Quadrilaterals and Polygons
Explore four-sided figures and beyond
Look around the room you are in right now. Chances are, you are surrounded by rectangles: windows, doors, books, screens, picture frames. Step outside, and you will see stop signs (octagons), yield signs (triangles), and countless buildings with walls made of rectangular and trapezoidal shapes. These multi-sided figures are called polygons, and understanding them means understanding the geometry of the built world.
You already have strong intuitions about these shapes. You know that a square is somehow “special” compared to a random four-sided figure, even if you could not articulate why. You sense that some shapes are more “regular” than others. In this lesson, we will give precise names to these intuitions and discover the elegant patterns that govern all polygons, from the humble triangle to the many-sided figures that tile our floors and decorate our buildings.
Core Concepts
What Is a Polygon?
A polygon is a closed figure made up of straight line segments. The word comes from Greek: “poly” (many) + “gon” (angle). So a polygon is literally a “many-angled” figure.
For a shape to be a polygon, it must follow these rules:
- It must be closed (the segments connect to form a complete boundary)
- It must have straight sides (no curves allowed)
- The sides can only meet at their endpoints (no crossing over each other)
The points where the sides meet are called vertices (singular: vertex), and the sides themselves are sometimes called edges.
We name polygons by their number of sides:
| Sides | Name | Example |
|---|---|---|
| 3 | Triangle | Yield sign |
| 4 | Quadrilateral | Most windows |
| 5 | Pentagon | The Pentagon building |
| 6 | Hexagon | Honeycomb cells |
| 7 | Heptagon | Some coins |
| 8 | Octagon | Stop sign |
| 9 | Nonagon | Less common |
| 10 | Decagon | Some architectural features |
| 12 | Dodecagon | Some clock faces |
| n | n-gon | General term for any polygon |
Convex vs. Concave Polygons
Imagine stretching a rubber band around a polygon. If the rubber band touches every vertex and lies flat against every side, the polygon is convex. If the rubber band would skip over some vertices (because they “poke inward”), the polygon is concave.
Here is another way to think about it:
- Convex polygon: All interior angles are less than $180°$. No part of the shape “caves in.”
- Concave polygon: At least one interior angle is greater than $180°$. Some part of the shape “caves in” (hence “con-cave”).
A star shape is concave because it has inward-pointing vertices. A regular hexagon is convex because every vertex points outward.
Regular Polygons
A polygon is called regular if two conditions are met:
- All sides have the same length (equilateral)
- All angles have the same measure (equiangular)
A regular triangle is an equilateral triangle. A regular quadrilateral is a square. Regular polygons have a beautiful symmetry: you can rotate them by a certain angle, and they look exactly the same.
The Interior Angle Sum Formula
Here is one of the most elegant patterns in geometry. The sum of the interior angles of any polygon depends only on how many sides it has.
For any polygon with $n$ sides:
$$\text{Sum of interior angles} = (n - 2) \cdot 180°$$
Why does this work? You can divide any polygon into triangles by drawing diagonals from one vertex. A quadrilateral splits into 2 triangles. A pentagon splits into 3 triangles. A hexagon splits into 4 triangles. In general, an n-gon splits into $(n - 2)$ triangles.
Since each triangle has interior angles that sum to $180°$, the total for the polygon is $(n - 2) \cdot 180°$.
For a regular polygon, where all angles are equal, each interior angle measures:
$$\text{Each interior angle} = \frac{(n - 2) \cdot 180°}{n}$$
The Exterior Angle Sum
If you walk around the outside of any convex polygon, turning at each vertex, by the time you return to your starting point, you will have turned through exactly $360°$, one complete rotation.
This means:
$$\text{Sum of exterior angles} = 360° \text{ (always, for any convex polygon)}$$
For a regular polygon, each exterior angle measures:
$$\text{Each exterior angle} = \frac{360°}{n}$$
Notice something beautiful: the interior and exterior angles at each vertex are supplementary (they add to $180°$), which connects these two formulas.
Quadrilaterals: The Four-Sided Family
A quadrilateral is any polygon with exactly four sides. The word means “four-sided” (quad = four, lateral = side). Quadrilaterals are special because they are the simplest polygons that can have parallel sides, and they form a rich hierarchy of related shapes.
