Integer triangle

Key takeaways

Integer triangle belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Integer triangle to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Integer triangle from memory before moving on to harder problems.

Reference excerpt

An integer triangle or integral triangle is a triangle all of whose side lengths are integers. A rational triangle is one whose side lengths are rational numbers; any rational triangle can be rescaled by the lowest common denominator of the sides to obtain a similar integer triangle, so there is a close relationship between integer triangles and rational triangles. Sometimes other definitions of the term rational triangle are used: Carmichael (1914) and Dickson (1920) use the term to mean a Heronian triangle (a triangle with integral or rational side lengths and area); Conway and Guy (1996) define a rational triangle as one with rational sides and rational angles measured in degrees—the only such triangles are rational-sided equilateral triangles.

General properties for an integer triangle

Integer triangles with given perimeter Any triple of positive integers can serve as the side lengths of an integer triangle as long as it satisfies the triangle inequality: the longest side is shorter than the sum of the other two sides. Each such triple defines an integer triangle that is unique up to congruence. So the number of integer triangles (up to congruence) with perimeter p is the number of partitions of p into three positive parts that satisfy the triangle inequality. This is the integer closest to p 2 / 48 {\displaystyle p^{2}/48} when p is even and to ( p + 3 ) 2 / 48 {\displaystyle (p+3)^{2}/48} when p is odd. It also means that the number of integer triangles with even numbered perimeters p = 2 n {\displaystyle p=2n} is the same as the number of integer triangles with odd numbered perimeters p = 2 n − 3. {\displaystyle p=2n-3.} Thus there is no integer triangle with perimeter 1, 2 or 4, one with perimeter 3, 5, 6 or 8, and two with perimeter 7 or 10. The sequence of the number of integer triangles with perimeter p, starting at p = 1 , {\displaystyle p=1,} is:

0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8 ... (sequence A005044 in the OEIS) This is called Alcuin's sequence.

Integer triangles with given largest side The number of integer triangles (up to congruence) with given largest side c and integer triple ( a , b , c ) {\displaystyle (a,b,c)} is the number of integer triples such that a + b > c {\displaystyle a+b>c} and a ≤ b ≤ c . {\displaystyle a\leq b\leq c.} This is the integer value ⌈ 1 2 ( c + 1 ) ⌉ ⋅ ⌊ 1 2 ( c + 1 ) ⌋ . {\displaystyle \lceil {\tfrac {1}{2}}(c+1)\rceil \cdot \lfloor {\tfrac {1}{2}}(c+1)\rfloor .} Alternatively, for c even it is the double triangular number 1 2 c ( 1 2 c + 1 ) {\displaystyle {\tfrac {1}{2}}c{\bigl (}{\tfrac {1}{2}}c+1{\bigr )}} and for c odd it is the square 1 4 ( c + 1 ) 2 . {\displaystyle {\tfrac {1}{4}}(c+1)^{2}.} It also means that the number of integer triangles with greatest side c exceeds the number of integer triangles with greatest side c − 2 by c. The sequence of the number of non-congruent integer triangles with largest side c, starting at c = 1, is:

1, 2, 4, 6, 9, 12, 16, 20, 25, 30, 36, 42, 49, 56, 64, 72, 81, 90 ... (sequence A002620 in the OEIS) The number of integer triangles (up to congruence) with given largest side c and integer triple (a, b, c) that lie on or within a semicircle of diameter c is the number of integer triples such that a + b > c , a2 + b2 ≤ c2 and a ≤ b ≤ c. This is also the number of integer sided obtuse or right (non-acute) triangles with largest side c. The sequence starting at c = 1, is:

0, 0, 1, 1, 3, 4, 5, 7, 10, 13, 15, 17, 22, 25, 30, 33, 38, 42, 48 ... (sequence A236384 in the OEIS) Consequently, the difference between the two above sequences gives the number of acute integer sided triangles (up to congruence) with given largest side c. The sequence starting at c = 1, is:

1, 2, 3, 5, 6, 8, 11, 13, 15, 17, 21, 25, 27, 31, 34, 39, 43, 48, 52 ... (sequence A247588 in the OEIS)

Area of an integer triangle By Heron's formula, if T is the area of a triangle whose sides have lengths a, b, and c then

4 T = ( a + b + c ) ( a + b − c ) ( a − b + c ) ( − a + b + c ) . {\displaystyle 4T={\sqrt {(a+b+c)(a+b-c)(a-b+c)(-a+b+c)}}.}

Since all the terms under the radical on the right side of the formula are integers it follows that all integer triangles must have 16T2 an integer and T2 will be rational.

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Integer triangle

Start with the simplest possible case. Write down what Integer triangle claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Integer triangle before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Integer triangle ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Integer triangle

In research: Integer triangle appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Integer triangle in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Integer triangle is common in secondary-school and first-year university syllabi. It links to neighbouring topics Arithmetic problems of plane geometry, Discrete geometry, Squares in number theory, so understanding it makes those chapters shorter.
In everyday life: Look for Integer triangle outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Integer triangle in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Integer triangle means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Integer triangle out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Integer triangle in simple terms?

An integer triangle or integral triangle is a triangle all of whose side lengths are integers. A rational triangle is one whose side lengths are rational numbers; any rational triangle can be rescaled by the lowest common denominator of the sides to obtain a similar integer triangle, so there is a…

Why does Integer triangle matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Integer triangle?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Integer triangle.