Free worksheets with worked answers
Eighteen original problems move from simple scale statements to unit conversion, changing scale and area. Every question includes explicit labels; every answer shows the calculation.
A three-page student pack and a separate three-page worked-answer pack. Download only the student file first if you want to solve without seeing the answers.
SC01–SC06: six rectangles with a direct “1 cm represents … m” scale. Find the actual width and height.
SC07–SC12: convert metres and centimetres, work with ratios, and use inch-to-foot scale statements.
SC13–SC18: calculate rectangle and triangle areas, change a drawing scale, and square a length factor.
SC07 / Drawing length
Convert to one unit before applying the ratio.
The drawing line is 4 cm. Reversing the check gives 4 × 200 = 800 cm, the original actual length.
SC11 / Inch-to-foot scale
At 1/4 inch to 1 foot, divide the drawing length by the number of inches used for one foot.
The statement is not 1:4. In matching units, 0.25 in : 12 in simplifies to 1:48.
SC18 / Area factor
Two independent lengths grow by the same factor.
Multiplying the area by 3 would apply the length factor only once. The new area is 108 cm².
For a ratio such as 1:75, one drawing unit represents 75 of the same real-world unit. For an inch-to-foot statement, retain the units on both sides.
Actual to drawing: divide by the ratio denominator. Drawing to actual: multiply. For a stated number of inches per foot, multiply feet to get drawing inches, or divide drawing inches to get feet.
A result in centimetres is not yet a result in metres. Convert explicitly so the final unit matches the question.
Length uses a linear factor. Area uses its square. A triangle still needs half the base times height after the side lengths are converted.
Use the numbers printed beside each figure. The illustrations are intentionally compact and do not reproduce the stated drawing dimensions. This keeps the task focused on scale reasoning and avoids turning an uncalibrated print into the answer.
For a physical layout exercise with real grid dimensions, use the quarter-inch room worksheet. Its 12 ft × 8 ft room is actually drawn at 1/4 inch to 1 foot in the file.
| Mistake | What goes wrong | A useful check |
|---|---|---|
| Mixing units inside a ratio | Writing 1/4 inch : 1 foot as 1:4 | Convert the foot to 12 inches; the ratio becomes 1:48. |
| Multiplying when you should divide | A reduced drawing becomes larger than the actual object. | At 1:200, a drawing length must be 200 times smaller in matching units. |
| Scaling an area once | A factor of 3 in length is treated as a factor of 3 in area. | Both dimensions change: 3 × 3 = 9. |
A few useful answers
No. The levels describe the calculations in this collection. Beginner uses direct scale statements, intermediate introduces conversions, and challenge adds area and changing scale. Choose by the learner’s current knowledge.
No. Every problem says to use the labels. These diagrams are illustrations, not full-size measuring figures. A separate graph-paper exercise is available for physical measurement.
No. The student pack has three question pages. The separate answer pack has three pages of worked solutions, with matching problem IDs SC01 through SC18.
Yes. Single-page SVGs are included. Keep attribution, link to CC BY 4.0 and say what you changed. If you change the numbers, recalculate the answer before sharing.
A scale statement such as 1 cm represents 4 m is a ratio between drawing length and real length. To find a drawing length, divide the real length by the real-world value represented by one drawing unit. To find a real length, multiply the drawing length by that same value. Keep the direction explicit: drawing to real uses multiplication, real to drawing uses division.
Mixed units cause most errors. Convert the real length to the unit named in the scale before dividing. If the scale is 1 cm represents 4 m and the real length is 600 cm, first convert 600 cm to 6 m, then divide by 4 m to get 1.5 cm on the drawing. Dividing 600 by 4 without converting would give 150 cm, which is wrong by a factor of 100.
When a drawing is redrawn at a different scale, every linear dimension changes by the same factor, but area changes by the square of that factor. If a plan is redrawn so that each drawing length becomes three times longer, each drawn area becomes nine times larger. Volume, if relevant, changes by the cube of the linear factor.
This matters when comparing two plans of the same object. A plan at 1:100 and a plan at 1:200 show the same real object, but the 1:100 drawing has linear dimensions twice as large and areas four times as large. The real object does not change; only the representation does.
Before committing to a scaled drawing, run a quick consistency check. Multiply the intended drawing length by the scale factor and confirm it returns the original real length. If the numbers do not match, the error is usually a unit conversion or an inverted ratio.
Also check the direction of the scale. A scale of 1:50 means one drawing unit represents 50 real units, so the drawing is smaller than the real object. Writing the ratio the wrong way round produces a drawing that is far too large and is easy to spot by comparing the drawing length with the real length.
A real wall is 900 cm long. A plan uses the scale 1 cm represents 3 m. Find the drawing length, then state how the drawn area would change if the plan were redrawn at 1 cm represents 1.5 m.
Convert 900 cm to 9 m. Divide 9 m by 3 m per centimetre to get 3 cm on the drawing. The new scale represents half the real length per centimetre, so each drawing length doubles. The linear factor is 2, so the drawn area becomes four times larger. Check: 3 cm times 3 m per centimetre returns 9 m, matching the real wall.
Convert the real length to the unit used in the scale statement, then divide by the real length represented by one drawing unit. Label the result with the drawing unit.
Area depends on two linear dimensions. If both are doubled, the product increases by two times two, which is four. The same reasoning gives nine for a factor of three.
Yes. A 1:100 plan and a 1:200 plan can describe the same object. The 1:100 drawing has linear dimensions twice as large and areas four times as large, but the real object is unchanged.
