What Is Frustum of Cone? | The Shape Behind Tapered Objects

A cone frustum is the slice of a cone between two parallel cuts, leaving two circular faces with different radii.

Paper cups, buckets, funnels, lampshades—this shape shows up all over. A frustum of a cone is what you get when the pointy tip of a cone is removed by a cut that’s parallel to the base. You still get that steady taper, just with a flat top instead of a vertex.

Below you’ll get a clear definition, the parts you should label, the formulas that show up most, and a few clean solution setups that are easy to reuse.

Frustum of Cone Meaning In Plain Words

Start with a right circular cone: one circular base, one vertex, and a curved side that narrows evenly. Now make a straight cut across the cone so the cutting plane is parallel to the base. Remove the small top cone. What remains is a frustum of the cone.

• Two parallel circular faces: a larger bottom circle and a smaller top circle.
• One curved side: the shortened conical surface that connects the circles.

When the cone’s axis is perpendicular to the bases, you get a right frustum. If the axis is tilted, you get an oblique frustum. Most school questions use the right case.

Parts Of A Frustum Of A Cone You Should Label First

Before you reach for a formula, label the diagram. A frustum problem feels hard when labels are missing or inconsistent.

Radii: Big R And Small r

Use R for the bottom radius and r for the top radius. Keep the same choice all the way through.

Height: The Perpendicular Distance Between The Bases

The height h is measured at a right angle to the bases. It is not the slanted edge length on the side.

Slant Height: The Side Length Along The Surface

The slant height s runs along the curved surface from the top rim down to the bottom rim. In a center cross-section, it becomes the hypotenuse of a right triangle.

Curved Surface And Total Surface

• Curved (lateral) surface area: the curved side only.
• Total surface area: curved side plus the two circular faces.

How A Frustum Relates To The Original Cone

A frustum is a cone with its tip removed, and that link gives a reliable way to reason about it: similar triangles.

Slice a right cone through its center. The full cone becomes a big triangle. The removed top piece becomes a smaller, similar triangle. Similarity ties radii and heights together, and it’s also why the frustum volume formula contains a middle term that mixes both radii.

For a standard reference that states the core relationships in one place, Wolfram’s page on Conical Frustum summarizes the geometry and formulas.

Formulas For A Right Circular Frustum Of A Cone

Most tasks fall into three buckets: slant height, surface area, and volume. Use consistent units, and keep π until the last step unless the question forces a decimal.

Slant Height

s = √(h² + (R − r)²)

Curved Surface Area

A_curved = π(R + r)s

Total Surface Area

A_total = π(R + r)s + πR² + πr²

Volume

V = (1/3)πh(R² + Rr + r²)

How To Solve Typical Frustum Questions Step By Step

You don’t need tricks. You need a steady order of steps and two fast checks.

Sketch, Label, Convert

Draw the shape, then write R, r, and h in the right spots. If a diameter is given, halve it at once and rewrite it as a radius.

Choose The Target

Volume needs R, r, and h. Surface area often needs the slant height s too, so compute s first when it’s missing.

Check Units And Scale

Area uses squared units, volume uses cubed units. Also compare your frustum volume to a cylinder with radius R and height h; the frustum must be smaller.

Worked Setups You Can Reuse

These short setups show what goes where. Copy the structure, then finish the arithmetic as your class expects.

Setup A: Volume From Radii And Height

Given R = 8 cm, r = 5 cm, h = 12 cm:

• Start: V = (1/3)πh(R² + Rr + r²)
• Substitute: V = (1/3)π(12)(8² + 8·5 + 5²)
• Simplify inside the brackets, then finish and attach cm³.

Setup B: Slant Height Before Curved Area

Given R = 10, r = 6, h = 9:

• First: s = √(9² + (10 − 6)²)
• Then: A_curved = π(10 + 6)s

If you want a quick refresher on why cone volume uses one-third of a matching cylinder, Khan Academy’s explanation of Volume of a pyramid or cone is a clear read.

Right And Oblique Frustums: What Changes

A lot of learners think “frustum” automatically means “right frustum.” The word frustum only tells you the solid sits between two parallel cuts. The axis can be straight up, or it can lean.

