Quadrilaterals and Polygons

A polygon is the simplest way we have of enclosing a flat region with straight edges — and almost every built structure you have ever seen, from a brick wall to a soccer ball's stitching to a smartphone screen, is a polygon or an assembly of them. Once you can name a shape, count its sides, and predict its angles without measuring a single one, geometry stops feeling like memorization and starts feeling like a set of rules you can trust.

This page teaches you to classify these shapes, to derive (not just recall) how their angles add up, and to reason about the special families — parallelograms, trapezoids, and regular polygons — from their defining properties. The goal is that when you meet a shape you have never seen, you can still say something true about it.

Learning Objectives

• Define a polygon and distinguish convex from concave and regular from irregular.
• Classify quadrilaterals into the parallelogram family, trapezoids, and kites.
• Derive and apply the interior-angle-sum formula $(n-2)\times 180^\circ$.
• Explain why the exterior angles of any convex polygon always sum to $360^\circ$.
• State and use the defining properties of parallelograms, rectangles, rhombi, squares, and trapezoids.
• Compute interior angles, exterior angles, and side counts for regular polygons.

Quick Answer

A polygon is a closed plane figure made of straight line segments; a quadrilateral is a polygon with four sides. The interior angles of any $n$-sided polygon sum to $(n-2)\times 180^\circ$, because the shape can be split into $n-2$ triangles. The exterior angles of any convex polygon always sum to exactly $360^\circ$, no matter how many sides it has. Quadrilaterals form a hierarchy: squares are special rectangles and special rhombi, which are special parallelograms, which are special trapezoids (under the inclusive definition). A regular polygon has all sides and all angles equal, so each interior angle is $\frac{(n-2)\times 180^\circ}{n}$.

Where It Came From

Polygons were not born from abstract curiosity — they came from the practical crises of building and dividing land. Ancient Egyptian surveyors, the harpedonaptae ("rope-stretchers"), re-drew field boundaries every year after the Nile's flood erased them, and to do that they needed reliable ways to lay out right angles and rectangular plots. Getting the corners wrong meant taxing a farmer for land he did not own.

The Babylonians and Egyptians already worked confidently with rectangles and trapezoids for area (a trapezoidal field is one of the oldest worked problems in mathematics). But the systematic theory of polygons — proofs about their angles rather than rules of thumb — is the achievement of the Greeks. Euclid's Elements (around 300 BCE) treats triangles and quadrilaterals with proofs, and the very last book, Book XIII, is devoted to constructing the regular solids, which are built from regular polygons.

The other great driver was tiling. Craftsmen laying mosaic floors and, centuries later, Islamic artists decorating walls needed to know which shapes could cover a surface with no gaps and no overlaps. That question — why do triangles, squares, and hexagons tile the plane but regular pentagons do not? — is answered entirely by the angle-sum facts on this page. When several tiles meet at a point, their angles must total $360^\circ$; a regular pentagon's $108^\circ$ angle simply will not divide evenly into that. So the practical need to make a floor forced people to understand interior angles exactly.

Naming and Classifying Polygons

A polygon is a closed figure formed by three or more straight line segments (its sides or edges) joined end to end, meeting at vertices. We name polygons by their side count:

Sides	Name	Sides	Name
3	Triangle	8	Octagon
4	Quadrilateral	9	Nonagon
5	Pentagon	10	Decagon
6	Hexagon	12	Dodecagon
7	Heptagon	$n$	$n$-gon

Two big distinctions matter:

• Convex vs. concave. A polygon is convex if every interior angle is less than $180^\circ$, so it has no "dents" — any line segment between two interior points stays inside. If even one interior angle exceeds $180^\circ$ (a reflex angle), the polygon is concave (a classic example is an arrowhead or a star).
• Regular vs. irregular. A regular polygon has all sides equal (equilateral) and all angles equal (equiangular). A square is regular; a non-square rectangle is equiangular but not equilateral, so it is irregular.

Worked example — is it convex? A quadrilateral has interior angles $100^\circ$, $80^\circ$, $200^\circ$, and $80^\circ$? First check the sum: $100 + 80 + 200 + 80 = 460^\circ$, which is not $360^\circ$, so this is impossible — such a quadrilateral cannot exist. Corrected: angles $100^\circ$, $80^\circ$, $100^\circ$, $80^\circ$ sum to $360^\circ$ and all are below $180^\circ$, so it is a valid convex quadrilateral.

The Interior Angle Sum

Here is the single most useful fact about polygons, and it comes with a proof you can reconstruct any time you forget the formula.

