has a perimeter of 24 cm, an area of 36 cm² and a diagonal of approximately 8.49 cm.

The reverse problem is also useful: if you know the diagonal, the side can be recovered before calculating the perimeter and area.

Calculate the dimensions of a square.

Rectangle

A rectangle is usually described by its length and width. Its area is their product, its perimeter is twice their sum, and its diagonal follows the Pythagorean theorem.

For a rectangle measuring 10 cm by 4 cm:

• area: 40 cm²;
• perimeter: 28 cm;
• diagonal: approximately 10.77 cm.

Open the rectangle calculator.

Triangle: the important part is what you know

Triangles illustrate why a geometry solver is more useful than a collection of isolated formulas. The method depends on the available information.

Base and height

When the base (b) and corresponding height (h) are known, the area is (bh/2). A base of 12 cm and a height of 7 cm give an area of 42 cm².

This information alone does not determine every side or angle, so a reliable calculator should not pretend that it does.

Three sides

If all three sides are known, Heron’s formula determines the area. The same data also make it possible to calculate the perimeter and angles.

Calculate a triangle from three sides.

Two sides and the included angle

With two sides and their included angle, the area follows (A=ab\sin(C)/2). The third side can then be obtained with the law of cosines, and the remaining angles can be derived.

Calculate a triangle from two sides and an angle.

Right triangle

A right triangle has a fixed 90° angle, which makes many deductions possible. If the two legs are known, the hypotenuse follows the Pythagorean theorem. If one leg and the hypotenuse are known, the other leg can be recovered.

For legs of 3 cm and 4 cm, the hypotenuse is 5 cm and the area is 6 cm².

Open the right-triangle solver.

Trapezoid

For parallel bases (a) and (b) and height (h), the area is:

[ A=\frac{(a+b)h}{2} ]

For bases of 8 cm and 14 cm with a height of 5 cm, the area is 55 cm².

Calculate a trapezoid.

Parallelogram and rhombus

A parallelogram’s area is base × corresponding height. Its perimeter requires the two side lengths.

A rhombus can be approached through its side and height or through its diagonals. When the diagonals (d_1) and (d_2) are known, its area is (d_1d_2/2).

These cases show why the calculator asks which measurements are known before displaying inputs.

Ellipse

An ellipse with semi-major axis (a) and semi-minor axis (b) has area (\pi ab). Unlike a circle, its perimeter does not have a simple elementary closed form, so numerical approximations are used when needed.

Calculate an ellipse.

Annulus and circular sector

An annulus is the region between two concentric circles. With outer radius (R) and inner radius (r), its area is (\pi(R^2-r^2)).

A circular sector depends on a radius and central angle. Its area is the corresponding fraction of the full circle.

These shapes are useful examples of why angles and lengths must keep distinct units.

Regular polygon

Instead of having separate calculators for pentagons, hexagons, octagons and every other regular polygon, a generic regular-polygon model is enough. Specify the number of sides and the side length.

This is one of the main principles behind the Bethemesh calculator: consolidate related calculations rather than multiplying nearly identical tools.

Calculate a regular polygon.

3D geometry

For solids, distinguish length, surface area and volume. If dimensions are entered in centimetres, lengths are expressed in cm, areas in cm² and volumes in cm³.

Cube

For edge (a):

• volume: (a^3);
• total surface area: (6a^2);
• face diagonal: (a\sqrt2);
• space diagonal: (a\sqrt3).

A cube with a 4 cm edge has volume 64 cm³ and surface area 96 cm².

Calculate a cube.

Rectangular prism

With length (l), width (w) and height (h), volume is (lwh). Total surface area is (2(lw+lh+wh)).

Calculate a rectangular prism.

Sphere

A sphere is determined by its radius. Its surface area is (4\pi r^2) and volume is (4\pi r^3/3).

A sphere with radius 3 cm has surface area about 113.10 cm² and volume about 113.10 cm³. The numerical values happen to match for this radius, but the units and physical quantities are different.

Calculate a sphere.

Cylinder

A cylinder requires a radius and height. The base area is (\pi r^2), volume is (\pi r^2h), lateral area is (2\pi rh), and total surface area includes both circular bases.

Calculate a cylinder.

Cone

For radius (r) and height (h), cone volume is (\pi r^2h/3). The slant height can be derived with the Pythagorean theorem and then used for lateral and total surface areas.

Calculate a cone.

Square pyramid

A square pyramid combines a square base with triangular faces. Knowing the base side and vertical height determines the volume, while surface calculations also use the slant height.

Calculate a square pyramid.

Torus

A torus is described by a major radius and a tube radius. These two measurements determine its surface area and volume.

Calculate a torus.

Mixing units safely

Real problems do not always provide every dimension in the same unit. A radius may be given in centimetres while a height is given in metres. Converting first prevents errors by factors of 10, 100 or 1,000.

The geometry calculator handles units directly in its quantity fields. For independent conversions, use the Universal Converter.

Common mistakes

The most common geometry mistakes are not difficult formulas but mismatched data: using diameter as radius, confusing height with a slanted side, mixing cm and m, forgetting that areas use squared units, or assuming that insufficient information uniquely determines a triangle.

A good workflow is simple: identify the shape, list the measurements you actually know, normalize the units, choose a compatible solving method, then check whether the resulting unit matches the quantity being calculated.

Why use one geometry calculator?

A consolidated solver has an important advantage: the user does not need to decide in advance which isolated formula page to search for. The shape and known measurements determine what can be calculated.

That is why Bethemesh does not try to present hundreds of nearly identical calculators. The Geometry Calculator groups the most useful 2D and 3D cases in one interface and only displays results that are mathematically justified.