Geometry calculators
Area, volume, and the relationships between the sides of things. Each tool shows the formula and where it comes from, because the derivation is usually shorter than the formula looks, and it is what makes a result checkable rather than merely plausible.
Area CalculatorArea and perimeter for the common plane figures. A triangle given three sides is checked against the triangle inequality first, because three lengths that cannot close have no area.Perimeter CalculatorPerimeter and the area it encloses. For a fixed perimeter a circle encloses the most area of any shape, which is why the ratio is shown.Circle CalculatorEvery circle property from any one of four measurements, including sector, arc and segment. The circumference is exactly the derivative of the area with respect to radius, which is why growing a circle by a thin ring adds area equal to the circumference times the thickness.Triangle CalculatorEvery property of a triangle from its three sides. Three sides fix a triangle completely, so the angles, heights and both circle radii follow without further measurement, and any two sides must exceed the third or no triangle exists.Pythagorean Theorem CalculatorThe missing side of a right triangle, whichever one it is, with the angles and area that follow. The 3-4-5 form is how a corner gets squared on site without a set square, and the page checks whether a measured corner is true.Rectangle and Square CalculatorEvery measurement of a rectangle from its two sides, including the diagonal and the simplified aspect ratio. The diagonal is Pythagoras, which is why a 16:9 screen is measured across the corner.Trapezoid CalculatorTrapezoid area from the two parallel sides and the height. The area is the median times the height, which is the same rule as a rectangle with the average width.Polygon CalculatorRegular polygon area, angles and radii. As the side count rises the shape approaches a circle, which the apothem and circumradius converging makes visible.Ellipse CalculatorEllipse area, perimeter and eccentricity. The area is exact and simple; the perimeter has no closed form at all, which is a genuinely surprising fact about a shape this ordinary.Volume CalculatorVolume of the common solids, in cubic metres, litres, cubic feet and gallons. All volumes scale with the cube of a linear dimension, so doubling any one measurement of a box doubles the volume but doubling all three multiplies it by eight.Surface Area CalculatorSurface area of the common solids, with paint quantity. Area scales with the square of a dimension while volume scales with the cube, which is why a larger container needs proportionally less material.Sphere CalculatorSphere properties from any one of five measurements. The surface area is the derivative of the volume with respect to radius, which is why growing a sphere by a thin shell adds volume equal to the area times the thickness.Cylinder CalculatorCylinder volume and area, with partial fill. A vertical cylinder is the one common shape that fills linearly, so half the height really is half the volume, which is not true of a horizontal tank or a cone.Cone CalculatorCone volume and surface, including truncated cones. The curved surface uses the slant height rather than the vertical height, which is the mistake that makes a paper cone come out too small.Pyramid CalculatorPyramid volume and surface for a rectangular base. A pyramid holds exactly one third of the prism enclosing it, and the face slant and edge slant are different lengths that are easy to confuse.Angle ConverterAngles across degrees, radians, gradians and arcseconds. Radians are the natural unit because they make the trigonometric derivatives come out clean, which is why every programming language expects them.Arc Length and Sector Area CalculatorArc length, sector area, chord and segment from a radius and an angle. The formulas are trivial in radians and awkward in degrees, which is the whole argument for radians.Golden Ratio CalculatorGolden ratio divisions from any one known dimension. Phi is the only number whose reciprocal is itself minus one, which is what makes the proportion self-similar.Scale Factor CalculatorScale factor with its area and volume consequences. Doubling a length quadruples the area and multiplies volume by eight, which is why scaling a recipe or a structure is never proportional.