GCSE · Geometry and measures

Cone frustums

Cut a similar cone from a larger cone and connect the remaining dimensions and surfaces.

Set the removed cone length scale to 0.5. Compare its volume with the full cone.

V=πh3(R2+Rr+r2)V=\frac{\pi h}{3}(R^2+Rr+r^2)
Cone frustums

Scroll the diagram sideways to see every label.

Check the volume by subtracting similar cones
VolumeValue
Original cone452.389 cm³
Removed similar tip56.549 cm³
Difference = frustum395.841 cm³

Removed tip: 50% of each original length; 12.5% of the original volume.

Large radius R
6 cm
Small radius r
3 cm
Frustum height
6 cm
Slant height
6.708 cm
Volume
395.841 cm³
Total surface area
331.042 cm²

R and r are the two circular radii; h in the formula is the remaining frustum height. Curved area is π(R + r) × slant height. Total surface area also includes both circular ends.

Predict, test and explain

Step 1 of 3

Predict

If a cone’s lengths halve, does its volume halve?

InvestigateSet the removed cone length scale to 0.5. Compare its volume with the full cone.

Explore and compare

Choose the investigation above. Predict what will change before you move a control.

Try it in the simulation. Change one thing at a time.

Notes and saved values stay in this tab. They disappear when you reload.

Hints, self-check and connections
Try a self-check

Fixed values, separate from the diagram controls.

If a cone’s lengths halve, does its volume halve?

Hint

Volume scales with the cube of the length scale factor because all three dimensions change.

Reveal answer

Volume scales with the cube of the length scale factor because all three dimensions change.

Build an explanation

Hint 1

Volume scales with the cube of the length scale factor because all three dimensions change.

Hint 2

Use the labels and units in the live model to check your method.

Hint 3

Explain which lengths, areas or volumes are being compared.

Worked example

This example uses fixed values, separate from the diagram controls.

A cone of radius 6 cm and height 12 cm loses a similar tip with half the original lengths. Find the frustum volume.

  1. Full cone volume = (1/3)π × 6² × 12 = 144π cm³.
  2. The removed cone has radius 3 cm, height 6 cm and volume 18π cm³.
  3. Subtract the two volumes.

Answer: 126π cm³.

Connect this idea

Watch out: Volume scales with the cube of the length scale factor because all three dimensions change.