Constructing a triangle · Code with Martin

Draw a triangle from three sides by the SSS rule: sketch, plan, construction, check.

From three sides a triangle is born. With a compass.

Today you turn from a counter into a drawer. The task: draw a triangle when you know the lengths of all three sides. That is called construction by the SSS rule (side–side–side).

Why a compass? Because a circle is the set of points at the same distance from the centre. When you look for a point that is 4 cm from one end of a segment and 5 cm from the other, you draw two circles — and where they meet, that point is. That is exactly where the third vertex of the triangle belongs.

Drawing with us always has four parts: a sketch (a small freehand picture with labels), a plan (what I will do and why), construction (precise drawing) and a check (I measure whether the given lengths fit).

It sounds ceremonial, but each part has a reason. The sketch sorts in your head what belongs where. Without it you draw blind.

Ema draws a triangle with sides 7, 5 and 4 cm and starts precisely at once — and after the third rubbed-out line she gives up. Sofie hands her paper: “A sketch first. Crooked is fine, just with numbers.” Ema scribbles a little triangle, labels the sides and suddenly sees: the longest side down, two arcs up. The construction then takes two minutes. “That rubbing cost me more time than the whole sketch,” she admits. Sofie nods: “A sketch is a building plan. A bricklayer does not build without a drawing either, even if they can lay bricks.”

💡

Before drawing, check the triangle inequality (the sum of the two shorter sides > the longest side). You save yourself drawing arcs that never meet.

SSS construction step by step

Given: sides a = 7 cm, b = 5 cm, c = 4 cm. (A reminder of labelling: side c sits opposite vertex C, it joins vertices A and B.)

Sketch: a small triangle freehand, label vertices A, B, C and the lengths by the sides.

Plan: I will draw side… which? The most comfortable is to start with the longest. I say: I draw segment AB of length 7 cm. Vertex C then must sit 5 cm from A (an arc of a circle with centre A and radius 5 cm) and at the same time 4 cm from B (an arc with centre B, radius 4 cm). C is the intersection of both arcs.

Construction:

Draw segment AB, |AB| = 7 cm.
With a compass opened to 5 cm make an arc with centre at A (above the segment).
Open the compass to 4 cm and make an arc with centre at B so it meets the first arc.
Mark the intersection C and join it to A and B.

Check: with a ruler remeasure all three sides. An error of a millimetre is the tax for the thickness of the lead; a centimetre is a mistake.

The arcs meet above the segment and below it — both intersections give a correct triangle, just mirrored. The question has two symmetric solutions; one in the exercise book and a note is enough.

Construction of an equilateral triangle with side 6 cm

Equilateral means a = b = c = 6 cm — I open the compass only once.

Method: I draw AB of length 6 cm. With centre at A I draw an arc of radius 6 cm, with the same opening an arc with centre at B. The intersection is C.

Check: all sides 6 cm and a protractor shows 60° at each vertex — a property of an equilateral triangle from the last lesson.

This construction is the oldest in geometry textbooks: two circles, one segment, done.

Why a construction with sides 3, 4 and 9 cm fails

I check the triangle inequality: 3 + 4 = 7. Is 7 > 9? No.

What would happen when drawing: I draw AB = 9 cm, an arc around A with radius 3 cm and an arc around B with radius 4 cm. The arcs stay apart — even if they lay stretched on the segment, they only reach 7 cm of nine. There is no intersection, there is no triangle either.

The lesson: check the inequality before drawing. A compass cannot make a point that maths has forbidden.

An isosceles triangle: base 5 cm, legs 7 cm

Sketch: base AB at the bottom, equal legs AC and BC.

Plan: C sits 7 cm from A and 7 cm from B — two arcs with the same radius.

Construction: AB = 5 cm, an arc with centre A (radius 7 cm), an arc with centre B (radius 7 cm), intersection C, join.

An extra check: C must sit exactly above the midpoint of the base — an isosceles triangle is symmetric. If C is visibly to the side, one compass opening slipped.

⚠️

Do not change the compass opening between planting and the arc. The most common source of crooked triangles: the compass slips a millimetre on the way to the paper. Open it by a ruler and draw the arc at once, in one smooth movement.

Practice: now you do it 🎲

Now try it for real. You will see all the questions at once, like on paper. Start with ten. When it goes well, add more.

Construction is drawn on paper. Here calculate the angles, so you know what should come out.

Practise the questions

On paper: draw by the ritual

Every construction has a sketch, a plan, construction and a check. Write all four parts.

Draw a triangle with sides 8 cm, 6 cm and 5 cm. Remeasure the result.
Draw an equilateral triangle with side 7 cm and with a protractor check the angles 60°.
Try a construction with sides 2 cm, 3 cm and 7 cm. Describe in your own words what happened to the arcs and why.
Draw an isosceles triangle with base 6 cm and legs 5 cm. Check that the vertex sits above the midpoint of the base.

Now you 💪

I know four parts: sketch, plan, construction, check.
I know why the third vertex is found as the intersection of two arcs.
Before construction I check the triangle inequality.

✅

Done when: Three triangles are drawn with the full ritual, the remeasured sides fit to a millimetre and by the impossible triple you have written why.

What to take from this lesson

SSS construction: a segment + two compass arcs, the intersection is the third vertex.
A circle = all points equally far from the centre. That is why a compass.
The ritual: sketch → plan → construction → check.
Check the side inequality before you draw.
An SSS question has two mirror solutions — drawing one is enough.