Constructing a triangle accurately means drawing it with a ruler and compasses, not just a protractor and ruler. Given the three side lengths (SSS) or two sides and the included angle (SAS), you can produce an exact triangle where every measurement is correct to within the accuracy of your instruments.
What equipment do you need and why does it matter?
You need:
- A sharp pencil (a blunt pencil makes lines too thick and introduces error)
- A 30 cm ruler with millimetre markings
- A pair of compasses (to mark exact distances as arcs)
- A protractor (for SAS and ASA methods only)
Compasses allow you to mark a distance precisely without measuring it each time — you set the gap and draw an arc, and every point on that arc is exactly that distance from the centre. This is why ruler-and-compasses constructions are more accurate than simply measuring and marking points.
Method 1: SSS — three sides given
SSS means you know all three side lengths. This is the most common construction in KS3.
Worked example: Construct triangle ABC with AB = 8 cm, BC = 6 cm, and AC = 5 cm.
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Draw the base. Draw a horizontal line and mark point A. Measure 8 cm along it and mark point B. Label both points. This gives you side AB.
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Set compasses for the second side. Open your compasses to exactly 6 cm (use the ruler to set this gap accurately). Place the point on B. Draw an arc above the line AB. Every point on this arc is exactly 6 cm from B, so C lies somewhere on it.
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Set compasses for the third side. Change the compasses to 5 cm. Place the point on A. Draw a second arc that crosses the first arc. Every point on this second arc is exactly 5 cm from A.
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Mark the vertex. The two arcs cross at point C — this is the only point that is 6 cm from B AND 5 cm from A at the same time. Mark it and label it C.
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Complete the triangle. Draw straight lines from C to A and from C to B. Erase any unwanted arc lines.
Check: Measure all three sides. Each should match the given lengths to within ±1 mm.
Method 2: SAS — two sides and the included angle
SAS means you know two sides and the angle between them (the included angle).
Worked example: Construct triangle PQR with PQ = 7 cm, angle PQR = 55°, and QR = 5 cm.
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Draw PQ. Draw a horizontal line and mark P and Q, 7 cm apart.
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Draw the angle at Q. Place your protractor at Q with its centre on Q and the baseline along QP. Mark a point at 55°. Draw a faint line from Q through this mark — this is the direction of QR.
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Mark R on the angle line. Open your compasses to 5 cm. Place the point on Q and mark R exactly 5 cm along the line you drew.
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Complete the triangle. Draw a straight line from R back to P.
Check: The angle at Q should be 55° and QR should be 5 cm.
Method 3: ASA — two angles and the included side
ASA means you know two angles and the side between them.
Worked example: Construct triangle DEF with DE = 9 cm, angle D = 40°, and angle E = 65°.
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Draw DE. Mark D and E exactly 9 cm apart.
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Draw the angle at D. Use a protractor at D to mark 40° above DE. Draw a faint line from D.
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Draw the angle at E. Use a protractor at E to mark 65° above ED. Note that you measure from the line ED, so the angle opens on the same side as step 2. Draw a faint line from E.
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Mark F. The two lines from D and E cross at point F. Mark and label it.
Check: Measure angle D (should be 40°), angle E (should be 65°), and angle F (should be 180° − 40° − 65° = 75°).
What makes a construction valid?
A construction is only valid if it could, in principle, be done with any radius of compasses and any precision of ruler — you must not use other measurements to "fix" a position that the given information doesn't determine.
In exams:
- Leave all construction arcs visible (don't rub them out) — they are evidence of your method and earn marks.
- Any length measurement must be within ±2 mm and any angle within ±2° to be credited.
Which method to use?
| Given information | Method | Unique triangle? |
|---|---|---|
| All three sides (SSS) | Compasses only | Yes (if valid) |
| Two sides + included angle (SAS) | Protractor + compasses | Yes |
| Two angles + included side (ASA) | Two protractor readings | Yes |
| Two sides + non-included angle (SSA) | Not a standard method | Sometimes two triangles possible — the ambiguous case |
SSA is not a reliable construction: you may get two different valid triangles, one, or none, depending on the numbers.
Frequently asked questions
Why can't I just use a protractor and ruler for SSS?
With SSS, you don't know any angles to start — you only have three lengths. A protractor can't help you here. The compasses method works because the two arcs locate the third vertex exactly using distances alone.
My two arcs don't cross — what went wrong?
If the two arcs don't cross, the three side lengths don't form a valid triangle. This happens when one side is longer than the sum of the other two. For example, sides 2 cm, 3 cm, and 7 cm can't form a triangle because 2 + 3 < 7. This is the triangle inequality: every side must be less than the sum of the other two.
Do I need to show construction arcs in the exam?
Yes — always leave your arcs visible. The mark scheme awards marks for the correct arcs as well as the completed triangle. Rubbing out the arcs removes evidence of your method and risks losing marks even if the finished triangle looks correct.
What is the difference between a construction and a sketch?
A sketch is a freehand drawing, roughly to proportion, used to set up a problem. A construction is an accurate drawing using instruments, where measurements are correct. In an exam, "construct" always means use a ruler and compasses (and protractor where needed) and leave all arcs visible.
Practise accurate triangle constructions step by step with Professor Pi at aitutors.me.