Geometry

3D shapes

Change lengths. Follow volume and surface area.

Predict, test and explain

Step 1 of 3

Predict

A cone and cylinder have the same radius and vertical height. How does the cone volume compare?

V=lwh=60 cm3V=lwh\quad = 60\,\text{cm}^3
l=5 cm,w=3 cm,h=4 cml=5\,\text{cm},\quad w=3\,\text{cm},\quad h=4\,\text{cm}

Scroll the diagram sideways to see every label.

Total surface areaS=2(lw+lh+wh)S=2(lw+lh+wh)S=94 cm2S = 94\,\text{cm}^2

Surface area includes every face and base.

InvestigateDouble every length. How do volume and surface area change?

Explore and compare
Can you break the rule?

Cuboids with equal volumes have equal surface areas.

Always, sometimes or never? Test it in the simulation.

Hint

Double one length and halve another. Keep the third length fixed.

Check the reasoning

Sometimes. Equal volume does not fix surface area. Identical dimensions give equal surface areas, while different dimensions may not.

2×3×4=1×3×8=24,52≠702\times3\times4=1\times3\times8=24,\quad52\ne70

One counterexample disproves “always”. Examples alone do not prove it.

Find two differently shaped cuboids with the same volume.

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

Hint

Save one cuboid. Double one length and halve another, keeping the third fixed.

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 cuboid has length 2 cm, width 3 cm and height 5 cm. Find its volume and total exterior surface area.

Hint

Multiply all three dimensions for volume. Add the areas of all six faces for surface area.

Reveal answer

Its volume is 30 cubic centimetres and its surface area is 62 square centimetres.

V=2×3×5=30 cm3,S=2(2×3+2×5+3×5)=62 cm2V=2\times3\times5=30\,\text{cm}^3,\qquad S=2(2\times3+2\times5+3\times5)=62\,\text{cm}^2

Build an explanation

Hint 1

Separate the space inside the solid from the surfaces around it.

Hint 2

Volume uses three dimensions; surface area adds the areas of all outer faces.

Hint 3

For a cuboid, volume = lwh and surface area = 2(lw + lh + wh).

Worked example

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

Find the volume and surface area of a 5 cm × 3 cm × 4 cm cuboid.

  1. Volume = 5 × 3 × 4 = 60.
  2. Surface area = 2(5 × 3 + 5 × 4 + 3 × 4).
  3. Double 15 + 20 + 12.

Answer: Volume 60 cm³; surface area 94 cm².

Connect this idea

Watch out: Turning the drawing changes the view, not the dimensions, volume or surface area.