In a right frustum, each straight line on the side (a generator) has the same length, so one slant height s works for the whole surface. That’s the case behind the surface area formulas above.

In an oblique frustum, the generators are not all equal, so one slant height can’t represent the full curved side. That’s why school questions tend to avoid oblique surface area.

Volume is kinder. As long as the two bases are parallel circles and h is the perpendicular distance between them, the volume still depends on R, r, and h, not on the tilt.

Net Of A Frustum And Why The Curved Area Formula Works

When teachers say “unroll the curved surface,” they mean you cut the frustum down one side and flatten the curved skin onto a plane. The flat shape is a ring-shaped sector: a slice of a donut.

Arc Length Match Is The Core Idea

The bottom rim of the frustum has circumference 2πR. The top rim has circumference 2πr. On the net, those become the outer and inner arc lengths of the sector.

The sector’s two straight edges are both slant heights. Multiply the average of the two circumferences by the slant height and you get the curved area, which is the same as:

A_curved = π(R + r)s

This “net” picture also explains a common check: if r = 0, the inner arc disappears and the net becomes a sector for a full cone’s curved surface.

Common Mistakes That Cost Marks

Most wrong answers come from a small set of habits. If you fix these, your accuracy jumps fast.

Mixing Radius And Diameter

If the problem states a diameter of 14 cm, the radius is 7 cm. Write that conversion on the page so it doesn’t vanish mid-work.

Using Slant Height In Volume

Surface area can use s. Volume never does. If you plug s into a volume formula, the answer will be too large.

Swapping R And r

The formulas still run when you swap the radii, so the error can hide. Reduce the risk by setting R ≥ r at the start.

Where You See Frustums In Real Objects

This shape shows up whenever you want a taper with a flat opening:

• Stacking paper cups
• Funnels and strainers
• Flower pots and buckets
• Tapered spacers in machines
• Some lighting shades

Formula Table For Fast Revision

This table gathers the expressions you’ll use most. It’s also a handy checklist of which measurements each formula needs.

Quantity	Formula	Inputs Needed
Slant height	s = √(h² + (R − r)²)	h, R, r
Curved surface area	Acurved = π(R + r)s	R, r, s
Total surface area	Atotal = π(R + r)s + πR² + πr²	R, r, s
Volume	V = (1/3)πh(R² + Rr + r²)	R, r, h
Height from slant height	h = √(s² − (R − r)²)	s, R, r
Radii difference	R − r	Needed for the slant triangle
Scale check (cylinder)	Vcyl = πR²h	Upper bound for frustum volume
No taper case	R = r	Shape becomes a cylinder

How To Get Frustum Volume From Two Cones

Some diagrams show the full cone and the small top cone that was removed. In that layout, you can build the frustum volume as a subtraction.

Write Two Cone Volumes

• V_full = (1/3)πH R²
• V_top = (1/3)πh_top r²
• V_frustum = V_full − V_top

Link The Missing Height With Similarity

Because the cut is parallel to the base, the two cones are similar, so:

r / R = h_top / H

Use that ratio to find the missing height, compute both cone volumes, then subtract. It’s longer than the direct frustum formula, yet it can match the way a question is drawn.

Second Table: Picking The Right Formula Fast

Use this table when you’re unsure where to start.

Question	Start With	Quick Check
Find volume	V = (1/3)πh(R² + Rr + r²)	Needs perpendicular height h
Find curved area	Acurved = π(R + r)s	Compute s first if missing
Find total area	Atotal = π(R + r)s + πR² + πr²	Add both circles
Find slant height	s = √(h² + (R − r)²)	Uses radii difference
Find height from s	h = √(s² − (R − r)²)	h must stay real
Find a missing radius	Similarity: r/R = htop/H	Works with a cut cone diagram
Check a result	Compare with πR²h	Frustum volume stays below

What Is Frustum of Cone? As A One-Line Exam Definition

In definition questions, keep it direct: a frustum of a cone is the part left after cutting a cone with a plane parallel to its base, leaving two circular, parallel faces of different size.

If the question also asks for a sketch, label R, r, h, and s right away. Those labels often earn method marks even before the calculations start.