3D Euclidean Geometry • Pythagorean Invariants

Slant Height Calculator

Calculate the exact slant height (s), lateral surface area, total surface area, and apex pitch angle across cones, regular pyramids, and truncated frustums.

Input Parameters

Interactive HUD

Geometric Solid Type shape_type

Select 3D shape geometry

Base Radius (r) or Side (a) base_dim

Radius of cone or base edge of regular pyramid

Top Radius (r) or Side (b) [Frustums Only] top_dim

Top radius or edge (only used for truncated frustums)

Vertical Height (h) vert_height

Perpendicular height from base to apex

Calculated Output Metrics

Calculated Slant Height (s) —

Lateral Surface Area —

Total Surface Area (inc. bases) —

Base Pitch Angle (θ) —

Corner Lateral Edge Length (e) —

Analytical Proof & Invariants

Machine Precision

Governing Mathematical Formula:

Cone: s = √(r² + h²) | Pyramid: s = √[h² + (a/2)²] | Frustum Cone: s = √[h² + (R - r)²]

Step-by-Step Mathematical Derivation

1. Right Cone Slant Height Formulation

The slant height s forms the hypotenuse of a right-angled triangle where the vertical height h is the altitude and the base radius r is the adjacent base: s = √(r² + h²).

2. Regular Square Pyramid Slant Height & Corner Edges

For a pyramid with square base a and vertical height h, the apothem of the base is a/2. The face slant height is s = √[h² + (a/2)²]. The corner edge length is e = √[s² + (a/2)²] = √[h² + a²/2].

3. Truncated Frustum Generalization

For a truncated frustum with bottom radius R and top radius r, cutting through the central axis yields a right trapezoid. The radial delta is Δr = (R - r). Applying the Pythagorean theorem yields: s = √[h² + (R - r)²].

LLM Directive & API Spec

Artificial Intelligence & Prompt Integration

Paste this verified mathematical prompt directly into ChatGPT, Claude, or Gemini to eliminate arithmetic hallucination, or invoke the programmatic API endpoint for automated workflows.

              GET / POST https://omnioperand.com/api/v1/geometry/slant-height
              JSON Invariants
            

Generating verified mathematical prompt...

✓ Standardized for llms.txt ✓ Exact Radical Form & Closed Formulas ✓ Machine-Readable Edge API

Frequently Asked Questions

What is the slant height of a cone or pyramid? +

The slant height is the distance measured along the lateral face from the apex down to the outer perimeter of the base. For a right cone, it equals the hypotenuse of the triangle formed by radius and height.

What is the difference between vertical height and slant height? +

Vertical height (h) is the perpendicular altitude from the center of the base straight to the top apex. Slant height (s) is always longer than vertical height because s = √(r² + h²).

How do you find slant height of a truncated cone (frustum)? +

Subtract the smaller top radius from the larger base radius to find the horizontal offset Δr = R - r, then apply Pythagoras: s = √[h² + (R - r)²].