KS3 · Maths

Constructing triangles with ruler and compass

Could a triangle have edges of 5 cm, 6 cm and 12 cm? Don't work it out. Drag the corner below and let two circles answer.

Maths · Constructions

Where can the third corner go?

AC 5.0 cm. BC 8.0 cm. Relationship: Every point on the circle is the same distance from its centre, so AC reads 5.0 cm wherever C is.

AC5.0 cmBC8.0 cmDrag C round the circle

Every point on the circle is the same distance from its centre, so AC reads 5.0 cm wherever C is.

AB is a 12 cm base. The big circle is every point that is 5 cm from A, and C sits on it. Drag C round and watch BC. Can it ever read 6.0? 7.0? 8.0? The curves around B mark 6 cm, 7 cm and 8 cm from B.

Watch out: When BC reads 7.0, C sits on the base itself. That is a flat line, not a triangle.

Worked example · the 6, 8, 10 triangle

Problem

Construct a triangle with edges 6 cm, 8 cm and 10 cm using a ruler and compasses, then check it.

Maths · What makes it accurate?

What really makes the corner accurate?

You need to draw a triangle with edges 6 cm, 8 cm and 10 cm, and it has to be accurate.

Which idea is closest to what you think?

Measure 6 cm and 8 cm with the ruler from the ends of the base, then adjust until the angle looks right.A neat freehand sketch with the lengths labelled on it is just as good.Any pencil will do, as long as the compasses are set to the right length.Sharp pencil, tightened compasses set from lengths drawn to the side, and arcs from each end of the base. Let the circles do the measuring.

How sure are you?

I'm sureI think soI'm guessing

Measure 6 cm and 8 cm with the ruler from the ends of the base, then adjust until the angle looks right.

This is a common misconception

That feels careful, because rulers feel accurate. But look at what you're doing: you're guessing where the corner goes and fixing it by eye. In a construction the ruler measures the base, and the circles do the measuring for the other two edges: every point on an arc is the same distance from its centre, so where two arcs cross, both distances are right at once.

A neat freehand sketch with the lengths labelled on it is just as good.

This is a common misconception

Sketches are brilliant, but for a different job. A construction is an accurate drawing made using geometrical properties and geometry equipment. A freehand sketch isn't done to scale. Sketch first to plan the triangle, then construct it.

Any pencil will do, as long as the compasses are set to the right length.

This is a common misconception

The pencil matters. A blunt pencil makes thick lines and loses accuracy. The compasses matter too: tighten them, with the pencil tip level with the needle point, and hold them with one hand. If they open or close mid-drawing, the start and end of an arc may not join.

Sharp pencil, tightened compasses set from lengths drawn to the side, and arcs from each end of the base. Let the circles do the measuring.

This matches the evidence

Yes. That is what turns a drawing into a construction. The base is measured with the ruler, and every other length comes from compasses set against a line you can check and reset.

The ruler measures the base. The compasses measure everything else.

Maths · Spot the slip

Where does this construction go wrong?

Construct triangle PQR with PQ = 7 cm, PR = 9 cm and QR = 10 cm. Click the first line where this student goes wrong.

A student's method, with no sketch drawn first — which line goes wrong?

Draw QR = 10 cm with the ruler. This is the base.
Set the compasses to 7 cm. Put the needle on R and draw an arc above the base.
Set the compasses to 9 cm. Put the needle on Q and draw an arc crossing the first arc.
Label the crossing point P. Join P to Q and to R with the ruler as a straight edge.
Measure the finished triangle: PQ = 9 cm and PR = 7 cm.

Maths · More than one answer

One description, how many triangles?

Someone says: 'I drew an isosceles triangle with edges of 6 cm and 9 cm.' Walk through the choices you would have to make.

Which length is the equal pair? → Which crossing point is the third corner?

2 × 2 = 4 possible drawings, and the tree ends 4 times.