Every quadrilateral has:
- 4 vertices
- 4 sides
- 4 interior angles that sum to $(4 - 2) \cdot 180° = 360°$
- 2 diagonals
Parallelograms
A parallelogram is a quadrilateral where both pairs of opposite sides are parallel. We often write $\square ABCD$ to denote parallelogram ABCD.
Parallelograms have several important properties:
- Opposite sides are congruent (equal in length)
- Opposite angles are congruent (equal in measure)
- Consecutive angles are supplementary (add to $180°$)
- Diagonals bisect each other (they cut each other in half)
These properties are not just observations; they can be proven, and they also work in reverse. If you can show that a quadrilateral has any one of these properties (along with some basic conditions), you can prove it is a parallelogram.
Special Parallelograms
Three special types of parallelograms deserve their own names because they have additional properties.
Rectangle: A parallelogram with four right angles ($90°$ each).
- All the properties of a parallelogram, plus:
- All four angles are $90°$
- Diagonals are congruent (equal in length)
Rhombus: A parallelogram with four congruent sides.
- All the properties of a parallelogram, plus:
- All four sides are equal in length
- Diagonals are perpendicular (they meet at right angles)
- Diagonals bisect the vertex angles
Square: A parallelogram that is both a rectangle AND a rhombus.
- All four angles are $90°$
- All four sides are equal in length
- Diagonals are congruent, perpendicular, and bisect each other and the vertex angles
A square is the most “special” quadrilateral because it has the most constraints and the most symmetry.
Trapezoids
A trapezoid is a quadrilateral with exactly one pair of parallel sides. (In some countries, “trapezoid” means at least one pair of parallel sides, which would include parallelograms. In the United States, the standard definition requires exactly one pair.)
The parallel sides are called bases, and the non-parallel sides are called legs. The height of a trapezoid is the perpendicular distance between the bases.
An isosceles trapezoid has legs that are congruent (equal in length). In an isosceles trapezoid:
- The base angles are congruent
- The diagonals are congruent
Kites
A kite is a quadrilateral with two pairs of consecutive sides that are congruent. Think of it like this: if you look at the shape, you can find two sides next to each other that match in length, and the other two sides next to each other also match (but the two pairs do not need to be equal to each other).
Properties of a kite:
- One diagonal is the perpendicular bisector of the other
- One pair of opposite angles (the ones between the non-congruent sides) are congruent
- The diagonal connecting the vertices where unequal sides meet bisects the other diagonal at right angles
The Quadrilateral Hierarchy
Here is how all these quadrilaterals relate to each other:
Quadrilateral
|
+-------------+-------------+
| |
Parallelogram Trapezoid Kite
| |
+-----+-----+ Isosceles
| | Trapezoid
Rectangle Rhombus
| |
+-----+-----+
|
Square
Every square is a rectangle, every rectangle is a parallelogram, and every parallelogram is a quadrilateral. This hierarchy helps you understand that more specific shapes inherit all the properties of the more general categories above them.
Notation and Terminology
| Term | Meaning | Example |
|---|---|---|
| Polygon | Closed figure with straight sides | Triangle, quadrilateral, hexagon |
| Convex | All interior angles less than $180°$ | Regular polygons, rectangles |
| Concave | Has at least one interior angle greater than $180°$ | Star shapes, arrow shapes |
| Regular polygon | All sides and all angles are equal | Equilateral triangle, square |
| Quadrilateral | Polygon with exactly 4 sides | Any four-sided figure |
| Parallelogram | Opposite sides are parallel | $\square ABCD$ |
| Rectangle | Parallelogram with 4 right angles | Doors, windows |
| Rhombus | Parallelogram with 4 congruent sides | Diamond shape |
| Square | Rectangle and rhombus combined | Chess board squares |
| Trapezoid | Exactly one pair of parallel sides | Some table tops |
| Isosceles trapezoid | Trapezoid with congruent legs | Lampshade sides |
| Kite | Two pairs of consecutive congruent sides | Flying kites |
| Diagonal | Segment connecting non-adjacent vertices | Cuts across a shape |
| Base | A side of a polygon (especially parallel sides in trapezoids) | Bottom of a trapezoid |
| Vertex | Corner point where sides meet | Points $A$, $B$, $C$, $D$ |
Examples
Find the sum of the interior angles of a hexagon.
