Geometrical constructions

Measuring and drawing lines

Accurate drawing questions are marked by measuring your diagram, so precision earns the marks. Examiners allow a small tolerance, usually about 2 mm on a length, but no more.

• Use a sharp pencil and a ruler for every straight edge. A freehand line, even a neat one, is not accepted.
• Measure from the 0 mark on the ruler, not from the end of the ruler, which is often a few millimetres before the 0.
• Read lengths in centimetres to 1 decimal place: 7.4 cm is 7 cm and 4 mm.
• Mark points with a small, sharp cross or dot and label them.

When a length is given by subtraction, such as "$C$ is on $AB$ with $AC=2.9$ cm", measure from $A$, not from $B$. You can check a measured length with arithmetic: if $AB=7.6$ cm and $AC=2.9$ cm then $CB$ must be $7.6-2.9=4.7$ cm.

Measuring and drawing angles

A protractor has two scales, running in opposite directions. To measure an angle:

1. Put the centre of the protractor exactly on the vertex.
2. Line up the zero line with one arm of the angle.
3. Read from the scale that starts at $0^{°}$ on that arm.

Always estimate first. If the angle is obviously acute and you read ${136}^{°}$, you have used the wrong scale: the answer is ${180}^{°}-{136}^{°}={44}^{°}$. The two scales always add to ${180}^{°}$, which is why the wrong-scale reading is the "supplement" of the right one.

To draw an angle, draw one arm with the ruler, put the protractor centre on the end that is to be the vertex, mark a dot at the required number of degrees on the correct scale, then join the vertex to the dot.

Reflex angles. A semicircular protractor only reaches ${180}^{°}$. A reflex angle and the angle on the other side of the arms make a full turn, so draw (or measure) the other angle and use ${360}^{°}$: to draw ${237}^{°}$, draw ${360}^{°}-{237}^{°}={123}^{°}$ and mark the reflex angle outside it. You could also extend one arm to make a straight line, ${180}^{°}$, and measure the extra ${57}^{°}$ beyond it.

Constructing a triangle from three sides

"Construct" means draw accurately using only a ruler and a pair of compasses, leaving the construction arcs showing. The arcs are the evidence of method; rubbing them out loses marks.

To construct triangle $ABC$ given $AB$, $AC$ and $BC$:

1. Draw the base $AB$ with a ruler, to the exact length. Label both ends.
2. Open the compasses to the length $AC$ (measure the opening on the ruler). Put the point on $A$ and draw an arc where $C$ will roughly be.
3. Open the compasses to the length $BC$. Put the point on $B$ and draw an arc that crosses the first one.
4. The crossing point is $C$. Join $AC$ and $BC$ with a ruler, and label $C$.

Why it works. An arc centred at $A$ with radius 6 cm is part of the circle of every point exactly 6 cm from $A$. The vertex $C$ has to be 6 cm from $A$ and the right distance from $B$, so it must lie on both circles, which is where the arcs cross. The arcs also cross on the other side of $AB$; that gives the same triangle reflected, so either is correct.

The syllabus only asks you to construct triangles from three sides (and shapes made of such triangles). Constructions of perpendicular bisectors and angle bisectors are not required.

When a triangle cannot be constructed

The two arcs only meet if the two shorter sides are long enough to reach each other across the base. If the two shorter sides together are not longer than the longest side, the arcs never cross and there is no triangle.

For example, sides of 3 cm, 4 cm and 8 cm: $3+4=7$, which is less than 8, so the arcs from the ends of the 8 cm base fall short of each other. With 3 cm, 5 cm and 8 cm, $3+5=8$ exactly and the "triangle" collapses into a straight line.

So for any triangle, each side must be shorter than the sum of the other two. This is worth checking before you start a construction, and it is the reason behind questions that ask which lengths could make a triangle.

Constructing a rhombus and other shapes from triangles

Any shape that can be split into triangles with known sides can be constructed the same way, one triangle at a time. A rhombus is the classic example: a diagonal splits it into two congruent isosceles triangles, each with two sides equal to the side of the rhombus.

To construct a rhombus with side $s$ and one diagonal $d$:

1. Draw the diagonal, length $d$.
2. With the compasses set to $s$, draw arcs from both ends of the diagonal, above and below it: four arcs in all.
3. The arcs cross at one point above and one below. These are the other two vertices.
4. Join the four sides with a ruler. Leave all four arcs showing.

The other diagonal is not given; it comes out of the construction, and a question may ask you to measure it.

Nets of solids

A net is a flat pattern that folds along its edges to make a solid, with every face appearing exactly once.

To draw the net of a cuboid, unfold it like a box: a strip of four rectangles that wraps round the sides, with the top and bottom attached to one of them. Each edge that is folded must match the edge it meets: if two faces meet along a 3 cm edge in the solid, they must share a 3 cm side in the net.

Nets to know:

• Cube: six squares. There are 11 different cube nets. Six squares in a 2 by 3 block do not fold into a cube, and nor does any arrangement where two squares would fold onto the same face.
• Cuboid: six rectangles in three matching pairs.
• Prism: the rectangles for the sides in a row, with the two end faces attached. A triangular prism has three rectangles and two triangles.
• Pyramid: the base, with a triangle on each edge of the base. A square-based pyramid net looks like a star of four triangles round a square.
• Cylinder: a rectangle and two circles. The length of the rectangle is the circumference of the circles, because it wraps once round them.

To decide which faces of a cube net are opposite, use the rule that in a straight row of squares, faces one apart (with one square between them) are opposite when folded.

Using a net to find surface area and volume

The surface area of a solid is the total area of all its faces, and the net shows every face exactly once. So the surface area is the area of the net: find the area of each piece and add.

The net also tells you the dimensions you need for the volume. For the cuboid net above, the faces show that the cuboid is 3 cm by 2 cm by 4 cm, so its volume is $3\times 2\times 4=24{\text{cm}}^3$ and its surface area is $2(3\times 2+3\times 4+2\times 4)=2\times 26=52{\text{cm}}^2$.

For a prism, volume $=$ area of cross-section $\times$ length. For a pyramid, volume $=\frac{1}{3}\times$ base area $\times$ vertical height. Be careful with a pyramid net: the height of each triangle in the net is the slant height, not the vertical height of the pyramid. You find the vertical height with Pythagoras.