Mathematical relationships calculators

Bilinear interpolation blends four corner values on a rectangle. Coordinate intervals must be nonzero; a midpoint averages the four corners.

Addition of logarithms multiplies their positive arguments; subtraction divides them. Real bases must be positive and different from one.

Cosine is the horizontal coordinate on the unit circle. The inverse returns a principal angle from 0° through 180°; other angles can share its cosine.

The inputs are positive ellipse semiaxes, not full axis lengths. Ramanujan’s approximation gives the perimeter; equal semiaxes reduce exactly to a circle.

An isosceles trapezoid has equal legs. Half the difference between its parallel bases forms the horizontal leg of a right triangle with the height.

The altitude to the base bisects an isosceles triangle into two congruent right triangles. The equal leg must exceed the perpendicular height for a positive base.

Two points with distinct x coordinates define a line. Interpolation uses its value between the points; outside that interval the same calculation is extrapolation.

Perpendicular height determines parallelogram area with the base. Other modes use two sides and their included angle, or two diagonals and their included angle.

Percent means per hundred. A 25% share of 80 is 20; the percentage adjustment section applies an increase or decrease to its original value.

Perimeter is the sum of four edge lengths. The coordinate mode follows consecutive vertices around the boundary and adds their Euclidean distances.

Relative change divides the signed difference by the magnitude of the nonzero baseline. An increase from 80 to 100 is 0.25 or 25%.

A right triangle has perpendicular legs a and b, a hypotenuse c and complementary acute angles. Its area is half the product of the legs.

A ring torus is formed by revolving a circular tube about an external axis. Inner and outer radii are measured from that axis; require 0 ≤ inner radius less than outer radius.

Sine is the signed vertical coordinate on the unit circle. The inverse returns the principal angle between −90° and 90°; it does not recover every periodic solution.

Rise over run is the signed slope between two coordinate points. Vertical lines have zero run and no finite slope; distance remains positive.

A radius to a point of tangency is perpendicular to the tangent. The external point’s center distance d must be strictly greater than the positive radius r.

The bases of a trapezoid are parallel, so the two interior angles on each leg sum to 180°. The four interior angles sum to 360°.

Three noncollinear vertices determine a unique circumcircle. A right triangle’s circumcenter is the midpoint of its hypotenuse.

Two positive sides and an included angle strictly between 0° and 180° determine one triangle. The cosine rule determines its third side.

A triangle uses positive rise divided by run. The line selector distinguishes upward from downward direction, assigning the sign while keeping lengths positive.

A prism of constant triangular cross section has volume equal to its face area times perpendicular prism length. Other modes calculate that face area from triangle dimensions.