Pyramid Angle Calculator - Alpha, Beta, Gamma and Delta
Use this pyramid angle calculator to find alpha, beta, gamma, and delta for a right regular pyramid. Pick square through octagon, enter base side and height.
Pyramid Angle Calculator
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What Is a Pyramid Angle Calculator?
A pyramid angle calculator is a geometry tool that returns the four characteristic angles alpha, beta, gamma, and delta, plus the slant edge length and the base inradius and circumradius, for any right regular pyramid. The calculator is built for students, teachers, hobbyists, and engineers.
- • Classroom geometry: Verify alpha, beta, gamma, and delta for a homework problem on a right regular pyramid.
- • Model and craft planning: Pick the right base polygon and slant edge to lay out triangular faces for a model.
- • Architecture checks: Reproduce the angle set of a real pyramid such as the Great Pyramid of Giza.
- • Engineering checks: Confirm a roof, hopper, or pyramidal mold has the face slope and apex angle the spec calls for.
A right regular pyramid has a regular polygon base (all sides equal, all interior angles equal) and an apex directly above the center. The symmetry means all triangular faces are identical isosceles triangles, so the angle set is described by alpha, beta, gamma, and delta. The calculator covers square through octagon bases.
When the same problem also asks for the capacity or material volume of the solid, the Pyramid Volume Calculator uses the same side and height inputs and adds one third of base area times height.
How a Pyramid Angle Calculator Works
The calculator derives the base inradius and circumradius of a regular n-gon from the side length, then applies the arctan, arccos, and isosceles triangle identities to produce the four characteristic angles and the slant edge length.
- n: Number of sides in the regular base: 4 for square, 5 for pentagon, 6 for hexagon, 7 for heptagon, 8 for octagon.
- s: Length of one base side. All base sides are equal.
- h: Vertical height from the center of the base to the apex.
- r: Base inradius (apothem), equal to s / (2 * tan(pi/n)).
- R: Base circumradius, equal to s / (2 * sin(pi/n)).
For a square pyramid (n=4), r equals s/2 and R equals s/sqrt(2). For a hexagon, r = s*sqrt(3)/2 and R = s, so the worked example for side 2 gives alpha = arctan(3/sqrt(3)) = 60 degrees exactly. As n grows, r and R converge and the alpha-beta gap closes.
Example: Great Pyramid of Giza (square, s = 230.6, h = 146.7)
n = 4, s = 230.6 m, h = 146.7 m
r = 115.3, R = 163.0588. alpha = 51.83 deg, beta = 41.98 deg. L = 219.34. gamma = 58.29 deg, delta = 63.42 deg.
alpha = 51.83 deg, beta = 41.98 deg, gamma = 58.29 deg, delta = 63.42 deg, slant edge = 219.34 m.
These four angles match the published geometry of the Great Pyramid within rounding.
Example: regular hexagonal pyramid s = 2, h = 3
n = 6, s = 2, h = 3
r = 1.7321, R = 2. alpha = 60 deg, beta = 56.31 deg. L = 3.6056. gamma = 73.90 deg, delta = 32.20 deg.
alpha = 60.00 deg, beta = 56.31 deg, gamma = 73.90 deg, delta = 32.20 deg, slant edge = 3.6056.
For a regular hexagonal pyramid, alpha is the clean arctan(sqrt(3)) = 60 degrees, which gives an exact sanity check on the calculator.
According to Wolfram MathWorld, a regular polygon with n sides of length s has inradius r = s/(2*tan(pi/n)) and circumradius R = s/(2*sin(pi/n)).
When the user only needs alpha or beta and wants to confirm the right-triangle step, the Arctan Calculator returns the same arctan value in degrees, radians, or gradians.
Key Concepts Explained
These four terms decide which right triangle feeds which angle on a pyramid; mixing them up is the most common reason a textbook answer comes out wrong.
Regular Base (n-gon)
A regular base has all sides equal and all interior angles equal. The number of sides n controls r and R. Square (n=4) gives the largest gap; octagon (n=8) makes alpha and beta nearly equal.
Alpha (face median to base)
Alpha is arctan(h/r), measured at the apex between the median of one face and the base. The median drops to the midpoint of a base side, so the relevant leg is r.
Beta (apex to corner)
Beta is arctan(h/R), measured at the apex between a slant edge and the base. The slant edge ends at a base vertex, so the relevant leg is R. Beta is always smaller than alpha in a regular pyramid.
Gamma and Delta (face triangle)
Each face is an isosceles triangle with base s and equal sides L. Gamma is the base angle arccos((s/2)/L), and delta is the apex angle 180 - 2*gamma. Their sum is 180 degrees.
The right triangle that gives alpha and beta uses the height h, inradius r, and circumradius R. On each face, the isosceles triangle uses the base side s and the slant edge L, so s/2 and L form the right triangle that defines gamma.
If the user wants to step through the right triangle that gives alpha or beta by hand, the Right Triangle Calculator solves the same pair of legs and hypotenuse and shows the matching angle.
How to Use This Calculator
The pyramid angle calculator returns alpha, beta, gamma, and delta for a right regular pyramid whenever a problem gives the base side and the vertical height. The defaults (n = 4, s = 230.6, h = 146.7) reproduce the Great Pyramid of Giza.
- 1 Pick the base polygon: Use the dropdown to choose square (4), pentagon (5), hexagon (6), heptagon (7), or octagon (8).
