How to Construct (Draw) an Equilateral Triangle

Written by

Malcolm McKinsey

January 12, 2023

Fact-checked by

Paul Mazzola

Construct an equilateral triangle

Equilateral triangles are easily constructed with a drawing compass, straightedge, and pencil because the 60° interior angles can be found using only the radius of a circle around the triangle (a circumscribed circle).

Suppose you are given the length of one side of the required equilateral triangle. You must construct two identical sides that, somehow, create three interior 60° angles.

Imagine a line segment CE:

1. Place the needle arm of your drawing compass on Point C, then adjust the pencil arm, so it touches Point E.

2. Swing an arc upward, making a light construction line.

3. Without adjusting the compass, relocate the needle to Point E.

4. Swing an arc upward from Point C, making another light construction line.

5. Where the two arcs intersect, you have the point of an equilateral triangle's third vertex. Call it Point A. Use your straightedge to connect Points A and C, Points A and E. You have your equilateral triangle ACE!

How to draw an equilateral triangle with a compass proof

Points C and E are centers of congruent circles with radii equal to CE. Lengths AC and CE are both radii of Circle C. Lengths AE and CE are, likewise, both radii of Circle E. By the transitive property:

Three congruent sides mean the triangle must be an equilateral triangle.

How to construct an equilateral triangle inscribed in a circle

Suppose you are given a circle, not a side. Suppose you are asked to construct an equilateral triangle inside a circle.

You know only the center point, Point S, but from that, you can copy the radius (distance from the center, Point S, to the circle itself).

1. Adjust your drawing compass the establish the radius.

2. Make a mark on the circle, anywhere you wish. Set the needle arm on that point.

3. Without changing the compass, swing two small arcs above and below your circle point, so the arcs cross the circle.

4. Relocate the drawing compass needle to one of those arcs.

5. Swing the compass again to make a small arc on the circle.

6. Repeat this two more times to create six points on the circle (your original point plus six small arcs intersecting it).

7. Use your straightedge to connect every other point on the circle. That is, draw line segments from, say, your first point on the circle to the third point, and another line segment from the third point to the fifth point. One more line, from the fifth point back to your starting point, and there you have it. You have inscribed an equilateral triangle inside the circle!

[insert drawing showing this, or animate the construction of the three sides]

Construct an equilateral triangle inscribed in a circle proof

Think of that equilateral triangle as itself made up of three smaller isosceles triangles, sharing Point S as a common vertex. That means three triangles each have a central angle (at Point S) of 120°, established by dividing the circle's full 360° by 3 (the number of central angles).

Each of these smaller isosceles triangles has two vertices on the circle and each shares congruent sides with the other triangles. That makes two of the three sides of the triangles congruent.

If two sides of an isosceles triangle are congruent, then the third side is, too, by the Side-Angle-Side theorem.

How to draw an equilateral triangle

This method allows you to control the size of your equilateral triangle, because you construct it from scratch.

1. Set your drawing compass to draw a circle; we'll call it Circle E.

2. Use your straightedge to establish a diameter (a line segment through the center Point E with endpoints on the circle itself). Label the endpoints Points B and R.

3. Without disturbing the compass, set the needle arm down on Point R and swing another complete circle. Where the new circle intersects the old circle at two points above and below diameter BR, label those Points A and S.

4. Connect Point B to A, Point A to S, and Point A to B. It BEARS noticing you have constructed an equilateral triangle!

Draw an equilateral triangle proof

You can prove the triangle is equilateral using your knowledge that any angle inscribed in a semicircle is a right angle, giving you right angles $\angle BAR$ and $\angle BSR$, which creates two congruent right triangles $\triangle BAR$ and $\triangle BSR$ sharing a common hypotenuse, BR.

This makes $AB\cong BS$ (corresponding sides of congruent triangles), which means $\triangle BAS$ is isosceles. That makes $\angle BAS$ and $\angle BSA$ congruent (base angles of an isosceles triangle are congruent). We know $\angle ABS$ is 60°, leaving the two remaining angles to sum to 120°, so they are each 60°. All three interior angles of $\triangle BAS$ are equal, making it an equilateral triangle.