Not Every Three Lengths Can Make a Triangle
Learn to name and classify triangles, test three lengths in seconds before drawing, construct a triangle from its sides, and see why the three lengths fix it completely.
Can any three lengths be joined into a triangle?
No. Try to build a triangle from sides of 2 cm, 3 cm and 9 cm and the two short sides will never reach across the long one — they fall short. There is a quick test that tells you in seconds whether three given lengths will work, before you touch a compass.
This page covers everything in the CBSE Class 7 Mathematics chapter's first half: naming and classifying triangles, the triangle inequality, constructing a triangle from three sides, and why those three lengths fix the triangle completely.
This page covers everything in the CBSE Class 7 Mathematics chapter's first half: naming and classifying triangles, the triangle inequality, constructing a triangle from three sides, and why those three lengths fix the triangle completely.
How are triangles named and classified?
A triangle has three vertices (corner points), three sides and three angles. A triangle with vertices A, B and C is written ; its sides are AB, BC and CA, and its angles are , and .
By sides: equilateral has all three sides equal, isosceles has exactly two equal, and scalene has all three different.
By angles: acute-angled has all three angles under , right-angled has one of exactly , and obtuse-angled has one over .
Every triangle carries one label from each list, so a triangle can be isosceles and right-angled at the same time.
For example, a set-square from your geometry box is a right-angled triangle, and the one with two angles is right-angled and isosceles together.
The pairing has a limit worth knowing: an equilateral triangle is always acute-angled, because its three equal angles must each be — so "equilateral and right-angled" describes nothing that exists.
By sides: equilateral has all three sides equal, isosceles has exactly two equal, and scalene has all three different.
By angles: acute-angled has all three angles under , right-angled has one of exactly , and obtuse-angled has one over .
Every triangle carries one label from each list, so a triangle can be isosceles and right-angled at the same time.
For example, a set-square from your geometry box is a right-angled triangle, and the one with two angles is right-angled and isosceles together.
The pairing has a limit worth knowing: an equilateral triangle is always acute-angled, because its three equal angles must each be — so "equilateral and right-angled" describes nothing that exists.
Formula
What is the triangle inequality?
The rule is:
In practice you only need to check the two shortest against the longest, since that is the only comparison that can fail.
Test 2 cm, 3 cm and 9 cm:
So no triangle is possible.
Test 5 cm, 6 cm and 9 cm:
So a triangle is possible.
The reason is simply that a straight line is the shortest path between two points. Going from one end of the 9 cm side to the other via the third vertex must be a longer journey than going straight — so the two shorter sides together have to exceed the long one.
The boundary case matters: if the two shorter sides add to exactly the longest, as with 4, 5 and 9, the three points fall on one straight line. They are collinear, and the figure encloses no area at all — so still no triangle.
In practice you only need to check the two shortest against the longest, since that is the only comparison that can fail.
Test 2 cm, 3 cm and 9 cm:
So no triangle is possible.
Test 5 cm, 6 cm and 9 cm:
So a triangle is possible.
The reason is simply that a straight line is the shortest path between two points. Going from one end of the 9 cm side to the other via the third vertex must be a longer journey than going straight — so the two shorter sides together have to exceed the long one.
The boundary case matters: if the two shorter sides add to exactly the longest, as with 4, 5 and 9, the three points fall on one straight line. They are collinear, and the figure encloses no area at all — so still no triangle.
How do you construct a triangle from three sides?
This is the SSS construction, and it needs only a ruler and compasses.
To build a triangle with sides 6 cm, 5 cm and 4 cm:
1. Draw the longest side as the base — here BC = 6 cm — using the ruler.
2. Open the compasses to 5 cm, place the point at B, and draw an arc above the base.
3. Open the compasses to 4 cm, place the point at C, and draw a second arc crossing the first.
4. Mark the crossing point A, and join AB and AC.
For an equilateral triangle, use the same length for all three arcs. For an isosceles triangle, use the same length twice.
Drawing the longest side first is a practical habit rather than a rule — it keeps both arcs comfortably on the page.
Always leave the arcs visible. They are part of the answer and carry marks, and rubbing them out makes the construction unverifiable even when the triangle is accurate.
To build a triangle with sides 6 cm, 5 cm and 4 cm:
1. Draw the longest side as the base — here BC = 6 cm — using the ruler.
2. Open the compasses to 5 cm, place the point at B, and draw an arc above the base.
3. Open the compasses to 4 cm, place the point at C, and draw a second arc crossing the first.
4. Mark the crossing point A, and join AB and AC.
For an equilateral triangle, use the same length for all three arcs. For an isosceles triangle, use the same length twice.
