Scale drawings, building plans, similarity and the trapezoidal rule: HSC Maths Standard 1 Year 12
“Interpret and use scale drawings, including building plans and elevation views, use similarity and scale factors to find lengths, calculate areas and volumes from plans, and use the trapezoidal rule to estimate the area of irregular shapes”
Use scale 1: by multiplying drawing lengths by (and dividing to go the other way), remembering building plans are in millimetres. Read floor plans and elevations to find real areas and volumes, use similar triangles to find unknown lengths, and estimate irregular areas with the trapezoidal rule for each strip.
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What this dot point is asking
You need to read and use scale drawings (including building plans and elevations), find lengths with scale factors and similar figures, calculate areas and volumes from plans, and estimate irregular areas with the trapezoidal rule.
The answer
Scales and scale factors
A scale of 1: means 1 unit on the drawing represents units in real life.
- Drawing to real: multiply by .
- Real to drawing: divide by .
Building plans use millimetres as the standard unit, so convert to metres (divide by 1000) for areas in m².
Plans and elevations
- Floor plans show rooms, walls, doors (with swing arcs) and windows from above, with dimensions and common symbols and abbreviations.
- Elevation views show the building from each side, giving heights, roof pitch and window positions.
- Use plans to calculate floor areas, wall areas (for painting), and volumes (for concrete slabs or rooms).
Similarity
Similar figures have equal corresponding angles and sides in the same ratio (the scale factor). Similar triangles are used to find heights and distances that cannot be measured directly (shadows, mirrors, surveying).
The trapezoidal rule
For an irregular shape with measurements and a distance apart:
For more strips, apply the rule to each strip and add the areas. Check the formula on your reference sheet.
- Convert plan measurements to real lengths.
- Convert to the units needed (m for m², m³).
- Split composite areas into rectangles and triangles, or use the trapezoidal rule for curved or irregular edges.
- Volume = area of cross-section × depth (for slabs, pools).
A concrete slab on a 1:100 plan measures 120 mm by 85 mm and is 100 mm thick in reality.
- Real dimensions: mm = 12 m; mm = 8.5 m.
- Area m².
- Volume m³ of concrete.
- Forgetting millimetres on plans
- A wall of 3600 means 3.6 m.
- Using the wrong h in the trapezoidal rule
- is the distance between measurements, not the total length.
- Scaling areas by the scale factor
- Convert lengths first, then calculate area.
Practice questions
Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.
foundation3 marksA floor plan has a scale of 1:50. The kitchen measures 64 mm by 90 mm on the plan. Find its real dimensions and floor area.Show worked solution →
Real length mm = 4.5 m; real width mm = 3.2 m.
Area m².
Marking guide: 1 mark for each dimension, 1 mark for the area.
core3 marksA 1.8 m tall person casts a 2.4 m shadow at the same time a tree casts a 14 m shadow. Use similar triangles to find the height of the tree.Show worked solution →
The triangles are similar (sun's rays at the same angle).
, so m.
Marking guide: 1 mark for identifying similar triangles, 1 mark for the ratio, 1 mark for 10.5 m.
exam5 marksA surveyor measures the width of an irregular block of land at 20 m intervals along a 40 m straight boundary: 18 m, 25 m and 21 m. Use two applications of the trapezoidal rule to estimate the area, then estimate the cost of turfing it at 14 dollars per square metre.Show worked solution →
Two applications, each with :
First strip: m².
Second strip: m².
Total area m².
Cost .
Marking guide: 1 mark for , 1 mark per strip (2 marks), 1 mark for the total, 1 mark for the cost.