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How to calculate real area from a scale drawing

Area scales by the square of the length scale

If one drawing unit represents $k$ real units, multiply each length by $k$ before finding area. Equivalently, multiply the drawing area by $k^{2}$ . Keep the units straight: a length scale does not multiply area only once.

Problem

A scale drawing uses $1$ cm to represent $5$ m. A rectangular bedroom measures $2$ cm by $3$ cm in the drawing. What is its actual area?

Answer

150 m^{2}

The drawing area is 6 cm², but the real area is 25 times its numerical value in square metres.

Step-by-step solution

1. Convert the lengths

The actual width is $2 \cdot 5 = 10$ m and the actual height is $3 \cdot 5 = 15$ m.

2. Multiply for area

A = 10 \cdot 15 = 150 m^{2} .

3. Check by scaling area

The drawing area is $2 \cdot 3 = 6$ square centimetres. Since the length scale factor is $5$ , the area factor is $5^{2} = 25$ . The numerical area is $6 \cdot 25 = 150$ square metres, confirming the result.

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