EAL
GCSE Maths · Similar shapes and scale factors
Before you watch

Where this fits
in engineering

A scale model proves a design cheaply. The trap is assuming everything scales the same way, because plate area and weight do not follow length.

FabricationBuilding a structure by cutting, forming and welding plate and section, rather than casting or machining it.
HopperA funnel-shaped vessel that holds material and feeds it out of the bottom under gravity.
PrototypeA smaller version built to the same proportions, used to prove a design before full-size manufacture.
Fabrication bay, prototype rig

Scaling
Up

Before Ashfield builds a full-size steel hopper, it welds a quarter-scale model to check the geometry. The model works. Now somebody has to order the plate and book the crane.

Model height
0.5 m
Model mass
9 kg
The overhead crane is rated at 500 kg. Is that enough?
Step 1
Step 2
Step 3

Find the scale factor from the lengths

Area grows by the scale factor squared

Volume and mass grow by the cube

Every length multiplies by k = 4. Heights, widths, weld runs, all of them.
Four across and four down is 16 squares. Area always uses k².
Four along, four across, four deep: k³ = 64. This is the one people get wrong.
0.5 m model 2 m full size Length × 4 0.5 m → 2 m Surface area: the plate you buy × 16 1.2 m² → 19.2 m² Volume and mass: the crane's problem × 64 15 L → 960 L 9 kg → 576 kg

What the crane actually lifts

Mass of the full-size hopper
9 × 4³
= 9 × 64
= 576 kg
Rating on the overhead crane
500 kg
76 kg short
Four times longer is sixty-four times heavier.
Order the plate, but book a different crane.

Three things to carry into the exam

  • Find k from a pair of matching lengths first
  • Lengths use k, areas use k², volumes and masses use k³
  • Given an area or volume ratio, square-root or cube-root it to get back to k
Two similar tanks hold 40 litres and 1,080 litres. The small one has a surface area of 0.6 m². Find the surface area of the large one.
1080 ÷ 40 = 27, so k = 3
0.6 × 3² = 5.4 m²
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The engineering

context and terms

Fabricators build a scale model before committing plate to a large vessel, to check the geometry and the weld sequence cheaply. The trap is assuming everything scales the same way. Make it four times longer and the plate order goes up sixteen times, while the weight goes up sixty-four. Get that wrong and the steel is under-ordered and the lifting gear is under-rated, which is a safety problem rather than an arithmetic one.

Fabrication
Building a structure by cutting, forming and welding plate and section, rather than casting or machining it.
Hopper
A funnel-shaped vessel that holds material and feeds it out of the bottom under gravity.
Prototype
A smaller version built to the same proportions, used to prove a design before full-size manufacture.
Scale factor
The number every length is multiplied by to get from one similar shape to the other.
Safe working load
The maximum a crane or sling is rated to lift. Exceeding it is not a margin, it is a failure.

Questions

4 questions · 10 marks
Write your answer, then check it.
  1. A model hopper is built to a scale of 1:4. The model is 0.5 m tall. Find the height of the full-size hopper. [1]
  2. The model has a surface area of 1.2 m². Find the surface area of the full-size hopper. [2]
  3. The full-size hopper holds 960 litres. Find the capacity of the model. [2]
  4. Two similar tanks have capacities of 40 litres and 1,080 litres. The smaller tank has a surface area of 0.6 m². Find the surface area of the larger tank. [5]

Controls. Tap the stage or press space to play and pause. Drag the amber circle along the line to scrub, or use the arrow keys to jump five seconds. The chapter buttons drop you straight into each stage of the working, which is the bit worth pausing on with a class. Captions start off; CC turns them on. Everything is inside this one file, so it runs offline. Screen-record the stage at 1920×1080 if you need an MP4 with a voiceover.

Demo developed by Lee Guthrie for EAL