- 2 Enter the base side length: Type the length of one base side. Use the same length unit for side and height.
- 3 Enter the vertical height: Type the perpendicular distance from the center of the base to the apex. Do not enter the slant edge.
- 4 Read the four angles: The result panel lists alpha, beta, gamma, and delta in degrees, plus the slant edge, inradius, and circumradius.
- 5 Sanity check the face sum: Verify 2 * gamma + delta = 180 degrees. Drift larger than 0.01 degrees means a unit mix-up or a wrong selector.
- 6 Reset for a new pyramid: Use Reset to restore the Great Pyramid defaults (n = 4, s = 230.6, h = 146.7).
For a square roof pyramid with a 4 m base side and a 3 m vertical height, pick square (n = 4), enter s = 4 and h = 3. The result is alpha = 56.31 deg, beta = 46.69 deg, gamma = 60.95 deg, and delta = 58.10 deg.
When the angles look right and the next step is the volume of the solid, the Square Pyramid Volume Calculator uses the same base side and height to return the interior capacity.
Benefits of Using This Calculator
Putting all four angles plus the slant edge on one form removes the most common pyramid-problem error path: confusing r and R, or mixing alpha and beta.
- • All four named angles at once: Alpha, beta, gamma, and delta appear together so the user can compare face center slope, corner slope, and face triangle angles.
- • Five base polygons on one page: The same side-and-height input feeds square, pentagon, hexagon, heptagon, and octagon bases.
- • Built-in r and R output: Inradius and circumradius are returned alongside the angles, so the user can see which leg feeds alpha versus beta.
- • Giza cross-check defaults: The default n = 4, s = 230.6, h = 146.7 matches a published pyramid example.
- • Mobile-friendly input: The polygon dropdown plus two short numeric fields sit on a compact responsive layout.
These benefits add up for users running many pyramid problems in a row. The slant edge and the two base radii are reported in the same view as the angles, and gamma and delta are always listed together so the 180 degree interior sum can be read off the page and checked before the answer is written down.
When the user only needs the inradius and circumradius of the regular base for a separate problem, the Hexagon Calculator returns the same r and R from a single side length input.
Factors That Affect Your Results
Four factors and two limitations control how reliable a pyramid angle result is for real geometry work.
Regularity of the base polygon
The formulas assume all base sides are equal and the apex sits above the center. An irregular polygon or an off-center apex breaks the r and R identities.
Polygon side count
The number of base sides n decides how close r and R are to each other. For a square the gap is large; for an octagon the gap is small.
Slenderness of the pyramid
When the height is much larger than the base side, alpha and beta both approach 90 degrees and delta shrinks. When the height is much smaller than the base side, alpha and beta both approach 0 degrees and delta approaches 180 degrees.
Input unit consistency
Mixing units (meters for side and feet for height) gives r, R, and the slant edge in mixed units. Use the same length unit for both inputs.
- • The formulas cover only right regular pyramids. An oblique pyramid needs a vector approach, and an irregular base needs a piecewise face computation.
- • The selector is fixed to square through octagon. For n greater than 8, the same identities still work, but the user has to extend the dropdown.
A practical sanity check after every calculation is to confirm that 2 * gamma + delta = 180 degrees. Drift larger than the rounding error means a wrong selector, swapped inputs, or mixed units. Comparing alpha and beta is the second check: beta must be smaller than alpha, because R is always longer than r.
According to Omni Calculator, the Great Pyramid of Giza has alpha = 51.83 deg, beta = 41.98 deg, gamma = 58.29 deg, and delta = 63.42 deg when the base side is 230.6 m and the height is 146.7 m.
According to Wolfram MathWorld, an isosceles triangle with base b and equal sides L has base angle arccos((b/2)/L) and apex angle 180 - 2*arccos((b/2)/L).
When the user is more interested in the base area, face area, or volume of the same regular pyramid than in the angles, the Hexagonal Pyramid Calculator uses the same base side and height to return the surface and capacity outputs.
Frequently Asked Questions
Q: What is the base angle of a pyramid?
A: The base angle is the angle between a triangular face and the base plane. For a right regular pyramid, the median of a face forms alpha with the base, equal to arctan(h/r) where h is the vertical height and r is the base inradius.
Q: What is the formula for the face angle of a pyramid?
A: Alpha is arctan(h/r) for the face median, and beta is arctan(h/R) for the slant-edge corner. r and R come from s, h, and the number of base sides n.
Q: How do you calculate the apex angle of a pyramid?
A: Compute the slant edge from apex to base vertex as sqrt(h^2 + R^2), then delta = 180 - 2 * arccos((s/2)/slantEdge). For s = 230.6 m and h = 146.7 m, delta = 63.42 degrees.
Q: What is the difference between alpha, beta, gamma, and delta?
A: Alpha is face median to base. Beta is slant edge to base. Gamma is the base angle of the triangular face. Delta is the face apex angle, equal to 180 minus 2*gamma.
Q: How do you find the angles of a square pyramid?
A: Set n = 4, then r = s/2, R = s/sqrt(2), L = sqrt(h^2 + R^2). alpha = arctan(h/r), beta = arctan(h/R), gamma = arccos((s/2)/L), and delta = 180 - 2*gamma.
Q: Why is the beta angle smaller than the alpha angle in a regular pyramid?
A: Beta uses the circumradius R and alpha uses the inradius r. For any regular polygon, R is larger than r, so arctan(h/R) is always smaller than arctan(h/r).