Solution:
A hexagon has 6 sides, so $n = 6$.
Using the interior angle sum formula: $$\text{Sum} = (n - 2) \cdot 180°$$ $$\text{Sum} = (6 - 2) \cdot 180°$$ $$\text{Sum} = 4 \cdot 180°$$ $$\text{Sum} = 720°$$
The interior angles of any hexagon, whether regular or irregular, always add up to $720°$.
Why it makes sense: A hexagon can be divided into 4 triangles, and $4 \times 180° = 720°$.
Find the measure of one interior angle of a regular octagon.
Solution:
A regular octagon has 8 sides, so $n = 8$. Since it is regular, all angles are equal.
First, find the total sum of interior angles: $$\text{Sum} = (n - 2) \cdot 180° = (8 - 2) \cdot 180° = 6 \cdot 180° = 1080°$$
Since all 8 angles are equal in a regular octagon: $$\text{Each angle} = \frac{1080°}{8} = 135°$$
Each interior angle of a regular octagon measures $135°$.
Quick check: This is between $90°$ (a square’s angle) and $180°$ (a straight line), which makes sense visually. The more sides a regular polygon has, the larger each angle becomes, approaching $180°$ as the shape gets closer to a circle.
In quadrilateral $PQRS$, you know that $\overline{PQ} \cong \overline{RS}$ and $\overline{QR} \cong \overline{SP}$. Prove that $PQRS$ is a parallelogram.
Solution:
We are told that opposite sides are congruent: $PQ = RS$ and $QR = SP$.
One of the key theorems about parallelograms states: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
This is exactly what we have been given. Therefore, $PQRS$ is a parallelogram.
Understanding the reasoning: Why does having congruent opposite sides force the sides to be parallel? Draw diagonal $\overline{PR}$. This creates two triangles: $\triangle PQR$ and $\triangle RSP$. Since $PQ = RS$, $QR = SP$, and $PR = PR$ (shared side), the triangles are congruent by SSS. This means $\angle QPR = \angle SRP$ (alternate interior angles), which proves $\overline{PQ} \parallel \overline{RS}$. Similarly, you can show the other pair is parallel.
In trapezoid $ABCD$, $\overline{AB} \parallel \overline{CD}$. If $\angle A = 65°$ and $\angle B = 72°$, find $\angle C$ and $\angle D$.
Solution:
In a trapezoid with $\overline{AB} \parallel \overline{CD}$:
- $\angle A$ and $\angle D$ are co-interior angles (same-side interior angles), so they are supplementary.
- $\angle B$ and $\angle C$ are co-interior angles, so they are also supplementary.
Finding $\angle D$: $$\angle A + \angle D = 180°$$ $$65° + \angle D = 180°$$ $$\angle D = 180° - 65° = 115°$$
Finding $\angle C$: $$\angle B + \angle C = 180°$$ $$72° + \angle C = 180°$$ $$\angle C = 180° - 72° = 108°$$
Verification: Check that all four angles sum to $360°$: $$65° + 72° + 108° + 115° = 360°$$
It checks out. The missing angles are $\angle C = 108°$ and $\angle D = 115°$.
In rhombus $WXYZ$, the diagonals $\overline{WY}$ and $\overline{XZ}$ intersect at point $M$. If $WY = 24$ cm and $XZ = 10$ cm, find: a) The length of each side of the rhombus b) The measure of $\angle WXY$ (one of the vertex angles)
Solution:
Part a) Finding the side length:
In a rhombus, the diagonals bisect each other at right angles. This means:
- $WM = MY = \frac{24}{2} = 12$ cm
- $XM = MZ = \frac{10}{2} = 5$ cm
- $\angle WMX = 90°$
Triangle $WMX$ is a right triangle with legs $WM = 12$ cm and $XM = 5$ cm.
