What Are Geometrical Constructions?
Geometrical constructions are the drawing of exact geometric figures - bisectors, perpendiculars, specific angles, and polygons - using only a compass and a straightedge, with no reliance on measured lengths. The straightedge draws straight lines but its markings are never used; the compass draws circles and arcs and carries equal distances from place to place. Because nothing is measured, a construction is exact in principle, not just accurate to the nearest millimetre.
The compass and the straightedge each do one job. The compass fixes a radius and swings arcs - that is how equal distances get transferred. The straightedge connects two points with a line. Every classical construction is some combination of those two moves.
How This Differs From Just Using a Compass
It is easy to confuse geometrical constructions with using the compass. The distinction is worth drawing clearly. Geometrical constructions are the standard procedures - the bisectors, perpendiculars, and angles you can produce. The compass drawing topic covers the tool itself: how the compass is held, how to set a radius, and why every arc it draws has a constant distance from its centre. In short, one is the recipe and the other is the instrument. This article is the recipe; for the instrument, see the compass-drawing companion piece.
Construction 1: The Perpendicular Bisector
A perpendicular bisector is a line that cuts a segment exactly in half and meets it at a right angle. Every point on it is equidistant from the two endpoints, which is why it appears constantly in later geometry.
Steps to construct the perpendicular bisector of segment $AB$:
Open the compass to more than half the length of $AB$.
With the point on $A$, draw an arc above and below the segment.
Keeping the same radius, place the point on $B$ and draw two more arcs, crossing the first pair at $P$ (above) and $Q$ (below).
Draw the straight line through $P$ and $Q$.
The line $PQ$ is the perpendicular bisector; it crosses $AB$ at its midpoint $M$ at $90°$. It works because $P$ and $Q$ are each the same distance from $A$ and from $B$, so both lie on the line of equal distance - the perpendicular bisector proper.
Construction 2: The Angle Bisector
An angle bisector is a ray that splits an angle into two equal angles.
Steps to bisect $\angle AOB$ (vertex at $O$):
With the point on the vertex $O$, draw an arc that crosses both arms of the angle, at $P$ and $Q$.
With the point on $P$, draw an arc in the interior of the angle.
Keeping the same radius, place the point on $Q$ and draw an arc crossing the previous one at $R$.
Draw the ray $OR$.
$OR$ bisects $\angle AOB$. It works because $OP = OQ$ and $PR = QR$, so the two halves are mirror images. This is the same reasoning behind the angle bisector and its equal-distance property.
Construction 3: A Perpendicular From a Point on a Line
To raise a perpendicular at a point $X$ on a line:
With the point on $X$, draw two arcs cutting the line at equal distances on either side, at $C$ and $D$.
Widen the compass. From $C$ and then $D$, draw arcs of equal radius meeting above the line at $E$.
Draw the line $XE$.
$XE$ is perpendicular to the original line at $X$. This is really a perpendicular bisector of the short segment $CD$, reused as a construction move.
Construction 4: A 60° Angle
The $60°$ angle is the seed of equilateral triangles and, folded and bisected, of $30°$ and $15°$ angles too.
Steps to construct a 60° angle at point $P$ on ray $PQ$:
With the point on $P$, draw an arc crossing $PQ$ at $S$.
Keeping the same radius, place the point on $S$ and draw an arc crossing the first arc at $T$.
Draw the ray $PT$.
$\angle TPQ = 60°$. The reason is neat: $P$, $S$, and $T$ are all the same distance apart, so triangle $PST$ is equilateral, and every angle of an equilateral triangle is $60°$.
Examples of Geometrical Constructions
Each example applies the constructions above to a slightly fuller task. Problem statements are in bold; the steps are not.
Example 1
Construct the perpendicular bisector of a 6 cm segment, then state one property of the resulting line.
Draw $AB = 6$ cm. Open the compass past $3$ cm (more than half), swing arcs from $A$ and $B$ above and below, mark the crossings $P$ and $Q$, and join them.
The line $PQ$ meets $AB$ at its midpoint at $90°$. One key property: every point on $PQ$ is equidistant from $A$ and $B$.
Example 2
Construct a 30° angle.
First construct a $60°$ angle using the equilateral-triangle method. Then bisect that $60°$ angle with the angle-bisector construction.
Each half is $30°$, so bisecting the $60°$ angle gives the required $30°$.
Example 3: The tempting shortcut that misfires
A student is asked to construct a 60° angle and reaches for the protractor, measures 60°, and draws it.
The line looks right on the page. But the task was a construction, and a protractor reading is a measurement, not a construction. A construction must use compass and straightedge alone, so the protractor answer is not valid - and if the protractor is slightly off, so is the angle, with no guarantee of exactness.
