Calculate face angle, dihedral angle, lateral edge angle, and apex angle of a regular pyramid. Supports square, triangular, pentagonal, and hexagonal bases. Also find volume and surface area.
<p>The <strong>Pyramid Angle Calculator</strong> computes all the key angles and measurements of a regular right pyramid — one whose base is a regular polygon and whose apex sits directly above the centre of the base. Select from four base types (square, triangular, pentagonal, or hexagonal), enter the base side length and the pyramid height, and the calculator does the rest.</p>
<p>The tool determines the <strong>face angle</strong> (the angle between the base plane and a lateral face), the <strong>lateral edge angle</strong> (the angle a lateral edge makes with the base), the <strong>dihedral angle</strong> between the base and each triangular face, the <strong>face apex angle</strong> at the top of each triangular face, and the <strong>face base angles</strong>. It also computes the slant height, lateral edge length, base apothem, volume, base area, lateral area, and total surface area.</p>
<p>Eight presets include the Great Pyramid of Giza, a regular tetrahedron, and pyramids of varying steepness and base type for instant exploration. Angle comparison bars let you see how the different angles relate, and a stacked bar breaks the surface area into base and lateral components. A reference table summarises the combinatorial properties of pyramids from 3-sided to 8-sided bases.</p>
<p>This calculator is ideal for geometry students exploring 3D solids, architects designing pyramidal structures, 3D modellers computing face normals, and anyone curious about the surprising geometry hidden in everyday pyramid shapes.</p>
Pyramid Angle problems often require several dependent steps, and a small arithmetic slip can propagate through every derived value. This calculator is tailored to that workflow: you enter base side length, pyramid height, base type, and it returns face angle (base–face), lateral edge angle, dihedral (base–face), face apex angle in one consistent pass. It is useful for homework checks, worksheet generation, tutoring walkthroughs, and fast field/design estimates where you need reliable geometry results without rebuilding the full derivation each time.
Slant height = √(h² + apothem²). Lateral edge = √(h² + circumradius²). Face angle = arctan(h / apothem). Lateral edge angle = arctan(h / circumradius). Volume = ⅓ × base area × h. Lateral area = ½ × perimeter × slant height.
Result: Face angle ≈ 71.57°
Apothem of square base = 10/2 = 5. Slant height = √(225 + 25) = √250 ≈ 15.81. Face angle = arctan(15/5) = arctan(3) ≈ 71.57°. Volume = ⅓ × 100 × 15 = 500.
This pyramid angle tool links the entered values (base side length, pyramid height, base type) to the target geometry relationships used in class and practice problems. Instead of solving each intermediate step manually, you can validate setup and arithmetic quickly while still tracing which measurements drive the final result.
Formula focus: the calculator formula
Pyramid Angle shows up in school geometry, technical drafting, construction layout checks, and early engineering design estimates. When values are changed repeatedly, the calculator helps you compare scenarios quickly and see how sensitive the shape is to each dimension.
Start with the primary outputs (face angle (base–face), lateral edge angle, dihedral (base–face), face apex angle) and then use the remaining cards/tables to confirm consistency with your diagram. Keep units consistent across inputs, and round only at the end if your assignment or project specifies a fixed precision.
The face angle (or slope angle) is the angle between the base plane and a lateral triangular face. It is measured from the base to the face along the apothem direction.
The dihedral angle is the angle between two planes. Here it measures the angle between the base plane and a lateral face of the pyramid.
It is the angle that a lateral edge (from a base corner to the apex) makes with the base plane. Use this as a practical reminder before finalizing the result.
The slant height is the distance from the apex to the midpoint of a base edge (along a face). The lateral edge is the distance from the apex to a base vertex — it is always longer than or equal to the slant height.
No. This calculator assumes a right regular pyramid where the apex is directly above the centre of the base. Oblique pyramids have asymmetric angles that require different formulas.
Square (4 sides), triangular (3), pentagonal (5), and hexagonal (6). The formulas generalise for any regular polygon base using the apothem and circumradius.