Count the Triangles Inside and the Angle Sum Follows
Learn to classify polygons as convex, concave or regular, count diagonals with the n(n-3)/2 formula, and find interior angle sums and missing angles.
Why do the angles of every quadrilateral add to 360 degrees?
Because a quadrilateral is two triangles stuck together.
Draw one diagonal across any four-sided figure and it splits into two triangles. Each triangle's angles add to , and between them those triangles use up every corner of the quadrilateral. So the total is .
That argument does not stop at four sides. From one vertex of a pentagon you can draw two diagonals, cutting it into three triangles, so its angles total . A hexagon splits into four triangles and totals .
The pattern is now visible: a polygon with sides splits into triangles. That single observation generates the whole formula, and it is worth carrying instead of the formula itself. This page covers the first part of the ICSE Class 8 Mathematics chapter on understanding shapes.
Draw one diagonal across any four-sided figure and it splits into two triangles. Each triangle's angles add to , and between them those triangles use up every corner of the quadrilateral. So the total is .
That argument does not stop at four sides. From one vertex of a pentagon you can draw two diagonals, cutting it into three triangles, so its angles total . A hexagon splits into four triangles and totals .
The pattern is now visible: a polygon with sides splits into triangles. That single observation generates the whole formula, and it is worth carrying instead of the formula itself. This page covers the first part of the ICSE Class 8 Mathematics chapter on understanding shapes.
What makes a polygon convex, concave or regular?
Convex means no interior angle is reflex; regular means all sides equal AND all angles equal.
Polygons are named by their number of sides: a triangle has , a quadrilateral , a pentagon , a hexagon , a heptagon , an octagon , a nonagon and a decagon . Beyond that, mathematicians simply say an *-gon*.
Convex against concave. In a convex polygon every interior angle is less than , and every diagonal lies entirely inside the figure. In a concave polygon at least one interior angle is reflex — greater than — and the shape appears dented inwards.
An arrowhead-shaped quadrilateral is the standard concave example. One of its angles is a reflex angle, and one of its diagonals runs outside the figure altogether. **Its angles still total , though, because the triangle-splitting argument does not care about dents. That is a genuinely useful fact: the angle sum formula holds for concave polygons too.
Regular needs both conditions. This is where marks are lost:
- A rhombus has four equal sides but unequal angles, so it is not regular.
- A rectangle has four equal angles but unequal sides, so it is not regular either.
- A square has both, so the square is the regular quadrilateral.
Worked example — checking a claim.** Is an equilateral triangle regular? All three sides are equal by definition, and the angles must then be each. Both conditions hold, so yes.
Why triangles are the exception. For a triangle, equal sides force equal angles and the other way round, so equilateral and regular mean the same thing. From four sides onwards the two conditions come apart — the rhombus and rectangle prove it — and each must be checked separately.
Polygons are named by their number of sides: a triangle has , a quadrilateral , a pentagon , a hexagon , a heptagon , an octagon , a nonagon and a decagon . Beyond that, mathematicians simply say an *-gon*.
Convex against concave. In a convex polygon every interior angle is less than , and every diagonal lies entirely inside the figure. In a concave polygon at least one interior angle is reflex — greater than — and the shape appears dented inwards.
An arrowhead-shaped quadrilateral is the standard concave example. One of its angles is a reflex angle, and one of its diagonals runs outside the figure altogether. **Its angles still total , though, because the triangle-splitting argument does not care about dents. That is a genuinely useful fact: the angle sum formula holds for concave polygons too.
Regular needs both conditions. This is where marks are lost:
- A rhombus has four equal sides but unequal angles, so it is not regular.
- A rectangle has four equal angles but unequal sides, so it is not regular either.
- A square has both, so the square is the regular quadrilateral.
Worked example — checking a claim.** Is an equilateral triangle regular? All three sides are equal by definition, and the angles must then be each. Both conditions hold, so yes.
Why triangles are the exception. For a triangle, equal sides force equal angles and the other way round, so equilateral and regular mean the same thing. From four sides onwards the two conditions come apart — the rhombus and rectangle prove it — and each must be checked separately.
Formula
How many diagonals does a polygon with n sides have?
A polygon with sides has
Where the formula comes from. A diagonal joins two non-adjacent vertices. Stand at any one of the vertices: you cannot join it to itself, nor to its two neighbours (those joins are sides, not diagonals), which leaves diagonals from that vertex. Doing this at every vertex gives — but each diagonal has now been counted twice, once from each end, so divide by .
Worked example 1 — a quadrilateral.
Two diagonals, which matches any rectangle you draw.
Worked example 2 — a pentagon.
Worked example 3 — a hexagon.
Worked example 4 — an octagon and a decagon.
Worked example 5 — working backwards. A polygon has 35 diagonals. How many sides?
You need two numbers three apart whose product is . Since , the answer is — a decagon.
