What Is a Frustum of a Cone? | Easy Definition With Formulas

A frustum of a cone is the solid left after cutting a cone with a slice parallel to its base, leaving two circular ends.

You’ve seen this shape a hundred times: a paper cup, a plant pot, a bucket-style tumbler, a lampshade. In geometry, it shows up when a cone gets trimmed but the math still needs to work.

Below, you’ll get the meaning in plain terms, the labels teachers expect on diagrams, and the formulas for volume and surface area. You’ll also get a few quick checks that keep answers sane.

Frustum Of A Cone Meaning In Plain Terms

A cone narrows until it reaches a single tip. If you slice off that tip with a flat cut that stays parallel to the circular base, the leftover piece is a frustum. It has:

• a larger circular base
• a smaller circular top
• a curved side that connects the two circles

Right Vs Oblique Frustums

Most school problems use a right circular frustum: the two circles sit directly above each other, and the axis is perpendicular to both bases. A tilted cut can create an oblique frustum. The idea stays the same, but the side leans and many classroom formulas no longer apply cleanly.

What Is a Frustum of a Cone? What You Label On The Diagram

Textbooks stick to a standard set of symbols. If you match those symbols to a sketch before you calculate, your work gets easier to follow and easier to grade.

Standard Measurements

• R = radius of the larger base
• r = radius of the smaller base
• h = vertical height (perpendicular distance between the bases)
• s = slant height (distance along the curved side from rim to rim)

One Relationship You’ll Use A Lot

On a right circular frustum, the slant height comes from a right triangle in a vertical cross-section. One leg is the height h. The other leg is the radius difference R − r. The slant height is the hypotenuse:

s = √((R − r)² + h²)

Setup Steps That Stop Most Mistakes

Before you plug numbers into anything, run this short routine:

1. Mark R on the larger circle and r on the smaller one.
2. Draw h straight between the bases, not along the side.
3. If you’re given diameters, divide by 2 to get radii.
4. Convert units once at the start so every length matches.

For a quick reference definition and the same symbol choices used in many math sources, see the Wolfram MathWorld entry for conical frusta.

Formulas You Need For A Frustum Of A Cone

Most assignments ask for one of three things: volume, curved side area, or total surface area. Here are the standard formulas for a right circular frustum.

Volume

V = (πh/3)(R² + Rr + r²)

Lateral Surface Area

This is only the curved side, not the circles.

A_lateral = π(R + r)s

Total Surface Area

This adds the two circular faces.

A_total = πR² + πr² + π(R + r)s

Why The Volume Formula Works

If the volume formula feels random, there’s a simple way to see it: a frustum is what remains after a smaller cone is removed from a larger cone. Both cones share the same tip angle, so their cross-sections make similar triangles. That similarity links the missing heights to the radii.

Once you find the two cone volumes with V = (1/3)π(radius)²(height) and subtract, the algebra simplifies into the compact frustum form (πh/3)(R² + Rr + r²). The shape may look new, but the volume method is the same cone volume you already know.

If you want to see the similarity-triangle step drawn out, this Khan Academy lesson on frustum volume shows the “large cone minus small cone” setup with a clear diagram.

Worked Examples You Can Copy

These two examples cover the most common question types. Keep your own work in the same layout and teachers can scan it fast.

Example 1: Volume From R, r, And h

Let R = 6 cm, r = 3 cm, h = 10 cm.

• Compute the bracket: R² + Rr + r² = 36 + 18 + 9 = 63
• Plug in: V = (π×10/3)×63
• Simplify: 63/3 = 21, so V = 210π cm³

As a decimal, 210π ≈ 659.7 cm³. For liquids, cm³ equals mL, so that’s about 659.7 mL.

Example 2: Total Surface Area

Use the same measurements and find the slant height first:

• s = √((6 − 3)² + 10²) = √109 ≈ 10.44 cm
• A_lateral = π(6 + 3)(10.44) = 9π(10.44) ≈ 295.3 cm²
• Areas of circles: π(6²) + π(3²) = 45π ≈ 141.4 cm²
• A_total ≈ 295.3 + 141.4 = 436.7 cm²

Frustum Table: Symbols, Uses, And Quick Checks

This table keeps the moving parts straight. It’s also a fast way to spot what a word problem is really asking for.

Item	What It Means	Where It Shows Up
R	Larger base radius	All formulas
r	Smaller base radius	All formulas
h	Vertical height between bases	Volume, slant height step
s	Slant height along the side	Lateral and total area
V	(πh/3)(R² + Rr + r²)	Capacity, material fill
Alateral	π(R + r)s	Labels, wraps, side paint
Atotal	πR² + πr² + π(R + r)s	Coating that includes both ends
Unit check	Lengths in cm/m; areas in cm²/m²; volume in cm³/m³	Stops unit drift

Common Mistakes That Drop Points

If your answer looks off, it’s usually one of these.

Diameter Used As Radius

A diameter must be halved. Writing “r = d/2” on the page is a simple guardrail.

Slant Height Used In Volume

Volume needs the vertical height h. If you only have a slanted measurement, convert it to h with the same right-triangle idea used to find s.

Wrong Surface Area Type

Lateral area is the curved side only. Total surface area adds both circles. Read the prompt, then match it to the right formula name before you compute.

R And r Swapped

Swapping radii won’t break volume (the formula is symmetric), but it can confuse your work and makes it easier to mix up the triangle step for s. Label the larger radius as R every time.

Table Of Tasks And The Formula That Fits

Use this as a quick “which tool do I grab?” list when a problem is wordy.

Question Type	Use This	Sanity Check
How much can it hold?	V = (πh/3)(R² + Rr + r²)	Answer ends in cubic units
Area for a label or wrapper	Alateral = π(R + r)s	No circles added
Paint or coating on the whole outside	Atotal = πR² + πr² + π(R + r)s	Two circles included
Find s from R, r, h	s = √((R − r)² + h²)	s is longer than h
Given diameters	Convert to radii first	All radii are d/2
Need exact form	Leave π in the result	No rounding unless asked
Units don’t match	Convert once at the start	All lengths share one unit

A Short Practice Set With Answers

Try these to lock in the setup. Each one stays friendly on arithmetic.

Problem 1

R = 5 m, r = 2 m, h = 6 m. Find the exact volume.

Answer: V = (π×6/3)(25 + 10 + 4) = 78π m³.

Problem 2

A frustum has R = 4 cm, r = 3 cm, h = 9 cm. Find s to the nearest tenth.

Answer: s = √((4 − 3)² + 9²) = √82 ≈ 9.1 cm.

Problem 3

Lateral area is 60π cm², with R = 5 cm and r = 1 cm. Find s.

Answer: 60π = π(6)s, so s = 10 cm.