Pyramid Angle Calculator

A regular pyramid has a regular polygon for its base and an apex directly above the base center. That symmetry makes every base edge equal, every triangular lateral face congruent, and every lateral edge equal. This calculator uses the base side length a, perpendicular height h, and number of base sides n to derive two important angles plus the slant height and lateral edge. It works for triangular, square, pentagonal, hexagonal, and higher-sided regular pyramids. The first intermediate measurement is the base apothem, the perpendicular distance from the polygon center to the midpoint of a side. For a regular n-gon it equals a divided by twice the tangent of pi divided by n. The base circumradius, measured from the center to a vertex, equals a divided by twice the sine of pi divided by n. Combining the pyramid height with the apothem forms a right triangle whose hypotenuse is the slant height. Combining the height with the circumradius similarly gives the lateral edge. The face apex angle is the angle at the top of one triangular lateral face between its two equal lateral edges. Splitting that isosceles face down the middle produces a right triangle, so the full angle is twice the arctangent of half the base side divided by the slant height. The base-to-face dihedral angle measures the opening between the base plane and a lateral face along a base edge. In the center-to-midpoint cross-section, it is the arctangent of height divided by base apothem. All angle results are reported in degrees and lengths retain the input unit. If the side and height are entered in centimeters, both derived lengths are centimeters. The calculator assumes an ideal right regular pyramid. It does not describe an oblique pyramid, an irregular base, a truncated pyramid, or a shape whose apex is displaced from the center. Those geometries require additional coordinates or measurements. These calculations support geometry exercises, architectural models, fabrication layouts, roof concepts, packaging, and three-dimensional design. Slant height is especially useful for lateral surface area, while lateral edge length helps size ribs from the apex to base vertices. The face apex angle describes each triangular panel, and the dihedral angle describes how steeply a panel rises from the base. Together they provide a compact, internally consistent description of the regular pyramid.