A polygon is a closed plane figure built from straight line segments that meet only at their endpoints. The simplest is the triangle ($3$ sides); the family extends to four-sided, five-sided, all the way up to "$n$-gon" for any positive integer $n \geq 3$.
The Formal Definition
A polygon is a closed two-dimensional figure formed by a finite number of straight line segments (called sides) that intersect only at their endpoints (called vertices). For a shape to count as a polygon:
It must be closed — the sides form a continuous loop with no gaps.
It must lie in a single plane — every side is in the same flat surface.
The sides must be straight — no curves; a circle is not a polygon.
The sides must meet only at the vertices — they cannot cross or overlap.
A polygon with $n$ sides has $n$ vertices, $n$ interior angles, and $\dfrac{n(n-3)}{2}$ diagonals.
Quick reference.
Definition: closed plane figure with straight sides only.
Sides $=$ Vertices $=$ Interior angles $= n$.
Sum of interior angles: $(n - 2) \times 180°$.
Sum of exterior angles: $360°$ for any polygon (regular or irregular).
Each interior angle (regular): $\dfrac{(n - 2) \times 180°}{n}$.
Each exterior angle (regular): $\dfrac{360°}{n}$.
Number of diagonals: $\dfrac{n(n-3)}{2}$.
Grade introduced: CCSS-M 5.G.B.4 (classify polygons); NCERT Class 8 — Understanding Quadrilaterals.
Classification by Number of Sides
Polygon | Sides | Vertices | Interior angle sum |
|---|---|---|---|
Triangle | $3$ | $3$ | $180°$ |
Quadrilateral | $4$ | $4$ | $360°$ |
Pentagon | $5$ | $5$ | $540°$ |
Hexagon | $6$ | $6$ | $720°$ |
Heptagon | $7$ | $7$ | $900°$ |
Octagon | $8$ | $8$ | $1080°$ |
Nonagon | $9$ | $9$ | $1260°$ |
Decagon | $10$ | $10$ | $1440°$ |
$n$-gon | $n$ | $n$ | $(n - 2) \times 180°$ |
Each new side adds $180°$ to the interior angle sum — a clean linear pattern.
Classification by Regularity and Shape
Polygons fall into four main categories by shape.
Regular polygon. All sides equal and all interior angles equal. A square, an equilateral triangle, a regular hexagon. Has multiple lines of symmetry equal to the number of sides.
Irregular polygon. Sides and/or angles not all equal. The most common kind in real-world shapes — a kite, a scalene triangle, a typical floor plan.
Convex polygon. Every interior angle is less than $180°$. Every diagonal lies inside the shape. A regular pentagon is convex.
Concave polygon. At least one interior angle is greater than $180°$ (a "reflex" angle). At least one diagonal lies outside the shape. An arrow-shape or an L-shaped floor plan is concave.
A polygon can be both irregular and convex (a scalene triangle); regular shapes are always convex.
Three Properties of Polygon
These are the most-used results in school geometry.
Interior angle sum
For any polygon with $n$ sides:
$$\text{Sum of interior angles} = (n - 2) \times 180°.$$
Exterior angle sum
For any convex polygon — regardless of $n$:
$$\text{Sum of exterior angles} = 360°.$$
Number of diagonals
From any vertex, you can draw a diagonal to all other vertices except itself and its two adjacent neighbours — giving $(n - 3)$ diagonals per vertex. Dividing by $2$ to remove double-counting:
$$\text{Number of diagonals} = \frac{n(n-3)}{2}.$$
Three Worked Examples of Polygon — Quick, Standard, Stretch
Quick. What is the sum of the interior angles of a heptagon ($n = 7$)?
Apply $(n - 2) \times 180°$ with $n = 7$:
$$(7 - 2) \times 180° = 5 \times 180° = 900°.$$
Final answer: $900°$.
Standard (Wrong Path First — The Detour Students Take). Each interior angle of a regular polygon is $144°$. How many sides does it have?
The wrong path. A student tries: $\dfrac{180°}{144°} \approx 1.25$, then gets stuck and guesses "the polygon has 5 sides."
The flaw: dividing $180°$ by the angle doesn't pick up any $n$. The right approach uses the regular-polygon angle formula.
The rescue. Each interior angle of a regular $n$-gon is $\dfrac{(n - 2) \times 180°}{n}$. Set this equal to $144°$:
$$\frac{(n - 2) \times 180°}{n} = 144°.$$
Cross-multiply: $(n - 2) \times 180° = 144°n$. Expand: $180n - 360 = 144n$. Rearrange: $36n = 360$. So $n = 10$.