On Isosceles, with edges 6 cm and 9 cm. 2 branches to choose from.

Watch out: Different question: told only that the two equal edges are 8 cm? Then any point on the arc gives an equal edge, so there are many possible triangles, not four.

Maths · Three kinds of triangle

How does the set-up change?

For each triangle, tick what is true of constructing it. Then check the grid.

Scalene

Both circles have the same radiusAll three edges are the same lengthEvery angle checks as 60° with a protractor

Isosceles

Both circles have the same radiusAll three edges are the same lengthEvery angle checks as 60° with a protractor

Equilateral

Both circles have the same radiusAll three edges are the same lengthEvery angle checks as 60° with a protractor

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

What you need to know

Every point on a circle is the same distance from its centre. That is why compasses can do the measuring.
Draw one edge as the base. Draw an arc from each end of it. The third corner is where the arcs cross.
Three lengths make a triangle only when the two shorter edges add up to more than the longest.
The two circles cross at two points, one on each side of the base. Either can be the third corner.

The big picture

To build a triangle from three edge lengths, draw one edge as the base, then find the third corner where two arcs, one from each end of the base, cross. It only works when the circles can meet.

Key points

1A construction is an accurate drawing made with geometrical properties and geometry equipment. A freehand sketch is not.

2All the points on a circle are the same distance from its centre, so the third corner is where two circles of the right radii meet.

3The ruler draws the base. The compasses, set from lengths drawn to the side of the page, fix the other two edges.

4If the two shorter edges add up to the longest, the circles only touch on the base. If they add up to less, they miss. Either way there is no triangle.

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Worked example

Problem

Triangle ABC has AB = 9 cm, AC = 4 cm and BC = 6 cm. Can it be constructed, and where does each arc go?

⚠ Watch out

Thinking any three lengths make a triangle if you draw carefully enough. If the two shorter edges only add up to the longest, the circles just touch on the base, and no care with the ruler can make a triangle appear.

🧠

Memory hook

Ruler for the base, circles for the corner. If the circles can't meet, neither can the triangle.

✓

Check yourself

Without looking back: you are given edges of 4 cm, 7 cm and 11 cm. What do the two circles do, and is there a triangle?

Flashcards

(14)

What is a construction?

An accurate drawing made using geometrical properties and geometry equipment. A freehand sketch is not one, because it isn't done to scale.

What can a pair of compasses draw?

Circles and arcs, including a circle of a particular radius. An arc is part of a circle.

What is a radius?

Any line segment that joins the centre of a circle to any point on its circumference.

What do all the points on a circle have in common?

They are all the same distance from the centre.

Where is the third corner of a triangle built from three edge lengths?

Where the two circles (or arcs) cross: one centred on each end of the base.

What does the ruler measure when you construct a triangle?

The base. The compasses do the rest of the measuring.

Why draw the other two edge lengths to the side of the page?

So you can set the compasses accurately from them, and reset them if they slip.

Do you need to draw full circles?

No. Arcs long enough to cross will do. If they don't meet, extend them.

The two circles cross at two points. What does that give you?

Two triangles that are reflections of each other. Either crossing point can be the third corner.

A triangle is given in words, like triangle PQR. Why sketch it first?

The sketch shows which edge joins which corners, so you centre each circle on the right one.

When do three lengths make a triangle?

Only when the two shorter edges add up to more than the longest. If they add up to exactly the longest, the circles only touch on the base. If less, they miss.

How do you set up an isosceles construction efficiently?

Draw the edge that differs as the base, so the compasses are set once for both equal edges.

What does an equilateral construction use, and what can you check?

Circles with equal radii. Every angle should read 60° with a protractor.

A triangle is described only by its equal edges, say 8 cm. How many answers?

Many, because any point on the arc gives an equal edge.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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What really makes the corner accurate?

Where does this construction go wrong?

One description, how many triangles?

How does the set-up change?

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