How to Use a Mathematical Scale
In mathematics, “scale” is a term used to describe the relationship of an actual measurement compared to the measurement as it is represented, as in a paper and pencil drawing, for instance. Mathematical scale is represented numerically as a ratio. For example, a distance of one inch that is represented by one tenth of an inch on a paper and pencil drawing would be described as 1/10 scale.Sometimes people use the words "mathematical scale" to mean an architect's scale or an engineer's scale, which are both measuring instruments that aid in using the principles of mathematical scale to convert measurements while making a scale drawing. To use an architect's scale or an engineer's scale you'll need to become familiar with the markings on the scale and then learn to use the scale to convert measurements to make a simple drawing.
Become familiar with the scale

Notice the shape of the scale. Both an architect's scale and an engineer's scale are typically slightly longer than twelve inches. But instead of being flat with two sides, like a ruler, a scale is three sided, so that it is shaped like a long triangular solid. This gives the scale six edge faces on which you'll see graduated markings.

Look at the graduated markings on the six edge faces of the scale. Each edge has a different set of graduated markings according to a certain scale.

Understand the difference between an architects scale and an engineer's scale. An architects scale is usually marked with the following ratios—1:1, 1:2, 1:5, 1:100, 1:500, and 1:1250. An engineer's scale is marked with the following ratios—1:10, 1:20, 1:30, 1:40, 1:50, 1:60; on an engineer's scale the ends of each edge are marked with the last number of the ratio. For instance, the edge with the markings that represent a 1:10 scale is marked with a 10. For our lesson, we'll be using an engineer's scale.
Reproduce a drawing using an engineer's scale

Choose the scale ratio that suits your needs. For our example we'll reproduce a drawing using 1:10 scale. The finished drawing will be 1/10 the size of the original.

Choose an object or drawn figure to reproduce. For our example, we'll copy a drawing of a rectangle that measure four inches by six inches.

Use your ruler to measure a dimension of the original drawingin this case, one side of the rectangle. The side measures six inches.

Use the 1:10 edge of the engineer's scale. Notice the scale edge is marked “10”. Notice that each inch is divided into ten equal segments. Each segment represents one real inch in 1:10 scale.

Place the engineer's scale on your graph paper. From the zero position on the scale, count off six segments to represent six inches. Draw a line along the straight edge of the scale from zero to the end of the six segments you counted.

Repeat this process for the other sides of the original drawing, so that the original rectangle drawing is completely reproduced in 1:10 scale. The original rectangle measured 6" by 4". Our 1:10 drawing of the rectangle measures 3/5" by 2/5", which is exactly one tenth the size of the original.
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Tips
 Apply the principles of mathematical scale to make a drawing of a three dimensional object or a room or plot of land using the same procedures—measure the actual dimension and then use the scale to draw lines reduced by which ever ratio you choose. For large objects, you'll use a smaller scale—1/500 for instance, instead of 1/10.
 When using mathematical scale to make drawings of irregularly shaped objects, it may also be necessary to measure the angles at the points where linear dimensions meet.
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 M. J. Doran
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