Using the Pythagorean theorem to find $WX$: $$WX^2 = WM^2 + XM^2$$ $$WX^2 = 12^2 + 5^2$$ $$WX^2 = 144 + 25$$ $$WX^2 = 169$$ $$WX = 13 \text{ cm}$$
Since all sides of a rhombus are congruent, each side is 13 cm.
Part b) Finding $\angle WXY$:
Look at triangle $WMX$ again. We can find $\angle MXW$ using trigonometry: $$\tan(\angle MXW) = \frac{WM}{XM} = \frac{12}{5} = 2.4$$ $$\angle MXW = \tan^{-1}(2.4) \approx 67.4°$$
Since the diagonal $\overline{WY}$ bisects $\angle WXY$ in the rhombus (a property of rhombus diagonals), we have: $$\angle WXY = 2 \times \angle MXW = 2 \times 67.4° \approx 134.8°$$
Actually, let me reconsider. The diagonal of a rhombus bisects the vertex angles at the vertices it passes through. Diagonal $\overline{WY}$ passes through vertices $W$ and $Y$, so it bisects angles at $W$ and $Y$. Diagonal $\overline{XZ}$ passes through $X$ and $Z$, so it bisects angles at $X$ and $Z$.
So $\angle WXY$ is bisected by diagonal $\overline{XZ}$, meaning: $$\angle WXM = \angle YXM$$
We found $\angle WXM \approx 67.4°$, so: $$\angle WXY = 2 \times 67.4° \approx 134.8°$$
The vertex angle $\angle WXY \approx 134.8°$ (or more precisely, $2\tan^{-1}(2.4) \approx 134.76°$).
Summary: Each side is 13 cm, and $\angle WXY \approx 135°$.
The exterior angle of a regular polygon measures $24°$. How many sides does the polygon have? What is the measure of each interior angle?
Solution:
For a regular polygon, each exterior angle equals $\frac{360°}{n}$, where $n$ is the number of sides.
We are told each exterior angle is $24°$: $$\frac{360°}{n} = 24°$$ $$n = \frac{360°}{24°} = 15$$
The polygon has 15 sides (a regular pentadecagon).
The interior and exterior angles at each vertex are supplementary: $$\text{Interior angle} = 180° - 24° = 156°$$
Verification using the interior angle formula: $$\text{Each interior angle} = \frac{(n-2) \cdot 180°}{n} = \frac{(15-2) \cdot 180°}{15} = \frac{13 \cdot 180°}{15} = \frac{2340°}{15} = 156°$$
This confirms our answer. Each interior angle is 156°.
Key Properties and Rules
Polygon Angle Formulas
Sum of interior angles (any polygon with $n$ sides): $$\text{Sum} = (n - 2) \cdot 180°$$
Each interior angle of a regular polygon: $$\text{Each angle} = \frac{(n - 2) \cdot 180°}{n}$$
Sum of exterior angles (any convex polygon): $$\text{Sum} = 360°$$
Each exterior angle of a regular polygon: $$\text{Each angle} = \frac{360°}{n}$$
Parallelogram Properties
A quadrilateral is a parallelogram if and only if any ONE of these conditions holds:
- Both pairs of opposite sides are parallel (definition)
- Both pairs of opposite sides are congruent
- Both pairs of opposite angles are congruent
- One pair of opposite sides is both parallel AND congruent
- The diagonals bisect each other
Special Parallelogram Properties
| Shape | Angles | Sides | Diagonals |
|---|---|---|---|
| Parallelogram | Opposite angles congruent; consecutive angles supplementary | Opposite sides congruent and parallel | Bisect each other |
| Rectangle | All angles are $90°$ | Opposite sides congruent | Bisect each other; diagonals are congruent |
| Rhombus | Opposite angles congruent | All sides congruent | Bisect each other; perpendicular; bisect vertex angles |
| Square | All angles are $90°$ | All sides congruent | Bisect each other; congruent; perpendicular; bisect vertex angles |
Trapezoid Properties
Any trapezoid:
- One pair of parallel sides (bases)
- Co-interior angles (between a leg and the bases) are supplementary
Isosceles trapezoid:
- Legs are congruent
- Base angles are congruent
- Diagonals are congruent
Kite Properties
- Two pairs of consecutive congruent sides
- One diagonal is the perpendicular bisector of the other
- One pair of opposite angles (between unequal sides) are congruent
- The main diagonal (connecting vertices where equal sides meet) bisects the vertex angles it passes through
Real-World Applications
Floor Tiling Patterns
When you tile a floor, you need shapes that fit together without gaps or overlaps. This is called a tessellation. Squares, regular hexagons, and equilateral triangles are the only regular polygons that can tile a plane by themselves. That is why you see so many rectangular and hexagonal tiles in kitchens and bathrooms.
