Connected ideas

Comparing and scaling solids

Compare shapes and change their dimensions.

Predict, test and explain

Step 1 of 3

Predict

A cone and cylinder share radius 3 cm and height 4 cm. Which volume ratio is cylinder:cone?

VBVA≈0.333\frac{V_B}{V_A}\approx 0.333
Shape A: Cylinder; Shape B: ConeBoth drawings use the same fixed centimetre scale in this mode. Shape A and Shape B dimensions are labelled below. Blue or green lines mark radius; dashed red lines mark vertical height on round solids.
Shape A\text{Shape A}
Shape B\text{Shape B}
Cylinder\text{Cylinder}
Cone\text{Cone}
r=2, h=4 cmr=2,\ h=4\,\text{cm}
Dashed red: vertical height\text{Dashed red: vertical height}
r=2, h=4 cmr=2,\ h=4\,\text{cm}
Dashed red: vertical height\text{Dashed red: vertical height}

Scroll the diagram sideways to see every label.

Shape A

V≈50.27 cm3V\approx 50.27\,\text{cm}^3

S≈75.4 cm2S\approx 75.4\,\text{cm}^2

Shape B

V≈16.76 cm3V\approx 16.76\,\text{cm}^3

S≈40.67 cm2S\approx 40.67\,\text{cm}^2

SBSA≈0.539\frac{S_B}{S_A}\approx 0.539

At the same radius and vertical height, the cone has one third of the cylinder’s volume. Their total surface areas do not have the same ratio. S means total surface area, including all bases. Both drawings use a fixed shared scale within this mode.

Check the formulae and values
QuantityShape AShape B
Volume formulaV=πr2h=π×22×4V=\pi r^2h=\pi\times2^2\times4V=13πr2h=13π×22×4V=\frac13\pi r^2h=\frac13\pi\times2^2\times4
Total surface formulaS=2πr(r+h)S=2\pi r(r+h)S=πr(r+r2+h2),s≈4.47 cmS=\pi r(r+\sqrt{r^2+h^2}),\quad s\approx4.47\,\text{cm}
Volume (cm³)≈50.27\approx 50.27≈16.76\approx 16.76
Total surface area (cm²)≈75.4\approx 75.4≈40.67\approx 40.67

InvestigateCompare a cylinder and cone at the same radius and height. What changes?

Explore and compare

Compare two similar solids when every length doubles.

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

Hint

Save the first measurements, then change only the scale factor.

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.

A cylinder and a cone both have radius 3 cm and vertical height 4 cm. Find their volumes.

r=3 cm,h=4 cmr=3\,\text{cm},\quad h=4\,\text{cm}

Hint

Use pi times radius squared times height; the cone has one third of that volume.

Reveal answer

The cylinder has volume 36π cm³; the cone has volume 12π cm³. Matching radius and height gives a 3:1 volume ratio.

Vcylinder=36π cm3,Vcone=12π cm3V_{\text{cylinder}}=36\pi\,\text{cm}^3,\quad V_{\text{cone}}=12\pi\,\text{cm}^3

Build an explanation

Hint 1

In Compare solids, change one dimension at a time. A cone has one third of a cylinder’s volume when their radii and vertical heights match.

Hint 2

In Scale one solid, multiply every corresponding length by the same k > 0. Total surface area is multiplied by k².

Hint 3

Uniform scaling multiplies volume by k³. Doubling only a cylinder’s radius multiplies its volume by 4; doubling only its height multiplies it by 2.

Worked example

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

A 2 cm by 3 cm by 4 cm cuboid is enlarged by scale factor 3. Find its new dimensions, total surface area and volume.

  1. Multiply all three lengths by 3: the new dimensions are 6 cm, 9 cm and 12 cm.
  2. The original surface area is 2(2 × 3 + 2 × 4 + 3 × 4) = 52 cm².
  3. Multiply surface area by 3² = 9: 52 × 9 = 468 cm².
  4. The original volume is 2 × 3 × 4 = 24 cm³. Multiply it by 3³ = 27: 24 × 27 = 648 cm³.

Answer: Dimensions: 6 cm × 9 cm × 12 cm; surface area: 468 cm²; volume: 648 cm³.

Connect this idea

Watch out: A scale factor of 3 multiplies lengths by 3, areas by 9 and volumes by 27. Those powers apply to uniform scaling. Changing only one of a cylinder’s radius and height does not produce a similar cylinder. Different solids with equal volume can have different surface areas.