0580 · 4.3Scale drawings

Scale drawings and conversions

0:00A map can show a journey of several kilometres on a single sheet of paper. To make that useful, it must preserve both distance and direction. Scale tells us how distances have been reduced, while bearings describe directions precisely. By the end of this video, you will be able to draw a route and interpret the information it contains. You need a ruler, a protractor, and confidence with simple multiplication and division.

0:22Remember that one metre contains one hundred centimetres, and one kilometre contains one thousand metres. A full turn contains three hundred and sixty degrees. These facts let us connect measurements on paper with distances and directions in the real world. A scale drawing represents an object or place using a fixed relationship between drawing lengths and actual lengths. Suppose one centimetre represents two hundred metres. Every centimetre on the drawing then represents the same actual distance.

0:51A line three centimetres long represents six hundred metres because it contains three of those equal lengths. The drawing reduces distances, but it preserves angles and directions. A road pointing east still points east on the plan, provided the north direction is marked correctly. To interpret a drawing length, multiply it by the actual distance that one centimetre represents. Going the other way, divide a real distance by that same amount to get the length to draw.

1:18Choosing a scale is a balance between fitting the drawing onto the page and making it large enough to measure accurately. Tiny diagrams make small ruler errors more important. Use a ruler for all straight edges, and mark endpoints carefully with a sharp pencil. A rough sketch can organise your thinking, but it cannot replace a requested scale drawing.

Calculating distances from ratio scales

1:38Our first exam-style example uses a scale of one to twenty-five thousand. Two places are seven point two centimetres apart on the map. We must find their actual separation in kilometres. A ratio scale compares lengths in the same units, so one centimetre represents twenty-five thousand centimetres. Convert twenty-five thousand centimetres into metres by dividing by one hundred, giving two hundred and fifty metres. Multiply seven point two by two hundred and fifty to obtain one thousand eight hundred metres.

2:07Dividing by one thousand gives one point eight kilometres. The units change during the calculation, but the actual distance stays the same. Using the same scale, suppose we must draw a distance of six hundred metres. Each centimetre represents two hundred and fifty metres, so calculate 600 ÷ 250. This gives a drawing length of two point four centimetres. Draw that length with a ruler rather than estimating it by eye.

Bearings and reverse directions

2:32North, east, south and west give us four main directions. On a drawing, use the north arrow rather than assuming north must point towards the top of the page. East is a clockwise turn of ninety degrees from north. South lies directly opposite, after a half-turn of one hundred and eighty degrees. West comes next on the clockwise turn, at two hundred and seventy degrees. A bearing is an angle measured clockwise from north and written using three figures.

2:58Imagine standing at the starting point, facing north, then turning clockwise until you face your destination. The angle through which you turn is the bearing. This gives every direction a precise description, rather than just saying that somewhere is roughly east. A point due east lies exactly ninety degrees clockwise from north, so its bearing is zero, nine, zero degrees. A direction twenty-five degrees clockwise from north has bearing zero two five degrees.

3:25Those leading zeros matter because bearings use three figures. Due north is normally written as zero, zero, zero degrees. A bearing of three, six, zero degrees describes the same direction after a complete turn. Bearings greater than one hundred and eighty degrees require a turn past south. To draw a bearing of two four five degrees, place the protractor centre at the starting point. Align its reference line with north and check which numbered scale measures clockwise.

3:51A semicircular protractor cannot show the whole turn directly, so use the remaining angle on the other side. Subtract two hundred and forty-five degrees from three hundred and sixty degrees to obtain one hundred and fifteen degrees. Measure that angle anticlockwise from north, and mark the direction. The clockwise turn to this mark is still two hundred and forty-five degrees. Remove the protractor and use a ruler to draw the required line.

4:16The bearing of Q from P is measured at P, looking towards Q. For the bearing of P from Q, we stand at Q instead and look back towards P. Going back along the same line means facing the opposite way, which is a half-turn. Because the north lines point in the same direction, the two bearings differ by one hundred and eighty degrees. For our second example, the bearing of Q from P is zero two five degrees.

4:40Add one hundred and eighty degrees to twenty-five degrees, giving two hundred and five degrees. Therefore, the bearing of P from Q is two zero five degrees. This is called the back bearing because it describes the return direction. For a bearing of two four five degrees, adding one hundred and eighty would take us past three hundred and sixty degrees. Instead, subtract one hundred and eighty degrees from two hundred and forty-five degrees, giving sixty-five degrees.

5:03Write the back bearing as zero six five degrees. The rule is to add one hundred and eighty degrees to a bearing below that value, and to subtract it from a larger bearing.

Combined scale drawing example

5:13Our third exam-style example combines scale and bearings. Q is six kilometres from P on a bearing of zero six zero degrees. R is four kilometres due east of Q. Use a scale of one centimetre representing one kilometre to draw the route, then find the direct distance and bearing from P to R. Divide six kilometres by one kilometre per centimetre to obtain a drawing length of six centimetres.

5:37The second distance becomes four centimetres using the same scale. Place P towards the lower left of your drawing space, leaving room above and to the right. Draw a north line vertically upwards from P using your ruler. Centre your protractor at P and align it with north. Mark sixty degrees clockwise from north, then draw a six-centimetre line in that direction to locate Q. From Q, draw a four-centimetre line due east to locate R.

6:03Due east means perpendicular to north and towards the right on this drawing. Join P to R with a ruler because we need the direct distance, not the length of the two-stage route. Measure this new line, obtaining approximately nine point seven centimetres. Each centimetre represents one kilometre, so the direct distance is approximately nine point seven kilometres. The route through Q is longer because it changes direction rather than going straight to R.

6:29To find the bearing of R from P, place the protractor centre at P. Measure clockwise from the north line to the line towards R, obtaining approximately seventy-two degrees. Write the bearing as zero seven two degrees. Your measurements may differ slightly because scale drawings depend on the accuracy of your construction and measuring tools. Common mistakes include measuring from east, turning anticlockwise, and placing the protractor at the destination.

6:54Each changes the meaning of the angle being measured. In the phrase bearing of R from P, the starting point is P, so draw the north line at P. Also check the protractor scale, because reading its opposite numbering gives the wrong direction. Another frequent error is interpreting one to twenty-five thousand as one centimetre representing twenty-five thousand metres. A ratio scale uses matching units, so that interpretation makes the distance one hundred times too large.

7:22Keep units visible in your written working. Avoid measuring an illustration unless the question identifies it as a scale drawing or asks you to construct one. These methods must work without a calculator on Paper two, and remain useful on calculator Paper four. For calculations, follow the question’s accuracy instruction; otherwise, numerical answers generally use three significant figures. Keep intermediate values unrounded. For measured answers, use the stated tolerance or rounding instruction, and do not invent extra precision beyond what your instruments support.

7:53A scale drawing connects paper lengths with actual distances through one consistent scale. Bearings connect directions with clockwise angles measured from north at the starting point. Reverse directions differ by one hundred and eighty degrees. With a ruler, a carefully aligned protractor, and clear units, you can turn a written journey into a drawing and interpret it confidently.