Parallelograms of any shape can also tessellate, which is why brick patterns (essentially parallelogram arrangements) have been used in architecture for thousands of years. Understanding polygon angles helps designers create complex, beautiful patterns.
Building Foundations and Architecture
Most buildings have rectangular foundations because rectangles have right angles, which makes construction simpler and more stable. The four right angles ensure walls are vertical and floors are level.
However, many modern buildings use trapezoidal shapes for aesthetic reasons or to fit oddly-shaped lots. The Flatiron Building in New York City is essentially a triangular prism. Understanding quadrilateral properties helps architects calculate areas, ensure structural integrity, and create visually striking designs.
Road Signs and Safety Symbols
Traffic signs use specific polygon shapes for instant recognition:
- Octagon: Stop signs use this unique shape so drivers can recognize them even if covered in snow or facing away.
- Triangle: Yield signs (pointing down) warn of caution.
- Pentagon: School zone signs use this distinctive shape.
- Rectangle/Square: Most informational signs.
The choice of polygon shape is not arbitrary. It is designed so that each type of message has a unique silhouette that drivers can recognize instantly, even before reading the text.
Soccer Balls and Geodesic Structures
A classic soccer ball is made of 12 regular pentagons and 20 regular hexagons sewn together. This shape (called a truncated icosahedron) works because the angles of pentagons and hexagons can be combined to approximate a sphere.
The same principle underlies geodesic domes, like the Spaceship Earth sphere at EPCOT. Understanding how polygon angles fit together allows engineers to create curved surfaces from flat pieces.
Kites and Flight
The flying toy we call a kite is named after the geometric shape (or perhaps vice versa). The kite shape is aerodynamically useful because its symmetry creates stable flight. The diagonal that connects the unequal-side vertices acts as the spine, and the shape naturally balances around it.
Self-Test Problems
Problem 1: Find the sum of the interior angles of a decagon (10-sided polygon).
Show Answer
Using the formula $(n - 2) \cdot 180°$ with $n = 10$: $$\text{Sum} = (10 - 2) \cdot 180° = 8 \cdot 180° = 1440°$$
The interior angles of a decagon sum to $1440°$.
Problem 2: Each interior angle of a regular polygon measures $140°$. How many sides does it have?
Show Answer
For a regular polygon, each interior angle equals $\frac{(n-2) \cdot 180°}{n}$.
Set this equal to $140°$ and solve for $n$: $$\frac{(n-2) \cdot 180}{n} = 140$$ $$(n-2) \cdot 180 = 140n$$ $$180n - 360 = 140n$$ $$40n = 360$$ $$n = 9$$
The polygon has 9 sides (a nonagon).
Quick check: For $n = 9$: $\frac{(9-2) \cdot 180°}{9} = \frac{7 \cdot 180°}{9} = \frac{1260°}{9} = 140°$. Correct!
Problem 3: In parallelogram $ABCD$, $\angle A = 3x + 10$ and $\angle B = 2x + 20$. Find all four angles.
Show Answer
In a parallelogram, consecutive angles are supplementary: $$\angle A + \angle B = 180°$$ $$(3x + 10) + (2x + 20) = 180$$ $$5x + 30 = 180$$ $$5x = 150$$ $$x = 30$$
Now find the angles:
- $\angle A = 3(30) + 10 = 100°$
- $\angle B = 2(30) + 20 = 80°$
- $\angle C = \angle A = 100°$ (opposite angles are congruent)
- $\angle D = \angle B = 80°$ (opposite angles are congruent)
Verification: $100° + 80° + 100° + 80° = 360°$. Correct!
Problem 4: The diagonals of a rhombus measure 16 cm and 30 cm. Find the perimeter of the rhombus.
