Pyramid Angle Calculator
Calculate the principal angles and slanted measurements of a regular right pyramid from its base polygon, side length, and vertical height.
Base and measurements
Choose a regular polygon with 4 – 8 equal sides.
Changing units converts the current side and height.
Required positive decimal; use a period as the decimal separator.
Required vertical distance from the base centre to the apex.
Angles
The main face slope angle measured from the horizontal base.
Alpha is 51.83 degrees for a square pyramid.
Calculated measurements
| Measurement | Symbol | Value | Role in the geometry |
|---|---|---|---|
| Side | AB | 230.600 m | One edge of the regular base |
| Segment | MC | 115.300 m | Base apothem: midpoint of a side to centre |
| Height | OC | 146.700 m | Perpendicular base-to-apex distance |
| Slanted side | BO | 219.338 m | Lateral edge from a base vertex to apex |
| Slanted height | MO | 186.588 m | Face median from side midpoint to apex |
Angle comparison
Angle details
| Angle | Current value | Geometric meaning |
|---|---|---|
| Alpha (α) | 51.83° | Angle between the face median and the base |
| Beta (β) | 41.98° | Angle between a lateral edge and the base |
| Gamma (γ) | 58.29° | Either equal base angle of a triangular face |
| Delta (δ) | 63.43° | Apex angle inside one triangular face |
How to use the pyramid angle calculator
What this calculator does
This calculator solves a regular right pyramid: the base is a regular polygon, and the apex sits directly above the base centre. From a base side and perpendicular height, it calculates the base apothem, lateral edge, face slant height, and four commonly used angles. It is useful for geometric checks, model making, architectural sketches, roof-like forms, classroom exercises, and comparing pyramid proportions. It does not solve an oblique pyramid, an irregular base, construction tolerances, material thickness, or structural loads.
When to use it
Use it when you need the slope of a pyramid face, the inclination of a corner edge, the internal angles of a triangular face, or missing slanted dimensions. It also helps compare two designs that have different heights or base polygons while keeping a consistent unit system.
How to calculate
- The calculator opens with a ready-to-use square-pyramid demonstration based on a 230.6 m side and 146.7 m height. The results and a validated Excel workbook are available immediately.
- Select the Base polygon. Then choose the Length unit; changing it converts the current Side (AB) and Height (OC) rather than merely changing the label.
- Replace the demonstration values with your own positive decimals. Results, tables, and the angle chart update live. Use a period for decimals; commas, scientific notation, zero, and negative values are rejected to prevent ambiguous geometry.
- Read Alpha (α) as the principal face slope, then compare Beta (β), Gamma (γ), and Delta (δ). Review the calculated lengths in the measurement cards and table.
- Select Download Excel to export the current inputs and computed values as a real .xlsx workbook. Reset clears the demonstration and all results; export remains unavailable until a complete valid side and height are entered again.
Input guide
Base polygon is required and accepts Square, Regular pentagon, Regular hexagon, Regular heptagon, or Regular octagon. The startup example uses Square. Increasing the number of sides changes the base apothem and circumradius for the same side length, which changes every angle and slanted length. Do not use this control for an irregular polygon.
Length unit is required and supports metres, centimetres, millimetres, feet, and inches. The startup example uses metres. Both length inputs must use the selected unit. A common mistake is entering the side in one unit and the height in another; convert them first or use the unit selector to convert both together.
Side (AB) is a required positive decimal representing one base edge; 230.6 is the example value. A larger side with unchanged height generally makes the pyramid flatter, reducing alpha and beta. Enter only a plain decimal with a period, without a unit symbol or thousands separator.
Height (OC) is a required positive decimal from the base centre straight up to the apex; 146.7 is the example value. A larger height with unchanged side makes the pyramid steeper, increasing alpha and beta while changing the face angles. Do not substitute slanted height for vertical height.
Output guide
Alpha (α) is the angle between the face median and horizontal base; it is an exact trigonometric consequence of the inputs, displayed to two decimals. Beta (β) is the smaller corner-edge inclination. Gamma (γ) is each equal base angle of a triangular face, and Delta (δ) is that face's apex angle. The summary pills repeat the base type, alpha, and delta. Values near 0° indicate a very flat relationship; values near 90° indicate a near-vertical relationship.
Segment MC is the base apothem. Slanted side (BO) is the lateral edge from a base vertex to the apex. Slanted height (MO) is the median up one triangular face. The Calculated measurements table restates these lengths and their roles. The Angle comparison chart and Angle details table use the same four degree values, so the chart is a comparison rather than a separate estimate.
Worked example
For the startup square pyramid, AB = 230.6 m and OC = 146.7 m. A square's apothem is half its side, so MC = 115.3 m. Therefore α = arctan(146.7 ÷ 115.3) = 51.83°. The half-diagonal is 163.059 m, giving β = 41.98°. The lateral edge is √(146.7² + 163.059²) = 219.338 m. This gives γ = arccos(115.3 ÷ 219.338) = 58.29°, and δ = 180° – 2 × 58.29° = 63.43°. These values match the first-open results and workbook.
Learn more
The angle calculations use inverse tangent and inverse cosine. The NIST Digital Library of Mathematical Functions overview of inverse trigonometric functions provides the formal definitions, while the OpenStax review of right-triangle trigonometric functions gives practical background for the ratios used here.
Formulas and assumptions
Let n be the number of sides in the regular base, s the side length, and h the vertical height. The base apothem r and circumradius R are determined by the regular polygon:
r = s ÷ (2 tan(π ÷ n)) and R = s ÷ (2 sin(π ÷ n))
The face slant height is l = √(h² + r²), while the lateral edge is e = √(h² + R²). The principal angles then follow from α = arctan(h/r), β = arctan(h/R), γ = arccos((s/2)/e), and δ = 180° – 2γ. The calculation uses full internal precision and rounds only the displayed values.
Because every length formula is homogeneous, scaling both side and height by the same factor changes all lengths by that factor but leaves all four angles unchanged.
Interpreting the geometry
For every convex regular base in this calculator, beta is smaller than alpha. The base vertex is farther from the centre than the midpoint of a side, so the same vertical rise produces a shallower angle along the corner edge. Gamma and delta describe the triangular face itself: two gamma angles plus delta must equal 180°. This identity is also a useful manual check.
Use consistent units and geometric measurements rather than nominal dimensions. For SI length conventions and metre-based conversions, see the NIST guidance on SI units of length. For physical construction, account separately for cuts, joints, surface thickness, and local engineering requirements.