GCSE · Geometry and measures

Composite prisms and missing blocks

Remove a corner from a cuboid and compare volume with exposed surface area.

Change the notch width and depth. Compare cross-section area, perimeter and exposed surface area.

V=(wh−nt)L,S=2(wh−nt)+2(w+h)LV=(wh-nt)L,\quad S=2(wh-nt)+2(w+h)L
Composite prisms and missing blocks

Scroll the diagram sideways to see every label.

Subtract volume; count the new exposed faces
StepCalculation
Outer cross-section8 × 6 = 48 cm²
Removed corner3 × 2 = 6 cm²
Prism volume42 × 5 = 210 cm³
Surface area2 × 42 + 28 × 5 = 224 cm²

A corner cut runs through the full 5 cm length.

Cross-section area
42 cm²
Cross-section perimeter
28 cm
Volume
210 cm³
Exposed surface area
224 cm²

Both ends are L-shaped. Every boundary edge creates a rectangular side face. For this corner cut only, the two new inside lengths exactly replace the outside lengths removed.

Predict, test and explain

Step 1 of 3

Predict

Can removing a corner leave the cross-section perimeter unchanged?

InvestigateChange the notch width and depth. Compare cross-section area, perimeter and exposed surface area.

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.

Can removing a corner leave the cross-section perimeter unchanged?

Hint

For this rectangular corner cut, the two new inner edges replace equal lengths on the outside.

Reveal answer

For this rectangular corner cut, the two new inner edges replace equal lengths on the outside.

Build an explanation

Hint 1

For this rectangular corner cut, the two new inner edges replace equal lengths on the outside.

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 prism has an 8 cm by 6 cm cross-section with a 3 cm by 2 cm top corner removed. Its length is 5 cm. Find volume and total surface area.

  1. Cross-section area = 8 × 6 − 3 × 2 = 42 cm².
  2. Volume = 42 × 5 = 210 cm³. The cross-section perimeter is 28 cm.
  3. Surface area = 2 × 42 + 28 × 5.

Answer: Volume 210 cm³; surface area 224 cm².

Connect this idea

Watch out: For this rectangular corner cut, the two new inner edges replace equal lengths on the outside.