How this instrument works
A right pyramid has its apex sitting directly above the center of its footprint, so the line straight up from that center to the apex is perpendicular to the ground plane. Cut the pyramid with a vertical plane that runs through the apex and the midpoint of one edge, and the cross-section splits into two mirror-image right triangles. One leg of each is Height, running straight up from the center; the other is Half the base width, running out along the ground to that edge's midpoint. Angle (base to apex, from center) is the angle this triangle makes at that edge midpoint — the pitch of the slant line climbing from the ground to the apex, which is also the angle a triangular face makes with the ground.
Because the two legs are perpendicular by construction, this is ordinary right-triangle trigonometry: tan θ equals the leg opposite θ (Height) over the leg adjacent to it (Half the base width), so θ itself is atan(height ⁄ half-base). Hold the base fixed and stretch the pyramid taller and θ climbs toward 90°, a needle-thin spire in the limit; hold the height fixed and widen the base instead and θ sinks toward 0°, the pyramid flattening into little more than a floor plan. Neither extreme is reached by an actual solid — a true pyramid keeps a positive height and a positive base — but the formula tracks the approach smoothly in both directions.
This is specifically the face angle, not the edge angle, and the difference is easy to miss. Half the base width is measured to an edge's midpoint, the short way across; the distance to a corner is longer — for a square footprint, it's the half-diagonal, half-base times √2 — so sighting up a corner instead of a face always gives a shallower angle than sighting up the middle of a face. The Great Pyramid at Giza is the textbook illustration: its faces climb from the ground at roughly 51.8°, a figure historians recover from exactly this ratio of height to half the base length, while the angle up its corner edges is noticeably flatter.
- Enter the pyramid's Height — the straight-up distance from the base to the apex.
- Enter Half the base width — half of one base edge's length, measured from the center to that edge's midpoint, not to a corner.
- Read Angle (base to apex, from center) for θ = atan(height ⁄ half-base), shown in degrees by default.
- Switch that field's unit to radians or turns if your next step needs one of those instead.
- For a rectangular base, run the sheet twice using each side's own half-width to get both face angles separately.
Worked example — a half-base of 3, a height of 4
A garden pyramid frame has a square base 6 units on a side, so Half the base width is 3, and it stands 4 units tall at the apex. θ = atan(4 ⁄ 3) = 0.9272952180016122 radians, which this sheet reports by default as 53.13010235415598°, or 53.13° rounded — the same 3-4-5 ratio that shows up across right-triangle geometry, here measuring how steeply the frame's four triangular faces lean outward from the ground.
Sight up one of the frame's corner edges instead of the middle of a face, and the relevant horizontal leg is no longer 3 but the half-diagonal of the square base, 3√2 ≈ 4.2426. That swap lowers the angle to atan(4 ⁄ 4.2426) ≈ 43.31° — visibly shallower than the 53.13° face angle, even though both describe the same physical frame, because a corner sits farther from the center than an edge's midpoint does.
Questions
What does 'half the base width' mean on this pyramid?
It's half the length of one edge, measured from the footprint's center out to that edge's midpoint — not the full edge, and not the distance to a corner. For a square footprint with 6-unit sides, that's 3; a rectangular footprint has two pairs of sides, giving two different half-base values and therefore two different face angles.
Is this the same as the angle up a pyramid's edge?
No — it's the face angle, measured to an edge's midpoint. The edge, or corner, angle uses a longer horizontal leg, the half-diagonal of the base, so it comes out shallower for the same pyramid. A base 6 units square with a 4-unit height gives a 53.13° face angle but only about 43.31° up a corner.
Why divide height by half-base rather than the other way round?
Because tan θ is opposite over adjacent, and at the corner of this right triangle where θ sits, Height is the leg opposite θ while Half the base width is the leg touching it. Reversing the division computes atan(half-base ⁄ height) instead, which equals 90° minus this angle — the OTHER acute angle in the same triangle, not this one.
What happens to the angle as a pyramid gets taller and narrower?
It climbs toward 90° without ever reaching it. A pyramid with half-base 1 and height 1 already sits at exactly 45°; push the height up while holding the base steady and height ⁄ half-base grows without bound, so its atan approaches — but never touches — a vertical 90° spire.
What happens as a pyramid gets flatter and wider instead?
The angle shrinks toward 0°. With half-base 10 and height only 1, this formula gives about 5.71° — a squat, low pyramid whose faces barely rise off the ground. Widen the base further, relative to the height, and the angle keeps sinking toward a flat 0° without ever quite reaching it.
Does this formula assume the apex sits directly above the base's center?
Yes — that is what makes the pyramid 'right' rather than oblique, and it is what guarantees Height and Half the base width meet at a true 90° angle inside the triangle this formula solves. An oblique pyramid, with its apex off to one side, has no single face angle this way; each face would lean at its own angle.