Standard Ruler and Compass Constructions

What a ruler and compass construction does

A construction makes an accurate geometric figure using a ruler as a straight edge and a pair of compasses to draw arcs. The ruler joins points; equal compass openings create equal distances. Rather than estimating a midpoint or measuring an angle with a protractor, you locate points where carefully chosen arcs cross.

Bisecting a line segment at right angles

To bisect a line segment means to divide it into two equal lengths. A perpendicular bisector of segment $AB$ does this while crossing $AB$ at $90^\circ$.

Draw $AB$. Open the compasses to more than half its length. With the compass point at $A$, draw an arc above $AB$ and another below it. Without changing the opening, repeat from $B$, making the new arcs cross the first pair. Join the two arc-crossing points with the ruler. This is the perpendicular bisector: the equal-radius arcs place both crossing points equally far from $A$ and $B$, so their joining line passes through the midpoint of $AB$ at right angles.

Drawing a perpendicular through a given point

A perpendicular through a specified point is not always a perpendicular bisector of the whole line segment shown. What matters is that the new line passes through the given point and meets the given line at $90^\circ$.

If the given point $P$ is on the line, centre the compasses at $P$ and mark points $A$ and $B$ on the line, one on each side of $P$, at the same distance from it. From $A$ and $B$, draw equal-radius arcs that cross above the line; choose an opening greater than half of $AB$. Join their crossing point to $P$. Because $P$ is the midpoint of $AB$, this join is perpendicular to the line at $P$.

If $P$ is away from the line, centre the compasses at $P$ and draw an arc that cuts the line at two points, $A$ and $B$. Keeping a new opening the same for both points, draw arcs from $A$ and $B$ on the opposite side of the line from $P$ so that they cross at $C$. Join $P$ to $C$. Both $P$ and $C$ are equally far from $A$ and $B$, so $PC$ is the perpendicular bisector of $AB$ and meets the given line at right angles.

Dividing an angle into two equal parts

An angle bisector starts at the angle’s vertex and splits the angle exactly in half. For an angle with vertex $Q$, draw an arc centred at $Q$ that crosses both arms. From each crossing point, draw an arc inside the angle, using the same compass opening for these two arcs. Join $Q$ to the point where the inner arcs cross. The join divides the original angle into two equal angles. For instance, bisecting a $60^\circ$ angle produces two $30^\circ$ angles.

Constructing a 60° angle

Start with a ray from $A$. Set the compasses to a chosen width and draw an arc centred at $A$ that crosses the ray at $B$. Keep that opening and draw an arc centred at $B$ to meet the first arc at $C$. Join $A$ to $C$. The three lengths $AB$, $AC$ and $BC$ are equal because they were made with the same compass opening. They form an equilateral triangle, so the angle $BAC$ is $60^\circ$. The same steps can construct an equilateral triangle on a given base $AB$.

Perpendicular constructions

• Bisector of $AB$: use equal arcs from $A$ and $B$ above and below the segment; join their crossings. The compass opening must exceed half of $AB$.
• At a point $P$ on a line: mark equal distances either side of $P$, cross equal arcs from those marks, then join the crossing to $P$.
• From a point $P$ off a line: use an arc from $P$ to cut the line twice; cross equal arcs from those cuts on the far side; join their crossing to $P$.

Angle constructions

• Bisect an angle: arc from the vertex across both arms → equal arcs from those cuts → join the vertex to their crossing.
• Construct $60^\circ$: make three equal sides with one unchanged compass opening, then join the vertex to the arc crossing. The resulting triangle is equilateral.