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Home›Geometry and drawings›Triangles

Which three lengths make a triangle

Sticks of 3 cm, 4 cm and 8 cm cannot be joined into a triangle. The two short ones reach only 3 + 4 = 7 cm, one centimetre short of the long one.

Three sets of straws to judge first

Before any rule, a puzzle. You have three bundles of straws, cut to these lengths in centimetres: 3, 4 and 5; 3, 4 and 7; 3, 4 and 8. In which bundles can the three straws be joined end to end to close a triangle? Make a guess for each one, and if there are straws and scissors nearby, cut them and try. Most people get the first and the last right and hesitate over the middle one.

Here is what happens. With 3, 4 and 5, the two shorter straws lean towards each other and meet above the long one: a triangle. With 3, 4 and 8, lay the 8 down and stand the 3 and the 4 on its ends. However you tilt them, their tips cannot meet, because even lying flat along the 8 they cover only 7. With 3, 4 and 7 the tips meet only when both short straws lie flat on top of the long one. The “triangle” has no height and no area. Geometry does not count it as a triangle, because its three corners are on one line.

Top: 3 and 4 are longer together than 5, so they meet above it. Middle: 3 + 4 = 7 lies flat. Bottom: 3 + 4 is shorter than 8, so the tips can never meet.

Any two sides together are longer than the third

The straws follow one rule, called the triangle inequality. In every triangle, the sum of any two sides is greater than the third side: a + b > c, b + c > a and c + a > b. There are three inequalities, but you never need to check all of them. Only the longest side can cause trouble, since any side plus something positive beats a shorter side. So the quick test is: add the two shorter lengths and compare with the longest. Bigger means a triangle. Equal or smaller means no triangle.

Think of a walk home that stops at a convenience store on the way. Unless the shop sits exactly on the straight path, the detour is longer than walking straight home. Home, school and shop are the three corners of a triangle, and the straight route is one side, always shorter than the other two put together. That is why people cut across a park on the diagonal and wear a path into the grass.

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A low, flat triangle. Drag C down towards the side AB. Angle C climbs towards 180° and the triangle gets thinner: the two sides AC and CB together come closer and closer to the length of AB, but the lab will not let them equal it, because the triangle would vanish.

How long can the third side be?

The rule also fences in a missing side. Suppose two sides are 5 cm and 9 cm. The third side c must satisfy 5 + 9 > c, so c < 14. It must also satisfy c + 5 > 9, so c > 4. Together, 4 < c < 14. In general, the third side lies strictly between the difference and the sum of the other two. If the sides must be whole centimetres, the choices are 5, 6, 7 and so on up to 13, which is nine possible lengths.

Isosceles triangles give a classic trap. Two sides of an isosceles triangle are 4 and 9. Is the third side 4 or 9? Try 4: the sides 4, 4, 9 fail, because 4 + 4 = 8 is less than 9. So the third side must be 9, the sides are 4, 9, 9, and the perimeter is 22.

The longest side faces the largest angle

A second fact travels with the first. In any triangle, the longer of two sides faces the larger angle, so the longest side is always across from the largest angle and the shortest side across from the smallest. Pull a corner away from the opposite side and both the angle there and the far side grow together. This is how you know, without measuring, that the hypotenuse of a right triangle is its longest side: it faces the 90° angle, and the other two angles are smaller.

