Compute actual measures and reproduce a drawing at a new scale

Learning goal: Use a scale drawing to compute actual lengths and areas, then reproduce the same object at a different scale. Distinguish drawing units, actual units, length factors and area factors.

A scale drawing represents actual lengths using a consistent ratio. Suppose a classroom floor plan uses one drawing centimeter for two actual meters. This scale means every represented length follows that relationship, not just the longest wall. A line four centimeters long represents eight meters, and a line three centimeters long represents six meters.

The mixed units are part of the scale. If you express both lengths in centimeters, one drawing centimeter corresponds to 200 actual centimeters, a dimensionless length ratio of 1:200. Writing 1:2 without naming centimeters and meters would mean a very different scale if readers assumed like units. Preserve unit labels or convert consistently before forming a unitless ratio.

The plan shows a rectangular room measuring four by three drawing centimeters. Its actual dimensions are eight by six meters. Actual area is eight times six, or 48 square meters. The drawing area is twelve square centimeters. Each one-square-centimeter drawing region represents a two-by-two-meter actual square, or four square meters, so twelve such regions represent 48 square meters.

The area conversion is not two square meters per drawing square centimeter. The scale factor applies to both perpendicular dimensions. In like units, a length factor of k produces an area factor of k² for similar figures. With mixed units, explain the represented unit square to avoid writing a bare squared factor without its meaning.

Now reproduce the same room at one drawing centimeter for four actual meters. The actual room remains eight by six meters, so the new drawing must be two by 1.5 centimeters. Each drawing length is half its former value because the new scale represents twice as much actual length per centimeter. The drawing area becomes three square centimeters, one quarter of the original twelve. The room itself has not shrunk.

To reproduce it on paper, draw a two-centimeter segment, construct perpendicular adjacent sides of 1.5 centimeters and close the rectangle. Label the new scale and actual dimensions. Use a ruler and check the right angles. A browser image is a visual model, not a physical centimeter ruler: display size changes between screens, so actual construction requires measured paper or a tool with explicitly defined units.

An irregular floor plan can be decomposed before applying area reasoning. If an L-shaped plan is a four-by-three drawing-centimeter rectangle with a one-by-one corner removed at the original scale, drawing area is twelve minus one, or eleven square centimeters. Actual area is eleven times four, or 44 square meters. Equivalently, subtract a two-by-two-meter missing corner from the actual eight-by-six rectangle.

Do not subtract a missing length from an area. A one-centimeter drawing corner is not enough information until its two dimensions are known. The removed corner in this example is one by one, which defines a square area. Name the dimensional quantity before applying a scale conversion.

A route or diagonal on a plan must be measured or determined from adequate geometry, not guessed from the wall lengths. For example, a 2.5-centimeter route on the original scale represents five meters along that represented route. If the route bends, add its segment lengths before or after converting consistently. The scale does not imply that every pair of locations is connected by a straight walkable path.

Check a rescaled drawing by converting its dimensions back to actual units. Two centimeters times four meters per centimeter returns eight meters; 1.5 returns six. If the new drawing were eight by six centimeters at the new scale, it would represent a 32-by-24-meter room, a different object. A reverse check catches the common error of copying actual measurements directly into drawing units.

Worked length: A wall is 7.5 meters long. At one drawing centimeter for two meters, its drawn length is 7.5 ÷ 2 = 3.75 centimeters. Multiplication by two goes from drawing to actual; division by two goes from actual to drawing. The units identify the direction.

Worked area: A triangular region on the original plan has a three-centimeter base and two-centimeter perpendicular height. Actual base and height are six and four meters, so actual area is twelve square meters. Drawing area is three square centimeters, and each represents four square meters, giving the same twelve.

Worked rescaling: A route is 2.5 drawing centimeters at the original scale, representing five meters. At one centimeter for four meters, it becomes 5/4 = 1.25 drawing centimeters. The route's actual length remains five meters; only the representation changes.

Construction workshop: Reproduce the rectangular room at both scales on measured paper. Add a door opening with an actual width of 0.9 meter (or a width you measure) and show how its drawn width changes: 0.45 centimeter at the first scale and 0.225 centimeter at the second. Calculate actual perimeter and area from each drawing and confirm they agree. Record intended dimensions separately from measurement errors in the construction.

Independent performance: Create a scale plan of a simple rectangular or L-shaped space using measured or explicitly invented dimensions. State the scale, compute two actual lengths and the actual area, then redraw it at a new scale. Explain the length and area factors between drawings and verify actual dimensions by converting back. A partner checks that all parts, including any unoccupied or removed regions, are treated consistently.