Drawing the longest side first is a practical habit rather than a rule — it keeps both arcs comfortably on the page.
Always leave the arcs visible. They are part of the answer and carry marks, and rubbing them out makes the construction unverifiable even when the triangle is accurate.
Why do three lengths give only one triangle?
Because once the base is drawn, the other two vertices have nowhere else to go. The two arcs cross at exactly one point above the base, so the triangle is completely determined.
You can test this: give the same three lengths to several classmates and every triangle produced will be identical in shape and size, however differently they are turned on the page. Rotating or flipping a triangle does not make it a different triangle.
This is why SSS is enough on its own, while three angles are not — three given angles fix the shape but allow any size, so they produce infinitely many triangles.
And if the lengths fail the inequality, the arcs simply never meet. That is the visual signature of impossible measurements, and it is worth recognising: arcs that will not cross mean the numbers were wrong, not your drawing.
In the collinear case, where the two shorter sides add to exactly the longest, the arcs touch at a single point on the base itself — enclosing nothing.
You can test this: give the same three lengths to several classmates and every triangle produced will be identical in shape and size, however differently they are turned on the page. Rotating or flipping a triangle does not make it a different triangle.
This is why SSS is enough on its own, while three angles are not — three given angles fix the shape but allow any size, so they produce infinitely many triangles.
And if the lengths fail the inequality, the arcs simply never meet. That is the visual signature of impossible measurements, and it is worth recognising: arcs that will not cross mean the numbers were wrong, not your drawing.
In the collinear case, where the two shorter sides add to exactly the longest, the arcs touch at a single point on the base itself — enclosing nothing.
Exam tip
Exam tip: testing the lengths before you start drawing
Students spend several minutes on a construction with impossible measurements, then assume their compass work is at fault.
Spend five seconds first: add the two shorter lengths and compare with the longest. If the sum is greater, construct it. If it is smaller or equal, write no triangle is possible and give the reason — that is the full answer, and it earns the marks.
Write the check into your working, as , so the examiner sees the test rather than guessing you knew it.
And when a question asks whether a triangle with given sides exists, never answer from the look of the numbers. 6, 7 and 13 seem plausible until you notice , giving the collinear case and no triangle.
Spend five seconds first: add the two shorter lengths and compare with the longest. If the sum is greater, construct it. If it is smaller or equal, write no triangle is possible and give the reason — that is the full answer, and it earns the marks.
Write the check into your working, as , so the examiner sees the test rather than guessing you knew it.
And when a question asks whether a triangle with given sides exists, never answer from the look of the numbers. 6, 7 and 13 seem plausible until you notice , giving the collinear case and no triangle.
Did you know
Why must two sides together be longer than the third?
Because the straight route between two points is always the shortest one.
Walking from B to C directly covers the third side. Walking from B up to A and then down to C covers the other two sides, and that detour cannot possibly be shorter than going straight.
So the two sides together must exceed the third — and when they merely equal it, the "detour" has flattened onto the straight path itself, which is exactly the collinear case.
Walking from B to C directly covers the third side. Walking from B up to A and then down to C covers the other two sides, and that detour cannot possibly be shorter than going straight.
So the two sides together must exceed the third — and when they merely equal it, the "detour" has flattened onto the straight path itself, which is exactly the collinear case.
Key takeaways
Triangles and the inequality: quick revision
- A triangle has three vertices, sides and angles; it is classified by sides as equilateral, isosceles or scalene, and by angles as acute-, right- or obtuse-angled.
- Every triangle takes one label from each list, but an equilateral triangle is always acute since its angles are each .
- The triangle inequality says any two sides must together exceed the third; checking the two shortest against the longest is enough.
- If they add to exactly the longest, the points are collinear and no triangle forms; if less, the arcs never meet.
- The SSS construction draws the base then crosses two arcs, and three lengths fix exactly one triangle — unlike three angles, which fix only the shape.
You will remember all of this far better after answering five questions on it than after reading it twice.
- Every triangle takes one label from each list, but an equilateral triangle is always acute since its angles are each .
- The triangle inequality says any two sides must together exceed the third; checking the two shortest against the longest is enough.
- If they add to exactly the longest, the points are collinear and no triangle forms; if less, the arcs never meet.
- The SSS construction draws the base then crosses two arcs, and three lengths fix exactly one triangle — unlike three angles, which fix only the shape.
You will remember all of this far better after answering five questions on it than after reading it twice.