BackgroundEuclid proves the triangle inequality as Proposition 20 of Book I: in any triangle the sum of any two sides is greater than the remaining one. His proof leans on Proposition 19, which says that the greater angle has the greater side opposite it. In the D. E. Joyce edition, the notes connect the result with the idea that a straight line is the shortest path, and they retell a problem solved by Heron of Alexandria. Two points lie on the same side of a straight line, and you must go from one to the line and then to the other by the shortest route. The answer is the path where the angle in equals the angle out, like light bouncing off a mirror, and the proof uses Proposition 20.
  1. Can sticks of 2, 3 and 6 cm make a triangle?
    • No
    • Yes
  2. Can sticks of 5, 7 and 10 cm make a triangle?
    • Yes
    • No
  3. Can sticks of 4, 4 and 8 cm make a triangle?
    • No, they lie flat
    • Yes, an isosceles one
    • Yes, an equilateral one
    • Yes, a right one
  4. Which set of lengths makes a triangle?
    • 6, 8, 10
    • 1, 2, 3
    • 2, 5, 9
    • 3, 3, 7
  5. Two sides are 5 cm and 9 cm. The third side is a whole number of centimetres. How many lengths are possible?
  6. Two sides are 7 and 12. What is the smallest whole-number third side?
  7. Two sides are 3 and 10. What is the largest whole-number third side?
  8. An isosceles triangle has two sides of 4 and 9 (one of them is repeated). What is its perimeter?
  9. In triangle ABC, A = 80°, B = 60°, C = 40°. Which side is longest?
    • BC
    • AC
    • AB
    • All equal
  10. Why do people cut across a lawn on the diagonal instead of walking round two sides?
    • One side of a triangle is shorter than the other two together
    • The diagonal has a right angle
    • Grass makes paths longer
    • The diagonal is the longest side
Common mistakes
  • “3, 4 and 7 make a very flat triangle.” When the two short sides add up exactly to the long one, the three corners lie on one line. There is no height and no area, so it is not a triangle. The inequality has to be strict: greater than, not equal to.
  • “I checked 3 + 8 > 4, so 3, 4, 8 is fine.” Adding the longest side to anything always beats a shorter side, so that check proves nothing. Add the two shortest and compare with the longest: 3 + 4 = 7 < 8.
  • “Two sides are 5 and 9, so the third side is between 5 and 9.” The third side can be shorter than both or longer than both. The real limits are the difference and the sum: 4 < c < 14. A side of 13 or 4.5 is allowed.
Teacher's noteThe 2022 Korean curriculum document does not list the triangle inequality as a separate achievement standard; it surfaces in middle school when learners construct a triangle from three given lengths and must explain their process. That makes it a good discovery task. Give out straws of 3, 4, 5, 7 and 8 cm and let learners find which triples close. The 3, 4, 7 case is the one to discuss: some will press the straws into a thin triangle and call it a success. Ask what its area is.
Key points
  • Any two sides of a triangle add to more than the third.
  • Quick test: the two shortest sides must add to more than the longest.
  • Third side c of a triangle with sides a and b: |a − b| < c < a + b.
  • The longest side faces the largest angle.
Sources for this course (7)
  • Curriculum: 교육부 고시 제2022-33호 [별책 8] 수학과 교육과정 (ncic.re.kr/inv/org/download.do?year=2022&seq=10003559&orgType=ogi4, PDF read 2026-10-19: 초등학교 3~4학년군 [4수03-08] 이등변삼각형·정삼각형, [4수03-09] 직각삼각형·예각삼각형·둔각삼각형, [4수03-25] 삼각형과 사각형의 내각의 크기의 합을 여러 가지 방법으로 추론; 초등학교 5~6학년군 [6수03-01] 합동·대응점·대응변·대응각, [6수03-14] 삼각형의 넓이(높이가 삼각형의 외부에 있는 것도 다룸); 중학교 1~3학년군 [9수03-03] 삼각형의 작도, [9수03-04] 삼각형의 합동 조건, [9수03-05] 삼각형의 내각과 외각, [9수03-09] 이등변삼각형의 성질을 합동 조건으로 정당화하고 증명이라는 용어 도입, [9수03-12]·[9수03-13] 닮음·닮음비·삼각형의 닮음 조건, [9수03-15] 피타고라스 정리를 추측하고 정당화, 역은 직관적으로 이해하고 직각삼각형 판별에 사용; 용어와 기호 대변, 대각, 내각, 외각, 증명, 닮음비, △ABC, ≡, ∽)
  • Textbook: OpenStax, Contemporary Mathematics, 10.3 Triangles (triangle from three noncollinear points; acute, right, obtuse; side a opposite angle A; hypotenuse and legs; angle sum 180° by a parallel line and alternate interior angles; x, x + 17, 3x − 62 gives 45°, 62°, 73°; SSS, SAS, ASA, AAS; similar triangles, scaling factor; tower shadow 180 ft with a 9 ft truck and 15 ft shadow gives 108 ft; Thales and the pyramid: openstax.org/books/contemporary-mathematics/pages/10-3-triangles)
  • OpenStax Prealgebra 2e, 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem (a² + b² = c²; Pythagoras around 500 BCE; 13 ft ladder 5 ft from the wall; 10 inch bracket about 7.1 inches; similar triangles: openstax.org/books/prealgebra-2e/pages/9-3-use-properties-of-angles-triangles-and-the-pythagorean-theorem)
  • 9.4 Use Properties of Rectangles, Triangles, and Trapezoids (A = ½bh from half a rectangle; isosceles and equilateral; equilateral perimeter 93 in: .../pages/9-4-use-properties-of-rectangles-triangles-and-trapezoids)
  • OpenStax Precalculus 2e, 8.2 Non-right Triangles: Law of Cosines (Heron’s formula with semi-perimeter s; sides 10, 15, 7 give s = 16 and area about 29.4; Chicago lot 62.4, 43.5, 34.1 m gives about 711.4 m²: openstax.org/books/precalculus-2e/pages/8-2-non-right-triangles-law-of-cosines)
  • History for story blocks only: Euclid’s Elements, D. E. Joyce edition, Clark University (Book I Prop. 20 triangle inequality and Heron’s shortest-path problem; Prop. 26 and its guide on congruence theorems, SSA ambiguity and side-side-right angle; Prop. 32 exterior angle and angle sum, Proclus corollaries; Prop. 47 and its guide on Old Babylonians ca. 1900 to 1600 BCE, Plimpton 322, the Zhou bi hypotenuse diagram, Proclus; Prop. 48 converse; Book VI Prop. 4 equiangular triangles: mathcs.clarku.edu/~djoyce/elements/bookI/propI47.html)
  • MacTutor History of Mathematics, University of St Andrews, Heron of Alexandria (formula proved in Metrica Book I; eclipse of 13 March 62; side of 720 about 26 5/6: mathshistory.st-andrews.ac.uk/Biographies/Heron/) MacTutor, "Euclid of Alexandria" (born about 325 BC, died about 265 BC; dating for the Elements, about 2,300 years ago: mathshistory.st-andrews.ac.uk/Biographies/Euclid/, checked 2026-10-19).

Checked: 2026-10-19