Draw a Circle Through All Three Corners of Any Triangle With Just Compasses
Construct a pair of tangents from an external point, circumscribe a circle about a triangle with perpendicular bisectors, inscribe a circle with angle bisectors, and draw both circles on a regular hexagon, checking every radius by calculation.
Which ideas make circle constructions work?
Every construction in this chapter rests on a result you already know:
- The angle in a semicircle is a right angle — used to draw tangents
- Points on a perpendicular bisector are equidistant from the ends of a segment — used for the circumcircle
- Points on an angle bisector are equidistant from the arms — used for the incircle
With only a ruler and compasses, you can draw each circle accurately and then check its radius by calculation. This chapter covers tangents, circumcircles, incircles and circles on a regular hexagon.
- The angle in a semicircle is a right angle — used to draw tangents
- Points on a perpendicular bisector are equidistant from the ends of a segment — used for the circumcircle
- Points on an angle bisector are equidistant from the arms — used for the incircle
With only a ruler and compasses, you can draw each circle accurately and then check its radius by calculation. This chapter covers tangents, circumcircles, incircles and circles on a regular hexagon.
How do you construct a pair of tangents to a circle from an external point and measure their length?
Join the centre to the external point, bisect that segment, draw a circle on it as diameter, and join the external point to the two places where this circle cuts the given circle.
Steps for a circle of radius cm and a point that is cm from the centre :
- Draw the circle with centre , radius cm, and mark with cm
- **Bisect ** to find its mid-point
- **With centre and radius **, draw a circle cutting the first circle at and
- **Join and — these are the tangents
Why it works.** is an angle in a semicircle on diameter , so and is perpendicular to the radius.
Check by calculation.
For radius cm and cm: cm.
An everyday example. Two ropes pulled tight from a tent peg to just touch a round water tank are equal tangents.
The substance. Your measured length should match the calculated one to within a millimetre or two.
Steps for a circle of radius cm and a point that is cm from the centre :
- Draw the circle with centre , radius cm, and mark with cm
- **Bisect ** to find its mid-point
- **With centre and radius **, draw a circle cutting the first circle at and
- **Join and — these are the tangents
Why it works.** is an angle in a semicircle on diameter , so and is perpendicular to the radius.
Check by calculation.
For radius cm and cm: cm.
An everyday example. Two ropes pulled tight from a tent peg to just touch a round water tank are equal tangents.
The substance. Your measured length should match the calculated one to within a millimetre or two.
How do you circumscribe a circle about a triangle, and what is its circumradius?
Construct the perpendicular bisectors of any two sides; they meet at the circumcentre, which is equidistant from all three vertices, and the distance to any vertex is the circumradius.
Steps:
- Construct the triangle from the given measurements
- Draw the perpendicular bisectors of two sides, say and
- **Mark their meeting point — the circumcentre
- With centre and radius **, draw the circle through , and
Worked example 1. cm, cm, cm. Since , the triangle is right-angled at , so the circumcentre is the mid-point of and
Worked example 2. An equilateral triangle of side cm has
Where the circumcentre lies. Inside an acute triangle, at the mid-point of the hypotenuse of a right triangle, and outside an obtuse triangle.
An everyday example. A water tank equally far from three houses stands at the circumcentre of the triangle they form.
The substance. The third perpendicular bisector should pass through the same point — a quick check of accuracy.
Steps:
- Construct the triangle from the given measurements
- Draw the perpendicular bisectors of two sides, say and
- **Mark their meeting point — the circumcentre
- With centre and radius **, draw the circle through , and
Worked example 1. cm, cm, cm. Since , the triangle is right-angled at , so the circumcentre is the mid-point of and
Worked example 2. An equilateral triangle of side cm has
Where the circumcentre lies. Inside an acute triangle, at the mid-point of the hypotenuse of a right triangle, and outside an obtuse triangle.
An everyday example. A water tank equally far from three houses stands at the circumcentre of the triangle they form.
The substance. The third perpendicular bisector should pass through the same point — a quick check of accuracy.
How do you inscribe a circle in a triangle, and what is its inradius?
Construct the bisectors of any two angles; they meet at the incentre, which is equidistant from all three sides, and the perpendicular distance from it to any side is the inradius.
Steps:
- Bisect two angles, say and
- **Mark their meeting point — the incentre
- From , draw a perpendicular** to one side, meeting it at
- **With centre and radius , draw the circle touching all three sides
Worked example 1.** For the right triangle with sides , and cm:
where is the semi-perimeter.
Worked example 2. An equilateral triangle of side cm has
which is exactly half its circumradius.
An everyday example. The largest round rangoli that fits inside a triangular courtyard is centred at the incentre and touches all three walls.
The trap. **The radius is the perpendicular distance from to a side**, not the distance from to a vertex.
Steps:
- Bisect two angles, say and
- **Mark their meeting point — the incentre
- From , draw a perpendicular** to one side, meeting it at
- **With centre and radius , draw the circle touching all three sides
Worked example 1.** For the right triangle with sides , and cm:
where is the semi-perimeter.
Worked example 2. An equilateral triangle of side cm has
which is exactly half its circumradius.
