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Frustum Calculator

Calculate volume, lateral surface area, and total surface area of a frustum (truncated cone).

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Top Radius (r₁)

Bottom Radius (r₂)

Height (h)

Frustum Diagram

Frustum Formulas

Basic Formulas:

Slant Height: l = √[h² + (r₂ - r₁)²]
Volume: V = πh/3(r₁² + r₁r₂ + r₂²)
Lateral Area: LSA = π(r₁ + r₂)l

Total Surface Area:

Top Area: A₁ = πr₁²
Bottom Area: A₂ = πr₂²
Total: SA = πr₁² + πr₂² + π(r₁+r₂)l

Understanding Frustums

A frustum is the portion of a cone (or pyramid) that remains after cutting off the top with a plane parallel to the base. It's like a cone with the tip cut off.

Key Properties:

When r₁ = 0, the frustum becomes a complete cone
When r₁ = r₂, it becomes a cylinder
Volume formula uses the geometric mean of the two circular areas
The lateral surface unrolls into a ring sector

Volume Formula Derivation:

Frustum volume = Large cone volume - Small cone volume. After algebraic simplification, this gives the elegant formula: V = πh/3(r₁² + r₁r₂ + r₂²).

Frequently Asked Questions

What are real-world examples of frustums?

Lampshades, flower pots, buckets, drinking cups, cooling towers, volcanic calderas, and many architectural columns are frustum-shaped.

How do I find the height of the original cone?

If H is the full cone height: H/r₂ = (H-h)/r₁, so H = hr₂/(r₂-r₁). This uses similar triangles formed by the cross-section.

Why is the volume formula so complex?

The formula accounts for how the cross-sectional area changes from r₁ to r₂. It's actually the average of three areas: top, bottom, and their geometric mean.

What is a pyramidal frustum?

A pyramidal frustum has polygonal bases (like squares or hexagons) instead of circles. The volume formula is similar: V = h/3(A₁ + √(A₁A₂) + A₂).

Related Calculators

Cone Cylinder Pyramid Sphere

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