Final answer: $10$ sides — a regular decagon.
Sanity check: a regular decagon's interior angle is $\dfrac{8 \times 180°}{10} = \dfrac{1440°}{10} = 144°$ ✓.
The lesson — for "find $n$ from an interior angle" problems, write the formula and solve algebraically — don't try to divide.
Stretch. Find the number of sides of a polygon whose interior angles sum to $1440°$.
Use $(n - 2) \times 180° = 1440°$. Solve: $n - 2 = 8$, so $n = 10$.
Final answer: $10$ sides — a decagon.
Cross-check: the diagonal count is $\dfrac{n(n-3)}{2} = \dfrac{10 \times 7}{2} = 35$ for a decagon — a result that shows up in NCERT Class 8 mensuration problems on polygon properties.
Where Polygons Appear — From Honeycomb to Stop Signs
A few places polygons quietly do the work:
Honeycombs. Bees build their wax cells as regular hexagons because the regular hexagon tessellates the plane and minimises wax for a given storage volume — proven mathematically as the honeycomb conjecture by Thomas Hales in 1999.
Stop signs. A regular octagon. Eight sides distinguish it instantly from any other sign.
Computer graphics. Every 3D model in a video game is a mesh of triangles — the simplest possible polygons.
City planning. The pentagon-shaped Pentagon building (Arlington, $1943$) is one of the largest single office buildings ever constructed.
Architecture. Hexagonal floor tiles, octagonal towers, and pentagonal domes all use polygons because their angle sums make them stable.
The systematic study of polygons begins in Euclid's Elements Books I and IV (c. 300 BCE), where the constructions of regular pentagons, hexagons, and fifteen-sided polygons are proved. Two thousand years later, Carl Friedrich Gauss (1777–1855, Germany) extended this work — at age $19$ — by proving exactly which regular polygons can be constructed with only a compass and straightedge (the $17$-gon was his celebrated discovery).
Tripping Points to Avoid in Polygon
Mistake 1: Calling a circle a polygon
Where it slips in: A student lists "circle" alongside triangle and square as a polygon.
Don't do this: Treat any closed shape as a polygon.
The correct way: A polygon has straight sides only. A circle has a single curved edge — no sides, no vertices.
Mistake 2: Counting an open shape as a polygon
Where it slips in: Drawing four line segments that don't close up and calling it a quadrilateral.
Don't do this: Skip the closure step.
The correct way: A polygon must be closed — start and end at the same vertex with no gap.
Mistake 3: Using the regular-polygon angle formula on an irregular polygon
Where it slips in: A pentagon has angles $80°, 100°, 110°, 125°$, and a missing angle. Student divides $540°$ by $5$ to get $108°$.
Don't do this: Apply the regular formula when the polygon is clearly irregular.
The correct way: Use the angle-sum to find the missing angle by subtraction: $540° - (80° + 100° + 110° + 125°) = 125°$.
A real-world version of the mistake. When the Burj Khalifa (Dubai, $2010$) was designed, its floor plan is built around three petals that overlap — each petal contains a regular hexagonal substructure. Engineers had to apply the exact regular-polygon angle to each petal floor; treating any one as "approximately hexagonal" would have produced load misalignment between floors. The polygon angle rules are not classroom-only.
Conclusion
A polygon is a closed two-dimensional shape made of straight line segments.
A polygon with $n$ sides has $n$ vertices, $n$ interior angles summing to $(n-2) \times 180°$, exterior angles summing to $360°$, and $\dfrac{n(n-3)}{2}$ diagonals.
Polygons classify by side count (triangle, quadrilateral, ..., decagon), regularity (regular vs irregular), and angle shape (convex vs concave).
Use the regular-polygon formulas only when the polygon is genuinely regular; for irregular ones, work with the sum and subtract.
Polygons underpin tessellation, honeycomb design, 3D computer graphics, and most of school geometry from Grade 4 onwards.
Three Problems to Cement Polygons
Find the sum of interior angles of an octagon.
Each interior angle of a regular polygon is $135°$. How many sides does it have?
A pentagon has angles $100°, 110°, 115°, 90°$. Find the fifth angle.
If problem 2 gave $\dfrac{180°}{135°}$ as your approach, return to Mistake on the Standard example.
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