Geometry and function relationships calculators

The rectangle corner angles are 90°. Its diagonals bisect one another; adjacent intersection angles add to 180°. Enter the selected labeled angle in degrees.

Arctangent converts a real tangent value to its principal angle strictly between −90° and 90°. A tangent ratio does not identify the angle’s quadrant without other information.

A circular segment is bounded by a chord and its arc. The central angle in radians gives sector area minus the triangle area; minor and major segments use different angle branches.

Similar right triangles have equal ratios of corresponding sides. Match each leg to its counterpart; swapping legs changes those ratios.

Expand the squared offsets of a standard circle equation. C represents radius squared, while h and k are the center coordinates.

The center is the midpoint of opposite vertices, opposite co-vertices or the two foci. For the standard ellipse equation, it is the pair of horizontal and vertical offsets.

Hyperbolic sine and cosine use exponential growth rather than periodic angles. The reciprocal functions csch and coth are undefined at zero; inverse branches use real principal values.

These formulas apply to two circles whose boundaries cross twice. Each crescent equals its disk area minus their common overlap; concentric and separate disks need another geometric case.

An isosceles triangle has equal legs. Its altitude to the base splits the base in half and forms two right triangles; the two base angles are equal.

A logarithm base must be positive and different from one, and its argument must be positive. The change-of-base quotient gives the same value using any valid auxiliary base.

A right rectangular pyramid has its apex directly above the center of its rectangular base. Each pair of opposite faces has the same slant height; vertical height and slant height are different measures.

Choose two sides, one side and an acute angle, or area and a side. The legs are perpendicular and the two acute angles add to 90°.

A positive scale factor multiplies all three corresponding sides. The angles stay equal, while the area multiplies by the square of the scale factor.

Heron’s formula gives area from three positive sides satisfying every strict triangle inequality. A scalene triangle has three unequal sides; the right-triangle mode uses perpendicular legs.

Use degree or radian units consistently. Reciprocal ratios are undefined when the corresponding sine or cosine is zero; principal inverse branches represent only one periodic solution.

Choose a 45–45–90, 30–60–90, fixed-leg-ratio or 3–4–5 triangle. Scale its sides together; perimeter scales linearly and area scales quadratically.

Use the law of cosines with three sides or two sides and their included angle. Interior angles must be positive and sum to 180°; each side must be shorter than the other two combined.

Match each angle to the side opposite it. The law of cosines calculates the missing side from two known sides and their included angle; the angle-sum mode finds the third interior angle.

A full rotation is 360 degrees. Sine and cosine are defined for every real angle; tangent and reciprocal ratios have singularities where their denominators vanish. Principal inverse branches do not recover every periodic angle.

Evaluate the six trigonometric ratios for a signed angle, or calculate right-triangle measurements. Reciprocal ratios are undefined where their denominator is zero; inverse angles use principal branches.