An everyday example. The largest round rangoli that fits inside a triangular courtyard is centred at the incentre and touches all three walls.
The trap. **The radius is the perpendicular distance from to a side**, not the distance from to a vertex.
How do you construct a regular hexagon and draw its circumscribed and inscribed circles?
A regular hexagon's side equals the radius of its circumcircle, so step off the side six times around a circle; its incircle has the same centre and a radius equal to the perpendicular distance to a side.
Steps for side cm:
- **Draw a circle of radius cm
- Starting anywhere on it, step off arcs of cm six times
- Join the six points** — each interior angle is
- The circle you began with is the circumcircle, radius cm
- Drop a perpendicular from the centre to one side and draw the incircle with that radius
Calculating the radii. The hexagon is made of six equilateral triangles of side cm, so
An everyday example. Hexagonal floor tiles and the heads of nuts and bolts are regular hexagons, with a circle fitting neatly both inside and around each one.
The substance. **The incircle radius is always of the circumcircle radius** for a regular hexagon, about times as large.
Steps for side cm:
- **Draw a circle of radius cm
- Starting anywhere on it, step off arcs of cm six times
- Join the six points** — each interior angle is
- The circle you began with is the circumcircle, radius cm
- Drop a perpendicular from the centre to one side and draw the incircle with that radius
Calculating the radii. The hexagon is made of six equilateral triangles of side cm, so
An everyday example. Hexagonal floor tiles and the heads of nuts and bolts are regular hexagons, with a circle fitting neatly both inside and around each one.
The substance. **The incircle radius is always of the circumcircle radius** for a regular hexagon, about times as large.
Exam tip
What earns full marks on circle constructions?
Keep every construction arc visible, write the steps of construction briefly, and measure and state the radius or length asked for.
- Use a sharp pencil and accurate compasses
- Label the centre as , or as appropriate
- Circumcircle: perpendicular bisectors; incircle: angle bisectors
- Incircle radius: always drop a perpendicular to a side first
- Hexagon: side equals circumradius
- Check measurements against a quick calculation
The trap. Bisecting angles to find the circumcentre. Angle bisectors give the incentre; perpendicular bisectors give the circumcentre.
- Use a sharp pencil and accurate compasses
- Label the centre as , or as appropriate
- Circumcircle: perpendicular bisectors; incircle: angle bisectors
- Incircle radius: always drop a perpendicular to a side first
- Hexagon: side equals circumradius
- Check measurements against a quick calculation
The trap. Bisecting angles to find the circumcentre. Angle bisectors give the incentre; perpendicular bisectors give the circumcentre.
Did you know
Why do honeybees build hexagonal cells?
Only three regular shapes can cover a flat surface with no gaps: equilateral triangles, squares and regular hexagons.
For the same area of each cell, the hexagon has the shortest total wall length of the three. Less wall means less wax for the same storage space.
So a honeycomb of hexagons gives bees the most room for honey with the least building material — and each cell has a circle that fits neatly inside it.
For the same area of each cell, the hexagon has the shortest total wall length of the three. Less wall means less wax for the same storage space.
So a honeycomb of hexagons gives bees the most room for honey with the least building material — and each cell has a circle that fits neatly inside it.
Exam relevance
Do circle constructions matter for JEE Main?
Ruler-and-compass constructions themselves are a board-exam skill and are not tested in JEE Main.
What carries forward. The ideas behind them do. Class 11 Straight Lines finds the circumcentre as the intersection of perpendicular bisectors and the incentre of a triangle from its vertices, and Class 11 Conic Sections finds the circle through three points and the length of a tangent from an external point — both JEE Main chapters.
Board versus competitive emphasis. The board paper marks accurate drawing and measurement; JEE Main expects the same results as equations and formulas.
The trap that costs marks. Confusing the circumcentre with the incentre, which gives the wrong centre for a circle.
What carries forward. The ideas behind them do. Class 11 Straight Lines finds the circumcentre as the intersection of perpendicular bisectors and the incentre of a triangle from its vertices, and Class 11 Conic Sections finds the circle through three points and the length of a tangent from an external point — both JEE Main chapters.
Board versus competitive emphasis. The board paper marks accurate drawing and measurement; JEE Main expects the same results as equations and formulas.
The trap that costs marks. Confusing the circumcentre with the incentre, which gives the wrong centre for a circle.
Key takeaways
What must you be able to do from this part?
- **Tangents from **: bisect , draw a circle on as diameter, join to the crossing points; radius , gives cm
- Circumcircle: perpendicular bisectors meet at the circumcentre; right triangle , , has cm
- Incircle: angle bisectors meet at the incentre; radius is the perpendicular to a side; cm for the same triangle
- **Equilateral side **: cm, cm
- **Regular hexagon side **: cm, cm
Construct a triangle with sides , and cm, then predict both its circumradius and inradius before drawing the circles.
- Circumcircle: perpendicular bisectors meet at the circumcentre; right triangle , , has cm
- Incircle: angle bisectors meet at the incentre; radius is the perpendicular to a side; cm for the same triangle
- **Equilateral side **: cm, cm
- **Regular hexagon side **: cm, cm
Construct a triangle with sides , and cm, then predict both its circumradius and inradius before drawing the circles.