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Area and scale

The plane and space, grade 11 (3 units) Lesson 3 of 15Learning

Area and scale

Lengths and area
  • 1 m² = 10,000\displaystyle 10{,}000 cm²
  • 1 dunam = 1,000\displaystyle 1{,}000 m²
  • 1 km² = 1,000,000\displaystyle 1{,}000{,}000 m²

This is the plan of a room at a scale of 1:100\displaystyle 1:100, and every square on it is 1 cm by 1 cm:

  • On the plan the room is 4 cm by 3 cm, and its area is 12 cm².
  • In reality the room is 4 m by 3 m, and its area is 12 m² – 120,000\displaystyle 120{,}000 cm².
  • The lengths grew 100 times, but the area grew 1002=10,000\displaystyle {100^2}={10{,}000} times: every square centimetre on the plan is, in reality, a square 100 cm by 100 cm.

At a scale of 1:n\displaystyle 1:n, the real area S\displaystyle S is the area on the map s\displaystyle s times n2\displaystyle n^2, in the same unit:

S\displaystyle S =\displaystyle = n2⋅s\displaystyle n^{2}\cdot s
Example: the area of a park

On a map at a scale of 1:10,000\displaystyle 1:10{,}000, a park is a rectangle 3 cm by 2 cm.
What is the park’s real area, in dunams?

The solution below first turns each length into a real length, and only then multiplies: that keeps the numbers smaller.

  • A centimetre on the map is 10,000\displaystyle 10{,}000 cm = 100 m. So the park is 300 m by 200 m.
  • The area: 300⋅200=60,000\displaystyle {300\cdot200}={60{,}000} m².
  • A dunam is 1,000\displaystyle 1{,}000 m², so the area is 60 dunams.
Exercise 1

On a flat’s plan at a scale of 1:100\displaystyle 1:100, the flat’s area is 85 cm².
What is the flat’s real area, in m²?

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    Exercise 2

    On a map at a scale of 1:25,000\displaystyle 1:25{,}000, a field is a rectangle 4 cm by 2 cm.
    What is the field’s real area, in dunams?

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      Exercise 3

      A lake’s area is 12 km².
      What is its area on a map at a scale of 1:100,000\displaystyle 1:100{,}000, in cm²?

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        Area on two maps

        The same island is on two maps: one at a scale of 1:200,000\displaystyle 1:200{,}000 and one at 1:50,000\displaystyle 1:50{,}000.

        • 200,000÷50,000=4\displaystyle {200{,}000\div50{,}000}={4}: on the second map every length is 4 times as large.
        • So the island’s area on the second map is 42=16\displaystyle {4^2}={16} times as large.
        Exercise 4

        A wood’s area on a map at a scale of 1:20,000\displaystyle 1:20{,}000 is 3 cm².
        What is its area on a map at a scale of 1:10,000\displaystyle 1:10{,}000, in cm²?

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          Exercise 5

          On a plan at a scale of 1:50\displaystyle 1:50, a room is a rectangle 9 cm by 7 cm.
          Floor tiles cost 120 ₪ a square metre.
          How much will the tiles for the whole room cost?

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power