Constructions and Scale Drawings

The three instruments, and what 'accurate' means on paper

Everything in this topic is marked on what is on the page, so the instruments matter as much as the method. Cambridge lists the geometrical instruments you should have for both the Core and the Extended papers: a pair of compasses, a protractor and a ruler. Add a sharp pencil, a clean eraser and a spare pencil, because a blunt one puts about a millimetre of error into every line you draw.

The syllabus repeats one instruction under constructions and again under scale drawings: a ruler should be used for all straight edges. A freehand line is not a straight edge however careful you are, and any construction or scale drawing built on one will not measure correctly. Measure lengths to the nearest millimetre and angles to the nearest degree, and write the unit with every length you record.

Labelling is part of the answer. Put the capital letters from the question at the right vertices, so that triangle ABC really has A, B and C where the question expects them, and mark each measured length on the diagram as you draw it. On a scale drawing, write the scale on the diagram and draw a north line at every point you measure a bearing from. Finish with the answer in the unit the question asks for.

• A ruler for every straight edge - no exceptions and no freehand
• Measure to the nearest millimetre and the nearest degree, and write the measurements on the diagram
• Label the vertices with the letters the question uses, in the right places
• Leave every construction arc on the page

Constructing a triangle from three sides

This is the construction Cambridge asks for, and the words 'using a ruler and pair of compasses only' rule out the protractor. You are given three side lengths and you build the triangle so that all three come out exactly right. Take the sides $AB=7\text{ cm}$, $BC=5\text{ cm}$ and $AC=6\text{ cm}$.

The arcs are what make the construction exact. An arc of radius 5 cm drawn from B is the set of every point that is 5 cm from B, and an arc of radius 6 cm from A is every point 6 cm from A. Where the two arcs cross is the only point that is both, so that crossing is C. That is why the arcs must be drawn with the compasses and not estimated, and why you must not change the opening of the compasses while drawing the two arcs that are meant to be equal.

The two arcs usually cross in two places, one above the base and one below. Either triangle is correct, because the question is asking for the three side lengths rather than a particular position, but keep A on the left and B on the right so the letters match the question. Finish by checking with the ruler that BC really is 5 cm and AC really is 6 cm.

• Draw the longest side as the base with the ruler: AB = 7 cm exactly
• Open the compasses to 5 cm and, with the point on B, draw an arc above the base
• Open the compasses to 6 cm and, with the point on A, draw an arc above the base that crosses the first one
• The crossing point is C: join C to A and to B with the ruler
• Check both lengths with the ruler, and leave the arcs on the page

A rhombus from two triangles, and the standard compass moves

The syllabus's own example of a construction is a rhombus: construct a rhombus by drawing two triangles. A rhombus has four equal sides, so a diagonal cuts it into two triangles whose sides are the rhombus side, the rhombus side and the diagonal. For a rhombus with sides of 5 cm and a diagonal of 6 cm, construct the triangle with sides 5 cm, 5 cm and 6 cm exactly as in the last block, then construct the same triangle again on the other side of the 6 cm line. Two triangles, four sides of 5 cm, and the 6 cm line is the shared diagonal. The same idea builds a kite or a parallelogram from the lengths you are given.

Three compass constructions turn up throughout geometry, and the compass work in them is the same skill the triangle construction needs. A perpendicular bisector is built by drawing arcs of equal radius, more than half of AB, from A and then from B, above and below the line; the two crossings joined give the line that cuts AB in half at right angles. An angle bisector is built by drawing an arc from the vertex that cuts both arms, then equal arcs from those two crossings; the line from the vertex through their crossing cuts the angle in half. A perpendicular from a point to a line is built by drawing an arc from the point that cuts the line twice, then constructing the perpendicular bisector of those two points.

Learn these three as practice with the compasses rather than as exam content: Cambridge states in C4.2 and E4.2 that constructions of perpendicular bisectors and angle bisectors are not required. The construction you will be asked for is the triangle from three side lengths, and every one of these moves is the same two habits - an unmoved compass for equal radii, and arcs left on the page.

