Constructions

Why this matters

Before protractors and computers, builders set out right angles for walls and foundations with pegs and string using exactly these constructions. They are still the most accurate way to draw angles by hand — a compass does not have the small reading errors of a protractor.

The toolkit

Every construction in this lesson is built from two moves:

Equal arcs. Two points the same distance from a third lie on a circle round it.
Crossing arcs. Where two arcs of equal radius from A and B cross, the point is the same distance from A as from B.

Construction	Result
Equilateral-triangle arcs	$60^\circ$
Perpendicular at a point on a line	$90^\circ$
Perpendicular bisector of a line segment	$90^\circ$ through the midpoint
Perpendicular from a point to a line	$90^\circ$ from a point off the line
Angle bisector	half of any angle

Making other angles

Once you can make $60^\circ$ and $90^\circ$ and halve any angle, many more follow:

$30^\circ$ : bisect $60^\circ$ . $\;15^\circ$ : bisect $30^\circ$ .
$45^\circ$ : bisect $90^\circ$ . $\;22.5^\circ$ : bisect $45^\circ$ .
$120^\circ$ : two $60^\circ$ angles side by side, or $180^\circ - 60^\circ$ on a straight line.
$135^\circ$ : $90^\circ + 45^\circ$ , or $180^\circ - 45^\circ$ .

Constructing a triangle from angles

To construct a triangle with a 6 cm base and base angles of $45^\circ$ and $30^\circ$ : draw the 6 cm base (measuring this is allowed — it is given), construct $45^\circ$ at one end and $30^\circ$ at the other, and extend the arms until they meet.

Neat work matters. Use a sharp pencil, make the arcs long enough to cross clearly, and never rub them out.

Key rules and formulas

60° comes from an equilateral triangle

Keep the compasses the same width throughout. The three points X, A and Y are all one radius apart, so triangle XAY is equilateral and angle YXA is

6 0^{\circ}

Bisecting halves any angle

Arc from the vertex to cut both arms, then equal arcs from those two points. The line from the vertex through the crossing splits the angle exactly in half.

90° from a perpendicular

Arcs from X cut the line at A and B. Wider arcs from A and B cross at P and Q. PQ is perpendicular to the line.

Building other angles

Halve, add and subtract:

6 0^{\circ} \to 3 0^{\circ} \to 1 5^{\circ}

;

9 0^{\circ} \to 4 5^{\circ} \to 22. 5^{\circ}

; and

9 0^{\circ} + 4 5^{\circ} = 13 5^{\circ}

Worked examples

Worked example 1

Constructing 60° and 30°

Construct an angle of

3 0^{\circ}

at the point X on a line.

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

The perpendicular from a point to a line

A point X is above a line. Construct the perpendicular from X to the line.