The rescue is the equilateral-triangle method: swing one arc from $P$, a second of the same radius from where it meets the ray, and join through the crossing point. Because all three sides are equal by construction, the $60°$ is exact by geometry, not by a scale reading.
Example 4
Construct a perpendicular to a line at a given point on it.
Mark point $X$ on the line. Swing equal arcs from $X$ to cut the line at $C$ and $D$. From $C$ and $D$, swing larger equal arcs meeting at $E$ above the line. Join $XE$.
$XE$ stands at $90°$ to the line at $X$, because $XE$ is the perpendicular bisector of $CD$.
Example 5
Divide a given angle into four equal parts.
Bisect the angle once to get two halves. Then bisect each half again.
Two rounds of the angle-bisector construction split the original angle into four equal pieces - halving twice gives quarters.
Example 6
Construct an equilateral triangle on a given base $BC$.
With radius equal to $BC$, swing an arc from $B$ and another from $C$; they meet at $A$. Join $AB$ and $AC$.
Because $AB = BC = CA$ (all set to the same compass radius), triangle $ABC$ is equilateral, and each angle is $60°$.
Where Geometrical Constructions Earn Their Keep
Compass-and-straightedge work is far more than a school exercise - it is where geometry was first made rigorous.
Ancient rigour. The Greeks built their entire geometry on constructions. Euclid's Elements opens with the construction of an equilateral triangle, and every later proof relies only on what can be drawn this way. The restriction to two tools was a deliberate demand for certainty.
The impossible three. Some tasks defeated geometers for two millennia - trisecting an arbitrary angle, doubling a cube, squaring a circle - until nineteenth-century algebra proved they cannot be done with compass and straightedge. The limits of the tools became a deep result.
Which polygons are constructible. The Greeks could build many regular polygons, but it took Gauss to prove exactly which are possible - famously the 17-sided heptadecagon, which no Greek had managed.
The deeper "why" is that a construction is a proof you can see. Nothing is measured, so nothing depends on the accuracy of a ruler; the figure is correct by geometry itself. For the full story of what can and cannot be built, see Wolfram MathWorld's entry on geometric constructions.
The Mistakes Students Make Most Often In Geometrical Construction
Mistake 1: Changing the compass width mid-step
Where it slips in: Between drawing the arc from the first point and the arc from the second, the rusher nudges the compass and the radius shifts.
Don't do this: Set the compass to one width for $A$'s arcs and a different width for $B$'s arcs in a perpendicular bisector.
The correct way: Once a radius is set for a paired construction, keep it fixed until both arcs are drawn. Equal arcs are the whole reason the construction is exact.
Mistake 2: Reaching for the protractor or ruler markings
Where it slips in: When an angle or length is needed, the habit is to measure it, as in Example 3.
Don't do this: Measure $60°$ with a protractor, or use the numbers on the ruler to mark a midpoint.
The correct way: A construction uses the straightedge only to draw straight lines and the compass to carry distances. Numbers are never read off. If the method calls for a measurement, it is not a valid construction.
Mistake 3: Opening the perpendicular-bisector arcs too small
Where it slips in: The compass is opened to less than half the segment, so the arcs from the two ends never cross.
Don't do this: Set the radius shorter than half of $AB$ and then hunt for a crossing point that does not exist.
The correct way: Always open the compass to more than half the segment length before swinging the arcs - that guarantees the arcs from each end overlap above and below.
Conclusion
Geometrical constructions produce exact figures using only a compass and an unmarked straightedge.
The core four are the perpendicular bisector, the angle bisector, a perpendicular from a point, and a $60°$ angle.
Every construction works because the compass carries equal distances - arcs of equal radius do the proving.
No measurement is ever used; a construction is a proof you can draw.
Some tasks, like trisecting an arbitrary angle, are provably impossible with these tools alone.
To take geometrical constructions further with a teacher, explore Bhanzu's geometry tutor, a middle school math tutor for the step-by-step methods, or browse math classes online. Want structured practice with a live trainer? Try a free class.
A Practical Next Step
Work through these constructions to solidify your understanding, using compass and straightedge only.
Construct the perpendicular bisector of an 8 cm segment. (Answer to Question 1: the arcs from each end, radius more than 4 cm, meet at two points whose join crosses the segment at its midpoint at 90°.)
Construct a 45° angle. (Answer to Question 2: construct a 90° angle, then bisect it.)
Construct an equilateral triangle on a 5 cm base. (Answer to Question 3: swing 5 cm arcs from both ends of the base; their crossing is the apex.)
Read More
Angle Bisector Theorem - the ratio result that follows from bisecting an angle.
Points and Lines - the basic objects every construction is built from.
Pythagoras Theorem - a classic result whose proof can be drawn as a construction.
Coordinate Plane - where constructed points can be given coordinates.
Was this article helpful?
Your feedback helps us write better content