Pick any vertex of a convex $n$-gon and draw diagonals to every other vertex. This slices the polygon into triangles. A quadrilateral ($n=4$) splits into 2 triangles; a pentagon ($n=5$) into 3; in general an $n$-gon splits into $n-2$ triangles. Since every triangle's angles sum to $180^\circ$, and these triangles' angles together make up exactly the polygon's interior angles:

$S = (n-2)\times 180^\circ$

Worked example — a heptagon. For $n = 7$:

$S = (7-2)\times 180^\circ = 5 \times 180^\circ = 900^\circ.$

If that heptagon were regular, each interior angle would be $\frac{900^\circ}{7} \approx 128.57^\circ$.

Worked example — working backwards. A polygon's interior angles sum to $1440^\circ$. How many sides? Solve $(n-2)\times 180 = 1440$, so $n - 2 = 8$, giving $n = 10$. It is a decagon.

Exterior Angles: The Constant That Never Changes

At each vertex, the exterior angle is the supplement of the interior angle — the angle you turn through if you walk along the boundary. Imagine walking all the way around the polygon once and returning to your start facing the original direction: you have made one full turn, $360^\circ$. That is the whole argument:

$\text{Sum of exterior angles} = 360^\circ \quad (\text{any convex polygon}).$

Remarkably this does not depend on $n$. A triangle, a square, and a 100-gon all have exterior angles summing to $360^\circ$.

For a regular polygon, all exterior angles are equal, so each one is $\frac{360^\circ}{n}$. This gives the fastest route to many answers.

Worked example — sides from an exterior angle. A regular polygon has each exterior angle equal to $24^\circ$. Then $n = \frac{360^\circ}{24^\circ} = 15$. It is a regular 15-gon, and each interior angle is $180^\circ - 24^\circ = 156^\circ$. Check with the interior formula: $\frac{(15-2)\times 180}{15} = \frac{2340}{15} = 156^\circ$. Consistent.

The Quadrilateral Family

Quadrilaterals deserve special care because their names overlap in a hierarchy that confuses many students. Using the inclusive definitions (the modern standard):

• Parallelogram: both pairs of opposite sides parallel. Consequences: opposite sides equal, opposite angles equal, consecutive angles supplementary, and diagonals bisect each other.
• Rectangle: a parallelogram with four right angles. Its diagonals are equal in length.
• Rhombus: a parallelogram with four equal sides. Its diagonals are perpendicular and bisect the angles.
• Square: both a rectangle and a rhombus — four equal sides and four right angles.
• Trapezoid (US) / trapezium (UK): at least one pair of parallel sides (the bases). An isosceles trapezoid has equal legs and equal base angles.
• Kite: two distinct pairs of adjacent equal sides; its diagonals are perpendicular.

Worked example — parallelogram angles. In parallelogram $ABCD$, angle $A = 70^\circ$. Consecutive angles are supplementary, so angle $B = 180^\circ - 70^\circ = 110^\circ$. Opposite angles are equal, so angle $C = 70^\circ$ and angle $D = 110^\circ$. Check: $70 + 110 + 70 + 110 = 360^\circ$.

Worked example — trapezoid midsegment. A trapezoid has parallel bases of length $8$ cm and $14$ cm. The midsegment (median) connecting the midpoints of the legs has length equal to the average of the bases:

$m = \frac{8 + 14}{2} = 11 \text{ cm}.$

Regular Polygons and Tiling

Regular polygons combine equal sides and equal angles, which makes them the natural building blocks of symmetric design. The interior angle drives whether a shape can tile the plane by itself.

Worked example — why hexagons tile. A regular hexagon has interior angle $\frac{(6-2)\times 180}{6} = 120^\circ$. At a shared vertex, $\frac{360^\circ}{120^\circ} = 3$ hexagons meet exactly — no gap. This is why honeycombs and many floor tilings are hexagonal. For a pentagon, the interior angle is $108^\circ$, and $\frac{360}{108}$ is not a whole number, so regular pentagons cannot tile the plane alone. Only the triangle ($60^\circ$), square ($90^\circ$), and hexagon ($120^\circ$) tile by themselves among regular polygons.

Real-World Applications

• Architecture and construction: Ensuring a room is truly rectangular relies on the equal-diagonals property of rectangles — builders measure both diagonals of a foundation and adjust until they match.
• Tiling, flooring, and manufacturing: Choosing hexagonal or square tiles that fit gap-free is a direct application of interior-angle sums; the same math governs how metal panels or floor patterns interlock.
• Traffic engineering: Stop signs are regular octagons (interior angle $135^\circ$), a shape recognizable even when partly obscured or reversed.
• Computer graphics and games: 3D models are meshes of polygons (usually triangles and quadrilaterals); classifying and subdividing them efficiently is core to rendering.
• Cartography and GIS: Land parcels, voting districts, and map regions are stored as polygons; area and boundary calculations depend on the geometry here.

