Math Detours · Grade 7 · Sharpen · 8

Blueprints

A scale drawing shrinks every length by the same factor. Lengths scale once, areas scale twice, and that surprises people.

Name ____________________Date ____________

Tonight's idea

A scale drawing, such as a builder's blueprint, shows something big shrunk to fit on paper. Every length is shrunk by the same factor, so the shape stays true. The scale tells you the factor. This garden plan has the scale 1 cm : 2 m. Each centimetre on paper stands for 2 metres of ground.

LawnVegetable bedPatio4 cm3 cm4 cm2.5 cm1.5 cmScale 1 cm : 2 m1 cm²on the planstands for1 m²1 m²1 m²1 m²2 m2 m4 m²real
A garden plan. The lengths are marked, in case your printer changed the size.

Lengths are easy. The lawn is 4 cm wide on the plan, so the real lawn is \(4 \times 2 = 8\) m wide.

Areas hold a surprise. One square centimetre of plan stands for a patch 2 m long and 2 m wide. That is \(2 \times 2 = 4\) m², not 2 m². The scale gets used twice, once for each direction.

At 1 cm : 2 m, real length = plan length × 2, and real area = plan area × 2 × 2.

Where you met this: map scales and doubling every length.

Watch one: the vegetable bed

On the plan the bed is 3 cm by 2.5 cm. Real lengths: \(3 \times 2 = 6\) m and \(2.5 \times 2 = 5\) m. Real area: \(6 \times 5 = 30\) m².

Now go by area. The plan area is \(3 \times 2.5 = 7.5\) cm². Each square centimetre stands for 4 m², and \(7.5 \times 4 = 30\) m². The two routes agree. Multiplying by 2 only once would give 15 m², half the true size.

Your turn

  1. Use the garden plan. (a) How long and how wide is the whole garden? (b) Find the real length, width and area of the patio. (c) A fence goes all the way round the garden. How long is it?
  2. Same scale, 1 cm : 2 m, going the other way. (a) A shed is really 3 m by 5 m. What size is it on the plan? (b) A pond has a real area of 12 m². What area does it cover on the plan?
  3. A house plan has the scale 1 : 50. That means 1 cm on paper stands for 50 cm in the house. (a) A room is 9 cm by 7 cm on the plan. Find its real length and width in metres. (b) Find its real area in m². (c) How many cm² of real floor does 1 cm² of plan stand for?
  4. (a) On the garden plan the lawn covers 16 cm². A gardener doubles that and orders 32 m² of turf. How much is really needed, and what fraction did he order? (b) On another plan, 1 cm² stands for 25 m². What is its scale, written as 1 cm : ? m.

Stretch: two plans of one field

Plan A has the scale 1 cm : 4 m. On it a field covers 30 cm². Plan B shows the same field at 1 cm : 2 m. What area does the field cover on plan B? Explain the answer without working out the real area.

Answers

  1. (a) 7 cm by 4 cm on the plan, so 14 m by 8 m. (b) \(3 \times 2 = 6\) m by \(1.5 \times 2 = 3\) m, area 18 m². (c) \(14 + 8 + 14 + 8 = 44\) m.
  2. (a) Divide by 2: 1.5 cm by 2.5 cm. (b) Each cm² stands for 4 m², so \(12 \div 4 = 3\) cm².
  3. (a) \(9 \times 50 = 450\) cm and \(7 \times 50 = 350\) cm, which is 4.5 m by 3.5 m. (b) \(4.5 \times 3.5 = 15.75\) m². (c) \(50 \times 50 = 2500\) cm².
  4. (a) \(16 \times 4 = 64\) m² is needed. He ordered 32 m², which is half. (b) \(5 \times 5 = 25\), so each centimetre stands for 5 m: the scale is 1 cm : 5 m.

Stretch. 120 cm². On plan B every length is drawn twice as long as on plan A, so every area is \(2 \times 2 = 4\) times as big: \(30 \times 4 = 120\). Check through the real area: \(30 \times 16 = 480\) m², and \(480 \div 4 = 120\) cm².

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