Calculate volume and surface area
Formulas:
Volume: V = (1/3)πh(R² + Rr + r²)
Slant Height: s = √(h² + (R-r)²)
Surface Area: A = π(R+r)s + πR² + πr²
Measure the radius of the smaller circular top (r).
Why: The top radius determines the upper cross-sectional area. This is crucial for the volume formula.
Measure the radius of the larger circular base (R). Confirm R ≥ r.
Why: The base is larger in a frustum. The inequality ensures valid geometry. R = r becomes a cylinder.
Measure the perpendicular distance between the top and base (h).
Why: Height must be perpendicular, not along the slant. Slant measurements overestimate height.
Apply V = (1/3)πh(R² + Rr + r²) and surface area formulas.
Why: The R² + Rr + r² term accounts for both circular areas plus their geometric mean. This comes from calculus integration.
Check that volume is positive and all radii/heights are valid.
Why: Documentation ensures reproducibility. Verification catches input errors before proceeding with design.
Traffic Cone Dimensions
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