Worked example 6 — a triangle.
A triangle has no diagonals, and the formula says so correctly: every pair of its vertices is adjacent. A formula that handles its own smallest case is a formula you can trust.
The two counts not to confuse. From a single vertex there are diagonals, and those cut the polygon into triangles. The total number of diagonals in the whole polygon is . A hexagon has diagonals from one vertex, triangles, and diagonals altogether — three different numbers, and questions deliberately ask for each.
Where the formula comes from. A diagonal joins two non-adjacent vertices. Stand at any one of the vertices: you cannot join it to itself, nor to its two neighbours (those joins are sides, not diagonals), which leaves diagonals from that vertex. Doing this at every vertex gives — but each diagonal has now been counted twice, once from each end, so divide by .
Worked example 1 — a quadrilateral.
Two diagonals, which matches any rectangle you draw.
Worked example 2 — a pentagon.
Worked example 3 — a hexagon.
Worked example 4 — an octagon and a decagon.
Worked example 5 — working backwards. A polygon has 35 diagonals. How many sides?
You need two numbers three apart whose product is . Since , the answer is — a decagon.
Worked example 6 — a triangle.
A triangle has no diagonals, and the formula says so correctly: every pair of its vertices is adjacent. A formula that handles its own smallest case is a formula you can trust.
The two counts not to confuse. From a single vertex there are diagonals, and those cut the polygon into triangles. The total number of diagonals in the whole polygon is . A hexagon has diagonals from one vertex, triangles, and diagonals altogether — three different numbers, and questions deliberately ask for each.
How do you find a missing interior angle of a polygon?
**Work out the total with , then subtract the angles you were given.**
The interior angle sums worth knowing on sight:
- Triangle:
- Quadrilateral:
- Pentagon:
- Hexagon:
- Octagon:
- Decagon:
Worked example 1 — a pentagon. *Four angles of a pentagon are , , and . Find the fifth.*
The total must be , and the given angles add to
Worked example 2 — a hexagon. *Five angles of a hexagon are , , , and .*
The total is and the given angles sum to , so the sixth angle is .
Worked example 3 — angles in a ratio. *The angles of a quadrilateral are in the ratio .*
Write them as , , , :
The angles are , , and . Check: they sum to and reduce to . Correct.
**Worked example 4 — working backwards to .** *The interior angles of a polygon sum to . How many sides?*
A nonagon. As a bonus, it has diagonals.
The check that catches an impossible question. Any interior angle sum must be a **multiple of **, since it equals . So a question claiming a polygon whose angles total describes nothing at all: is not a whole number. Dividing first and confirming you get an integer is the quickest way to spot a misprint — or your own arithmetic slip.
One more boundary case. Since must be at least , the smallest possible sum is . There is no polygon with an angle sum of .
The interior angle sums worth knowing on sight:
- Triangle:
- Quadrilateral:
- Pentagon:
- Hexagon:
- Octagon:
- Decagon:
Worked example 1 — a pentagon. *Four angles of a pentagon are , , and . Find the fifth.*
The total must be , and the given angles add to
Worked example 2 — a hexagon. *Five angles of a hexagon are , , , and .*
The total is and the given angles sum to , so the sixth angle is .
Worked example 3 — angles in a ratio. *The angles of a quadrilateral are in the ratio .*
Write them as , , , :
The angles are , , and . Check: they sum to and reduce to . Correct.
**Worked example 4 — working backwards to .** *The interior angles of a polygon sum to . How many sides?*
A nonagon. As a bonus, it has diagonals.
The check that catches an impossible question. Any interior angle sum must be a **multiple of **, since it equals . So a question claiming a polygon whose angles total describes nothing at all: is not a whole number. Dividing first and confirming you get an integer is the quickest way to spot a misprint — or your own arithmetic slip.
One more boundary case. Since must be at least , the smallest possible sum is . There is no polygon with an angle sum of .
Exam tip
Exam tip: divide by 180 first and check it comes out whole
**Interior angle sum is ** — it always comes from splitting the polygon into triangles.
When you are given the sum and asked for , **divide by first**. A non-integer answer means a misprint or an arithmetic slip, because every valid sum is a multiple of .
**Diagonals total .** Do not confuse this with the diagonals from a single vertex, or the triangles they create. A hexagon has , and of these respectively.
For the reverse question, turn into and look for two numbers three apart.
Regular needs both equal sides and equal angles. A rhombus fails on angles, a rectangle fails on sides; only the square qualifies among quadrilaterals.
Concave polygons still obey the angle sum. A dented quadrilateral totals just like any other.
For ratio questions, write the angles as , , , and use the total.
Memorise , , and for four, five, six and eight sides — they appear constantly.
And add your angles back up at the end; the total is a free, complete check.
When you are given the sum and asked for , **divide by first**. A non-integer answer means a misprint or an arithmetic slip, because every valid sum is a multiple of .
