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Frustum of a Cone or Pyramid

Frustum of a Cone or Pyramid

A truncated cone or pyramid in which the plane cutting off the apex is parallel to the base.

Note: the word is frustum, not frustrum.

 

Frustum diagram with variables r (upper base radius), R (lower base radius), s (slant height), h (height), showing volume,...

 

 

See also

Slant height, volume, lateral surface area, surface area

Key Formula

V=h3(A1+A2+A1⋅A2)V = \frac{h}{3}\left(A_1 + A_2 + \sqrt{A_1 \cdot A_2}\right) For a conical frustum with circular bases: V=πh3(R2+r2+Rr)V = \frac{\pi h}{3}\left(R^2 + r^2 + Rr\right) Lateral surface area of a conical frustum: AL=π(R+r)ℓA_L = \pi(R + r)\ell
Where:
  • VV = Volume of the frustum
  • hh = Perpendicular height (vertical distance between the two parallel bases)
  • A1A_1 = Area of the larger base
  • A2A_2 = Area of the smaller base
  • RR = Radius of the larger circular base (conical frustum)
  • rr = Radius of the smaller circular base (conical frustum)
  • ℓ\ell = Slant height — the distance along the lateral surface between the edges of the two bases
  • ALA_L = Lateral surface area of the conical frustum

Worked Example

Problem: A conical frustum has a larger base radius of 6 cm, a smaller base radius of 3 cm, and a height of 8 cm. Find its volume.
Step 1: Write down the volume formula for a conical frustum.
V=πh3(R2+r2+Rr)V = \frac{\pi h}{3}\left(R^2 + r^2 + Rr\right)
Step 2: Substitute the given values: R = 6, r = 3, h = 8.
V=π⋅83(62+32+6⋅3)V = \frac{\pi \cdot 8}{3}\left(6^2 + 3^2 + 6 \cdot 3\right)
Step 3: Compute the expression inside the parentheses.
62+32+6⋅3=36+9+18=636^2 + 3^2 + 6 \cdot 3 = 36 + 9 + 18 = 63
Step 4: Multiply to get the volume.
V=8π3⋅63=504π3=168π≈527.8 cm3V = \frac{8\pi}{3} \cdot 63 = \frac{504\pi}{3} = 168\pi \approx 527.8 \text{ cm}^3
Answer: The volume is 168π≈527.8168\pi \approx 527.8 cm³.

Another Example

This example focuses on surface area rather than volume, and uses the slant height instead of the perpendicular height. It shows how the total surface area combines both circular bases with the lateral surface.

Problem: A conical frustum has a larger base radius of 10 cm, a smaller base radius of 4 cm, and a slant height of 10 cm. Find its total surface area.
Step 1: Identify the three parts of total surface area: the larger base, the smaller base, and the lateral surface.
Atotal=πR2+πr2+π(R+r)ℓA_{\text{total}} = \pi R^2 + \pi r^2 + \pi(R + r)\ell
Step 2: Calculate the area of the larger base.
πR2=π(10)2=100π\pi R^2 = \pi(10)^2 = 100\pi
Step 3: Calculate the area of the smaller base.
πr2=π(4)2=16π\pi r^2 = \pi(4)^2 = 16\pi
Step 4: Calculate the lateral surface area using the slant height.
AL=π(R+r)ℓ=π(10+4)(10)=140πA_L = \pi(R + r)\ell = \pi(10 + 4)(10) = 140\pi
Step 5: Add all three parts together.
Atotal=100π+16π+140π=256π≈804.2 cm2A_{\text{total}} = 100\pi + 16\pi + 140\pi = 256\pi \approx 804.2 \text{ cm}^2
Answer: The total surface area is 256π≈804.2256\pi \approx 804.2 cm².

Frequently Asked Questions

What is the difference between a frustum and a truncated cone?
A frustum is a specific type of truncated cone or pyramid where the cutting plane is parallel to the base. A truncated cone or pyramid, more generally, can be cut by a plane at any angle. So every frustum is a truncated solid, but not every truncated solid is a frustum.
How do you find the slant height of a frustum if you know the height and radii?
For a conical frustum, the slant height ℓ\ell can be found using the Pythagorean theorem. The slant height, the vertical height hh, and the difference of the radii (R−r)(R - r) form a right triangle. Use ℓ=h2+(R−r)2\ell = \sqrt{h^2 + (R - r)^2}.
Why does the frustum volume formula have a square root term?
The A1⋅A2\sqrt{A_1 \cdot A_2} term (or the RrRr term for cones) accounts for the way the cross-sectional area changes as you move from the large base to the small base. It is not a simple linear change, so you cannot just average the two base areas. The formula comes from integrating the continuously changing cross-sections, which introduces this geometric mean term.

Frustum vs. Complete Cone/Pyramid

FrustumComplete Cone/Pyramid
DefinitionThe portion remaining after the apex is cut off by a plane parallel to the baseThe full solid from base to apex
Number of basesTwo parallel bases (one large, one small)One base and one apex point
Volume formula (cone)V=πh3(R2+r2+Rr)V = \frac{\pi h}{3}(R^2 + r^2 + Rr)V=13πR2HV = \frac{1}{3}\pi R^2 H
When to useWhen the top is removed or the solid has two parallel ends of different sizesWhen the solid tapers to a single point

Why It Matters

Frustums appear frequently in geometry courses when you study volumes and surface areas of solids. Real-world objects shaped like frustums include buckets, lampshades, drinking cups, and sections of cooling towers. Understanding the frustum formula also reinforces how integration and cross-sectional area work, which connects directly to calculus concepts you will encounter later.

Common Mistakes

Mistake: Using the average of the two base areas instead of the full frustum formula to calculate volume.
Correction: The volume is NOT h2(A1+A2)\frac{h}{2}(A_1 + A_2). You must include the geometric mean term A1⋅A2\sqrt{A_1 \cdot A_2} and divide by 3, not 2. The correct formula is V=h3(A1+A2+A1A2)V = \frac{h}{3}(A_1 + A_2 + \sqrt{A_1 A_2}).
Mistake: Confusing the perpendicular height hh with the slant height ℓ\ell.
Correction: The height hh is the vertical distance between the two parallel bases. The slant height ℓ\ell is measured along the lateral surface. They are related by ℓ=h2+(R−r)2\ell = \sqrt{h^2 + (R - r)^2}. Volume uses hh; lateral surface area uses ℓ\ell. Mixing them up gives an incorrect answer.

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