Show Answer
In a rhombus, the diagonals bisect each other at right angles.
Half of each diagonal:
- $\frac{16}{2} = 8$ cm
- $\frac{30}{2} = 15$ cm
Each side of the rhombus is the hypotenuse of a right triangle with legs 8 cm and 15 cm: $$\text{side}^2 = 8^2 + 15^2 = 64 + 225 = 289$$ $$\text{side} = 17 \text{ cm}$$
The perimeter is: $$P = 4 \times 17 = 68 \text{ cm}$$
Problem 5: In isosceles trapezoid $EFGH$, the parallel bases are $\overline{EF}$ and $\overline{GH}$. If $\angle E = 68°$, find all four angles.
Show Answer
In an isosceles trapezoid, base angles are congruent.
$\angle E$ and $\angle F$ are the base angles at the top base $\overline{EF}$: $$\angle F = \angle E = 68°$$
The angles along each leg are supplementary (co-interior angles with parallel lines): $$\angle E + \angle H = 180°$$ $$68° + \angle H = 180°$$ $$\angle H = 112°$$
Similarly: $$\angle G = \angle H = 112°$$ (base angles at the bottom base are congruent)
Verification: $68° + 68° + 112° + 112° = 360°$. Correct!
The angles are: $\angle E = 68°$, $\angle F = 68°$, $\angle G = 112°$, $\angle H = 112°$.
Problem 6: A kite has one diagonal of length 12 inches and another of length 8 inches. If the diagonals intersect 3 inches from one end of the shorter diagonal, find the area of the kite.
Show Answer
The area of a kite is: $$A = \frac{1}{2} \times d_1 \times d_2$$
where $d_1$ and $d_2$ are the lengths of the diagonals.
$$A = \frac{1}{2} \times 12 \times 8 = \frac{1}{2} \times 96 = 48 \text{ square inches}$$
Note: The information about where the diagonals intersect does not affect the area calculation, but it would matter if you needed to find side lengths.
Problem 7: Prove that if a quadrilateral has diagonals that bisect each other, it must be a parallelogram.
Show Answer
Let quadrilateral $ABCD$ have diagonals $\overline{AC}$ and $\overline{BD}$ that intersect at point $M$, where $M$ is the midpoint of both diagonals.
This means: $AM = MC$ and $BM = MD$.
Consider triangles $\triangle AMB$ and $\triangle CMD$:
- $AM = MC$ (given)
- $BM = MD$ (given)
- $\angle AMB = \angle CMD$ (vertical angles)
By SAS (Side-Angle-Side), $\triangle AMB \cong \triangle CMD$.
Therefore, $AB = CD$ (corresponding parts of congruent triangles).
Similarly, by considering $\triangle AMD$ and $\triangle CMB$, we can show $AD = BC$.
Since both pairs of opposite sides are congruent, $ABCD$ is a parallelogram.
Summary
-
Polygons are closed figures with straight sides. We name them by their number of sides: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), and so on.
-
Convex polygons have all interior angles less than $180°$; concave polygons have at least one angle greater than $180°$ (they “cave in”).
-
Regular polygons have all sides congruent and all angles congruent.
-
The sum of interior angles of any n-sided polygon is $(n - 2) \cdot 180°$. For regular polygons, divide by $n$ to find each angle.
-
The sum of exterior angles of any convex polygon is always $360°$. For regular polygons, each exterior angle is $\frac{360°}{n}$.
-
A parallelogram has both pairs of opposite sides parallel. Key properties: opposite sides congruent, opposite angles congruent, consecutive angles supplementary, diagonals bisect each other.
-
Special parallelograms form a hierarchy:
- Rectangle: 4 right angles, congruent diagonals
- Rhombus: 4 congruent sides, perpendicular diagonals
- Square: All properties of both rectangle and rhombus
-
A trapezoid has exactly one pair of parallel sides. An isosceles trapezoid has congruent legs and base angles.
-
A kite has two pairs of consecutive congruent sides. Its diagonals are perpendicular, with one bisecting the other.
-
Understanding these shapes helps you recognize patterns in floor tiles, building design, road signs, and countless other applications. The geometry of polygons is literally all around you.