**Diagonals total .** Do not confuse this with the diagonals from a single vertex, or the triangles they create. A hexagon has , and of these respectively.
For the reverse question, turn into and look for two numbers three apart.
Regular needs both equal sides and equal angles. A rhombus fails on angles, a rectangle fails on sides; only the square qualifies among quadrilaterals.
Concave polygons still obey the angle sum. A dented quadrilateral totals just like any other.
For ratio questions, write the angles as , , , and use the total.
Memorise , , and for four, five, six and eight sides — they appear constantly.
And add your angles back up at the end; the total is a free, complete check.
Did you know
What happens to a polygon as you keep adding sides
Work out the interior angle of a regular polygon for a few values of and something curious emerges. Divide the total by :
- Triangle:
- Square:
- Hexagon:
- Decagon:
- Regular -gon:
The angles climb towards but never reach it. They cannot: an interior angle of exactly would mean the two sides meeting at that corner lie in a straight line, and the corner would have vanished.
So as grows, each corner gets flatter and less noticeable, while the figure as a whole grows rounder. A regular polygon with a hundred sides, drawn on paper, is almost impossible to distinguish from a circle by eye — and that resemblance is not a coincidence but the reason why a polygon with very many sides is a good practical stand-in for a circle.
There is a neat consistency check hiding in the list. The angles never reach , the number of triangles never reaches , and the two facts are the same fact: the polygon always loses exactly two triangles' worth of angle, no matter how large it grows.
- Triangle:
- Square:
- Hexagon:
- Decagon:
- Regular -gon:
The angles climb towards but never reach it. They cannot: an interior angle of exactly would mean the two sides meeting at that corner lie in a straight line, and the corner would have vanished.
So as grows, each corner gets flatter and less noticeable, while the figure as a whole grows rounder. A regular polygon with a hundred sides, drawn on paper, is almost impossible to distinguish from a circle by eye — and that resemblance is not a coincidence but the reason why a polygon with very many sides is a good practical stand-in for a circle.
There is a neat consistency check hiding in the list. The angles never reach , the number of triangles never reaches , and the two facts are the same fact: the polygon always loses exactly two triangles' worth of angle, no matter how large it grows.
Key takeaways
Polygons, diagonals and angle sums: quick revision
- A polygon with sides splits into triangles from one vertex, so its **interior angles sum to **.
- Sums to know: quadrilateral , pentagon , hexagon , octagon , decagon .
- Convex means every interior angle is under and every diagonal lies inside. Concave means one angle is reflex — but the angle sum formula still holds.
- Regular needs equal sides and equal angles. A rhombus fails on angles, a rectangle on sides; the square is the regular quadrilateral. For triangles alone, equilateral and regular coincide.
- Diagonals: , because each of vertices sends diagonals and every diagonal is counted twice.
- Quadrilateral , pentagon , hexagon , octagon , decagon . A triangle has , and the formula gives that correctly.
- Backwards: diagonals means , so .
- Three counts not to confuse for a hexagon: diagonals from one vertex, triangles, diagonals in total.
- Missing angle: a pentagon with has a fifth angle of .
- A hexagon with has a sixth angle of .
- **Ratio ** in a quadrilateral gives , so .
- **Finding from the sum**: gives , which also has diagonals.
- Every valid angle sum is a **multiple of **, so a claimed total of is impossible.
- Each interior angle of a regular polygon rises towards without reaching it — , , , , — which is why many-sided polygons look round.
Draw a heptagon and count its diagonals by hand, then check against . Trusting a formula you have verified once yourself is worth more than memorising three.
- Sums to know: quadrilateral , pentagon , hexagon , octagon , decagon .
- Convex means every interior angle is under and every diagonal lies inside. Concave means one angle is reflex — but the angle sum formula still holds.
- Regular needs equal sides and equal angles. A rhombus fails on angles, a rectangle on sides; the square is the regular quadrilateral. For triangles alone, equilateral and regular coincide.
- Diagonals: , because each of vertices sends diagonals and every diagonal is counted twice.
- Quadrilateral , pentagon , hexagon , octagon , decagon . A triangle has , and the formula gives that correctly.
- Backwards: diagonals means , so .
- Three counts not to confuse for a hexagon: diagonals from one vertex, triangles, diagonals in total.
- Missing angle: a pentagon with has a fifth angle of .
- A hexagon with has a sixth angle of .
- **Ratio ** in a quadrilateral gives , so .
- **Finding from the sum**: gives , which also has diagonals.
- Every valid angle sum is a **multiple of **, so a claimed total of is impossible.
- Each interior angle of a regular polygon rises towards without reaching it — , , , , — which is why many-sided polygons look round.
Draw a heptagon and count its diagonals by hand, then check against . Trusting a formula you have verified once yourself is worth more than memorising three.