Common Mistakes

1. Thinking every four-sided shape with a right angle is a rectangle. Why wrong: a rectangle needs four right angles and opposite sides parallel. A single right angle proves nothing. Correction: verify all four angles are $90^\circ$ (or the shape is a parallelogram with one right angle, which forces the rest).

2. Multiplying $180^\circ$ by $n$ instead of $(n-2)$. Why wrong: the polygon splits into $n-2$ triangles, not $n$. Using $n\times 180$ overcounts badly (a quadrilateral would wrongly get $720^\circ$). Correction: always subtract 2 first: $(n-2)\times 180^\circ$.

3. Believing the exterior angle sum grows with more sides. Why wrong: students assume "more sides, bigger total." But one full trip around the boundary is always one full turn. Correction: the exterior sum is fixed at $360^\circ$; it is each individual exterior angle that shrinks as $n$ grows.

Comparison and Connections

The parallelogram family is best seen as a chain of increasing restrictions. Each shape below inherits every property of the ones to its left.

Shape	Opposite sides parallel	All sides equal	All angles $90^\circ$	Diagonals equal	Diagonals perpendicular
Trapezoid	one pair	no	no	no	no
Parallelogram	both	no	no	no	no
Rectangle	both	no	yes	yes	no
Rhombus	both	yes	no	no	yes
Square	both	yes	yes	yes	yes

Note the two facts that distinguish a rectangle from a rhombus: equal diagonals signal a rectangle, while perpendicular diagonals signal a rhombus. A square, sitting at the bottom, has both.

Practice Questions

Recall

What is the sum of the interior angles of an octagon? Answer: $(8-2)\times 180^\circ = 1080^\circ$.

Understanding

Explain why the exterior angles of a triangle and a decagon both sum to $360^\circ$. Guidance: Walking once around any convex polygon returns you to your starting direction, a single $360^\circ$ turn, and the exterior angles are exactly the turns you make. Side count does not change the total turn.

Application

A regular polygon has an interior angle of $150^\circ$. How many sides does it have? Answer: Each exterior angle is $180^\circ - 150^\circ = 30^\circ$, so $n = \frac{360^\circ}{30^\circ} = 12$. It is a regular dodecagon.

Analysis

A quadrilateral has diagonals that bisect each other, are equal in length, and are perpendicular. Classify it as specifically as possible and justify. Answer: Diagonals bisecting each other $\Rightarrow$ parallelogram. Equal diagonals $\Rightarrow$ rectangle. Perpendicular diagonals $\Rightarrow$ rhombus. Being both a rectangle and a rhombus, it must be a square.

FAQ

Is a square a rectangle? Yes. Under inclusive definitions a square meets every requirement of a rectangle (parallelogram with four right angles) and simply adds equal sides. Every square is a rectangle, but not every rectangle is a square.

Is a trapezoid a parallelogram? Not usually. A trapezoid needs only one pair of parallel sides. A parallelogram needs both pairs parallel, so a parallelogram is a special trapezoid under the inclusive definition — but a generic trapezoid is not a parallelogram.

Does the interior-angle formula work for concave polygons? Yes, the formula $(n-2)\times 180^\circ$ still gives the correct total, provided you count reflex angles (those over $180^\circ$) properly. The exterior-angle-sum $360^\circ$ result, however, is stated for convex polygons.

Why can't regular pentagons tile a floor? Their interior angle is $108^\circ$, and $360^\circ$ is not a whole-number multiple of $108^\circ$, so tiles meeting at a point either leave a gap or overlap. Only triangles, squares, and hexagons tile the plane on their own.

What's the difference between a diagonal and a side? A side is an edge of the polygon connecting two adjacent vertices. A diagonal connects two non-adjacent vertices. An $n$-gon has $\frac{n(n-3)}{2}$ diagonals.

Quick Revision

• Polygon: closed figure of $\ge 3$ straight sides. Convex = all interior angles $< 180^\circ$.
• Interior angle sum: $(n-2)\times 180^\circ$.
• Exterior angle sum (convex): always $360^\circ$.
• Regular polygon: interior angle $= \frac{(n-2)\times 180^\circ}{n}$; exterior angle $= \frac{360^\circ}{n}$.
• Parallelogram: opposite sides/angles equal, diagonals bisect each other.
• Rectangle: equal diagonals. Rhombus: perpendicular diagonals. Square: both.
• Trapezoid midsegment $= \frac{b_1 + b_2}{2}$.
• Number of diagonals: $\frac{n(n-3)}{2}$.

Related Topics

Prerequisites

• Geometry overview

Related Topics

• Triangles and their angle sums
• Perimeter and area of plane figures

Next Topics

• Circles and their properties
• Coordinate geometry of polygons