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Pyramid Angles — Base-to-Face, Face-to-Face, Apex

Compute slant height, base-to-face angle, face-to-face (dihedral) angle, and apex angle for square and triangular pyramids. From the Great Pyramid to tetrahedrons.

Concept Fundamentals
s = √(h² + (a/2)²)
Slant Height
α = tan⁻¹(h/(a/2))
Base-to-Face
π − 2·sin⁻¹((a/2)/s)
Face-to-Face
γ = 2·tan⁻¹((a/2)/h)
Apex Angle

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Base-to-face angle increases with height—steeper pyramids Great Pyramid of Giza has base-to-face ~51.5° Regular tetrahedron face-to-face = arccos(-1/3) ≈ 109.47° Slant height forms right triangle with height and half-base Apex angle decreases as pyramid gets taller

Key quantities
s = √(h² + (a/2)²)
Slant Height
Key relation
α = tan⁻¹(h/(a/2))
Base-to-Face
Key relation
π − 2·sin⁻¹((a/2)/s)
Face-to-Face
Key relation
γ = 2·tan⁻¹((a/2)/h)
Apex Angle
Key relation

Ready to run the numbers?

Why: Pyramid angles determine structural stability, aesthetic proportions, and material joinery. The Great Pyramid's ~51.5° base-to-face angle has fascinated architects and historians for millennia.

How: Enter base side and height. Slant height s = √(h²+(a/2)²). Base-to-face angle = tan⁻¹(h/(a/2)). Face-to-face uses dihedral formula. Apex angle from geometry of triangular faces. Supports degrees or radians.

Base-to-face angle increases with height—steeper pyramidsGreat Pyramid of Giza has base-to-face ~51.5°
Sources:Wolfram MathWorld — PyramidKhan Academy — 3D Geometry

Run the calculator when you are ready.

Pyramid Angle CalculatorEnter base side and height to compute slant height and all pyramid angles
📐
3D GEOMETRYPyramid Angles

Pyramid Angle Calculator

Face-to-face, base-to-face, apex angles. Square and triangular pyramids.

Slant Height →Cone →

📐 Sample Examples — Click to Load

Pyramid Dimensions

pyramid_angle.sh
CALCULATED
$ pyramid_angles --base=10 --height=10
Slant Height
11.1803 cm
Base-to-Face
63.43°
Face-to-Face
126.87°
Apex Angle
53.13°
Share:
Pyramid Angles
Base=10 cm, Height=10 cm
63.43°
Slant: 11.18Face-Face: 126.87°Apex: 53.13°

3D Visualization

aha = 10, h = 10

Property Radar

Property Comparison

Angle Breakdown

📐 Calculation Breakdown

INPUT
Base Side
10 cm
Height
10 cm
GEOMETRY
Slant Height
11.1803 cm
√(h^{2} + (a/2)^{2})
ANGLES
Base-to-Face Angle
63.43°
\text{tan}⁻¹(h / (a/2))
Face-to-Face Angle
126.87°
π - 2·sin⁻¹((a/2)/s)
Apex Angle
53.13°
2·tan⁻¹((a/2)/h)

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

🏛️

The Great Pyramid base-to-face angle (~51.5°) may relate to the golden ratio or star positions

— Archaeology

🔺

Regular tetrahedron face-to-face angle is arccos(-1/3) ≈ 109.47°—same as methane (CH₄)

— Chemistry

🗼

Louvre Pyramid has 51° base-to-face angle—similar to Egyptian pyramids

— Architecture

📐

Slant height forms a right triangle with height and half-base; use Pythagoras

— Geometry

📋 Key Takeaways

  • • Base-to-Face Angle — angle between base and triangular face; steeper pyramids have larger values
  • • Face-to-Face Angle — dihedral angle between adjacent faces; square pyramids use π − 2·sin⁻¹((a/2)/s)
  • • Apex Angle — angle at the top vertex; gets smaller as pyramid gets taller
  • • Great Pyramid of Giza has base-to-face ~51.5° — studied for astronomical significance

💡 Did You Know?

🏛️The Great Pyramid base-to-face angle (~51.5°) may relate to the golden ratio or star positionsSource: Archaeology
🔺Regular tetrahedron face-to-face angle is arccos(-1/3) ≈ 109.47° — same as methane (CH₄)Source: Chemistry
📐Slant height forms a right triangle with height and half-base; use PythagorasSource: Geometry
🗼Louvre Pyramid has 51° base-to-face angle — similar to Egyptian pyramidsSource: Architecture
⛰️Steeper pyramids (tall relative to base) have larger base-to-face anglesSource: Trigonometry
📏Face-to-face angle determines how materials join at edges in constructionSource: Engineering

📖 How Pyramid Angle Calculations Work

Pyramid angles derive from the base dimensions and height. The slant height s=h2+(a/2)2s = \sqrt{h^2 + (a/2)^2} forms the hypotenuse of a right triangle.

Base-to-Face Angle

α=tan⁡−1(h/(a/2))\alpha = \tan^{-1}(h / (a/2)) — inverse tangent of height over half-base.

Face-to-Face (Square Pyramid)

β=π−2sin⁡−1((a/2)/s)\beta = \pi - 2\sin^{-1}((a/2)/s) — dihedral angle between adjacent triangular faces.

🎯 Expert Tips

Historical Accuracy

Compare your Giza pyramid angles with documented ~51.5° base-to-face.

Units

Use degrees for architecture; radians for calculus and physics.

Stability

Steeper pyramids (larger base-to-face) have different structural behavior.

Tetrahedron

Regular tetrahedron face-to-face is constant ~109.47° regardless of size.

⚖️ Comparison

FeatureThis CalculatorManualBasic Online
Square & triangular pyramids✅⚠️❌
Step-by-step✅✅❌
Charts & viz✅❌❌
AI analysis✅❌❌

❓ FAQ

What is the base-to-face angle?

The angle between the base plane and a triangular face. It determines how steep the pyramid appears.

Why is face-to-face different for triangular pyramids?

Regular triangular pyramids (tetrahedrons) have a constant dihedral angle arccos(-1/3) ≈ 109.47°.

How does height affect the angles?

Increasing height increases base-to-face angle (steeper) and decreases apex angle (sharper top).

What was the Great Pyramid angle?

Approximately 51.5° base-to-face — some researchers link it to astronomical or mathematical significance.

Can I use radians?

Yes — select radians in the angle unit dropdown for calculus and physics applications.

What is slant height?

The straight-line distance from apex to the midpoint of any base edge — the hypotenuse of the height/half-base triangle.

📊 Stats

51.5
Giza °
109.47
Tetrahedral °
4
Square faces
3
Tri faces

⚠️ Disclaimer: Results are mathematically precise. For critical construction, verify independently and account for measurement tolerances.

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