• Rhombus, side 5 cm, diagonal 6 cm = two triangles with sides 5 cm, 5 cm, 6 cm
• Perpendicular bisector: equal arcs from each end, joined through the two crossings
• Angle bisector: an arc cutting both arms, then equal arcs from those two points
• Perpendicular from a point: an arc cutting the line twice, then the perpendicular bisector of those two points

Nets: drawing them, then using the measurements

A net is a solid opened out flat so that every face lies in one plane. A cube opens into six squares joined edge to edge, a cuboid into three pairs of rectangles, a triangular prism into two triangles and three rectangles, and a square-based pyramid into a square with four triangles on its sides. Cambridge asks you to draw the nets of cubes, cuboids, prisms and pyramids, and then to use the measurements in a net to work out volumes and surface areas.

A net carries the length, the width and the height of the solid, because every edge of the net is an edge of the solid. Read those three measurements off the net and use the ordinary formulas: for a cuboid, volume is length times width times height, and the surface area is the sum of the areas of the six faces. The net is also the safest way to find a surface area, because every face is in front of you and none can be forgotten.

Watch the units. Lengths from a net are in centimetres or millimetres, so a volume comes out in cubic units and an area in square units. Cambridge expects the unit with the number: $30{\text{ cm}}^3$, not 30.

• Cube: 6 equal squares, so the surface area is $6\times {\text{side}}^2$
• Cuboid 5 cm by 3 cm by 2 cm: faces of 15, 10 and 6 ${\text{cm}}^2$ in pairs, so the surface area is $2\times (15+10+6)=62{\text{ cm}}^2$ and the volume is $5\times 3\times 2=30{\text{ cm}}^3$
• Triangular prism: 2 triangles and 3 rectangles, all three rectangles as long as the prism
• Square-based pyramid: 1 square and 4 triangles

Scale drawings: what a scale of 1 : n means

A scale drawing is the real thing with every length divided by the same number, so the drawing has the same shape as the real object. A scale of 1 : 50 000 means that 1 unit on the drawing stands for 50 000 of the same unit in real life, so 1 cm on the drawing is 50 000 cm on the ground. Turn that into something you can picture: 50 000 cm is 500 m, which is 0.5 km. A scale written as 1 cm to 2 km says the same thing in the other style, and 1 cm to 2 km is the same as 1 : 200 000, because 2 km is 200 000 cm.

Drawing to real life: multiply. Two villages are 7.5 cm apart on a map with a scale of 1 : 25 000, so the real distance is $7.5\times 25000=187500$ cm. Change that to metres and then to kilometres: 187 500 cm = 1875 m = 1.875 km.

Real life to the drawing: divide. Two points are 4.5 km apart and the map scale is 1 : 50 000. Convert the real distance to centimetres first, because a scale compares like with like: 4.5 km = 450 000 cm, and $450000÷50000=9$ cm on the map. Both directions start with that same step, getting both lengths into the same unit, which is where most lost marks in this topic come from.

• 1 : 200 - 1 cm on the drawing is 200 cm = 2 m in real life
• 1 : 25 000 - 1 cm is 25 000 cm = 250 m = 0.25 km
• 1 : 50 000 - 1 cm is 500 m = 0.5 km
• 1 cm to 2 km - the same as 1 : 200 000; 5.4 cm is $5.4\times 2=10.8$ km

Three-figure bearings

A bearing describes a direction as an angle. It is measured clockwise from north, always written with three figures, and runs from 000 to 360. Due north is 000, due east is 090, due south is 180 and due west is 270. The three figures matter: a direction 5 degrees east of north is written 005, and due east is written 090 rather than 90.

The bearing of A from B is measured at B. That one sentence is the whole of most bearing questions. Draw a north line through the point you are measuring from, then turn clockwise from that north line to the line joining the two points. North lines are always drawn vertical on the page, so two of them in one diagram are parallel - which is why bearing questions so often finish with the parallel-line angle facts.

The bearing of A from B and the bearing of B from A always differ by 180. If the bearing of B from A is 025, the bearing of A from B is $025+180=205$. If the bearing of A from B is 250, the bearing of B from A is $250-180=070$. Add 180 when the answer stays under 360, and subtract 180 when it does not. The second of the two is called the back bearing.

In a scale-drawing question you may also meet the compass directions in words. 'Point D is due east of point C' means that D lies on the line running east from C, so the bearing of D from C is 090, and the bearing of C from D is 270. Due north, south and west work the same way with 000, 180 and 270.

• North 000, east 090, south 180, west 270
• Bearing of B from A = 025, so the bearing of A from B = 205
• Bearing of A from B = 250, so the bearing of B from A = 070
• 'D is due east of C' means the bearing of D from C is 090
• Measure at the point named second: the bearing of A from B is measured at B