Same Math › Geometry · grades 9–10 › Compass and straightedge constructions
How do compass and straightedge constructions work?
A construction draws an exact figure with only a compass (for equal lengths) and a straightedge (for straight lines), no ruler marks or protractor. Every construction is a few arcs and lines in a fixed order. To bisect a segment, swing equal arcs from both ends, above and below, and connect the two crossings. The steps work because every arc makes equal lengths.
- Arcs: same compass width from each end.
- Line: through the two crossing points.
- It hits AB at the midpoint, at a right angle.
Why schools teach it
Why build without a ruler?
Geometry students make formal constructions (HSG.CO.D.12), including copying segments and angles, bisecting them, and drawing perpendicular and parallel lines, and they inscribe shapes in circles (HSG.CO.D.13).
The point is the reasoning. Every arc is a set of equal lengths, so each construction is a small proof: the perpendicular bisector works because each crossing point is the same distance from A and B.
At the kitchen table
What to say, and what to skip
Say this
What does the compass guarantee?
Equal lengths.Did the compass width change?
Keep it the same for both arcs.Is the compass wider than half the segment?
Otherwise the arcs won't cross.Which two points do you connect?
The crossings.
Skip this
- Measuring with a ruler.Constructions use the compass for lengths.
- Changing the compass width mid-step.The equal lengths are the whole point.
- Erasing the arcs.Teachers want to see them as the proof.
See it in action
Bisecting an angle
Arc from the vertex
crosses both sidesSame width, arcs from those two points
they cross inside the angleConnect the vertex to the crossing
the angle bisector
It works because it builds two congruent triangles (SSS).
Free practice sheet
Make a constructions sheet
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Parents also ask
Is a straightedge just a ruler?
A ruler can be one if you ignore the marks.
Which constructions come up most?
Copying a segment and an angle, bisecting a segment and an angle, perpendicular lines, parallel lines and the equilateral triangle.
What if my child's compass slips?
A compass with a locking wheel helps, as does a pencil sharpened short.
Are there apps for this?
Yes, many classes use GeoGebra or Desmos for digital constructions with the same steps.