Quadrilaterals: Types, Properties, Perimeter and Area

What is a quadrilateral?

A quadrilateral is a polygon with exactly four sides, four vertices (corners) and four interior angles. The word comes from the Latin quattuor (four) and latus (side); quadrangle and tetragon mean the same, and the Indonesian word is segiempat.

We name a quadrilateral by its vertices in order round the shape, such as ABCD. Its sides are AB, BC, CD and DA. Two sides that share a vertex are adjacent; AB and CD are opposite, and so are BC and DA. A diagonal joins two opposite vertices: AC and BD.

Quadrilaterals come in three kinds of outline:

• Convex. Every interior angle is less than 180^\circ and both diagonals lie inside the shape.
• Concave. One interior angle is more than 180^\circ and one diagonal lies outside. The concave kite in the second picture is called a dart or arrowhead.
• Crossed. Two opposite sides cross each other, giving a bow-tie. A crossed quadrilateral is not a simple polygon, so it has no single inside, no area in the usual sense, and the rules in this article do not apply to it.

Unless it says otherwise, "quadrilateral" below means a simple one: its sides meet only at the vertices.

Why do the angles of a quadrilateral add up to 360°?

The interior angles of every simple quadrilateral add up to 360^\circ, because one diagonal that lies inside it splits it into two triangles and each triangle has angle sum 180^\circ.

Draw the diagonal AC of a convex ABCD. The angle at A is split between the triangles ABC and ACD, and so is the angle at C, while B belongs to the first triangle and D to the second. The four angles of the quadrilateral are therefore exactly the six angles of the two triangles, regrouped:

\angle A+\angle B+\angle C+\angle D=180^\circ+180^\circ=360^\circ.

The same holds for a concave quadrilateral: use the diagonal that lies inside, and count the reflex angle (more than 180^\circ) as it is. In the dart of the first section the angles are about 37^\circ, 53^\circ, 37^\circ and 233^\circ, and 37+53+37+233=360. A polygon with n sides has angle sum (n-2)·180^\circ: a pentagon has 540^\circ.

This one fact solves many problems. If three angles are 70^\circ, 95^\circ and 120^\circ the fourth is 360-285=75^\circ. If the angles are in the ratio 1:2:3:4, call them x,2x,3x,4x as in turning words into algebra: then 10x=360, so x=36^\circ and the angles are 36^\circ,72^\circ,108^\circ,144^\circ.

What are the types of quadrilaterals?

The named quadrilaterals are the parallelogram, rectangle, rhombus, square, kite and trapezoid, and they form a family tree in which every special type keeps all the properties of the more general ones.

Read the table as a hierarchy, from the general to the special:

• A square is a rectangle (right angles), a rhombus (equal sides), a parallelogram and a kite, all at once.
• Every rectangle and every rhombus is a parallelogram, because their opposite sides are parallel.
• Every rhombus is a kite, since all four sides are equal and so the adjacent ones are equal in pairs.
• A rectangle that is not a square, a parallelogram that is not a rectangle and a kite that is not a rhombus are the genuinely different cases.
• A trapezoid is its own branch: with one pair of parallel sides it can never be a parallelogram.

So "is a square a rectangle?" has the answer yes: a rectangle is any quadrilateral with four right angles, and a square has them. The everyday habit of treating the names as separate boxes is exactly what the next section's table corrects.

What are the properties of each quadrilateral?

Each type is told apart by its sides, its angles, its diagonals and its symmetry, and the table gives all of them for a typical example of each type. A special case can have more: a particular parallelogram may happen to be a rectangle.

Some of these have a neat reason. A diagonal of a parallelogram cuts it into two congruent triangles, which is why opposite sides are equal, opposite angles are equal and the diagonals bisect each other. The angles next to one side of a parallelogram add up to 180^\circ, since they sit between parallel lines. In a kite the diagonal along the axis of symmetry cuts the other diagonal in half at a right angle, which is why the area is half the product of the diagonals. An isosceles trapezoid has equal base angles in each pair, so it has one line of symmetry, through the midpoints of its two bases. Try each type below and read its checklist.

The diagonals of a rectangle with sides l and w are equal and have length √(l^2+w^2), by the Pythagorean theorem. For a 6×4 rectangle that is √(52)=2√(13)≈7.21, after you simplify the radical. For a square with side s the diagonal is s√2, and because √2 is irrational, no square with whole-number sides has a whole-number diagonal.

For a rhombus the diagonals cross at right angles and halve each other, so each side is the hypotenuse of a right triangle whose legs are half of each diagonal. With diagonals 10 and 24 the legs are 5 and 12, the side is √(25+144)=13, and the perimeter is 4·13=52.

How do you find the perimeter and area of a quadrilateral?

The perimeter is the sum of the four sides, and the area comes from a formula for the type: base times height for a parallelogram, half the product of the diagonals for a rhombus or kite, and half the sum of the parallel sides times the height for a trapezoid.

The height is the perpendicular distance between the parallel sides, never the slanted side. Perimeter is a length, in meters; area is a count of unit squares, in square meters.

Each formula is the rectangle in disguise:

• Parallelogram. Cut a right triangle off one slanted end and slide it to the other end: you get a rectangle with the same base and height, so the area is bh.
• Trapezoid. Two copies, one turned upside down, fit together into a parallelogram with base a+b and height h. The trapezoid is half of it, so its area is ((a+b)h)/(2).
• Rhombus and kite. The diagonals are perpendicular, so the rectangle d_1× d_2 drawn round the shape has exactly twice its area: the area is \frac12d_1d_2. The same holds for every convex quadrilateral whose diagonals are perpendicular.

For the trapezoid in the picture, a=8, b=4 and h=4 give ((8+4)·4)/(2)=24. The \frac12 in these formulas is the usual multiplication of fractions: half of 48 is 24.

A word problem: a plot of land is a trapezoid whose parallel sides are 30 m and 20 m, 12 m apart, and land costs 80 dollars per square meter. The area is ((30+20)·12)/(2)=300 m², so the plot costs 300·80=24000 dollars.

How do you classify a quadrilateral and find its area from coordinates?

Place the vertices on a grid and use three exact tests: a zero cross product means parallel, a zero dot product means perpendicular, and equal squared distances mean equal lengths; the area is half the absolute shoelace sum.

For the side from P=(x_1,y_1) to Q=(x_2,y_2) write the vector \vecPQ=(u,v)=(x_2-x_1, y_2-y_1). For two vectors (u_1,v_1) and (u_2,v_2):

• they are parallel exactly when u_1v_2-v_1u_2=0 (the cross product);
• they are perpendicular exactly when u_1u_2+v_1v_2=0 (the dot product);
• a side has length √(u^2+v^2), so two sides are equal exactly when u^2+v^2 agrees; no square root is needed.

These tests use only multiplication and addition, so on integer or fraction coordinates they are exact, and they cope with vertical sides, where a slope would divide by zero.

Take A(0,0), B(6,0), C(8,4), D(2,4). The sides are \vecAB=(6,0), \vecBC=(2,4), \vecCD=(-6,0), \vecDA=(-2,-4). The cross products 6·0-0·(-6) and 2·(-4)-4·(-2) are both 0, so both pairs of opposite sides are parallel: a parallelogram. The squared sides are 36 and 20, so it is not a rhombus, and \vecAB·\vecBC=12\ne0, so it is not a rectangle.

For the area use the shoelace formula (Gauss's area formula, also called the surveyor's formula). List the vertices in order and add x_iy_i+1-x_i+1y_i over the four edges, going back to the first vertex at the end:

area=\tfrac12\left|\sum_i\bigl(x_iy_i+1-x_i+1y_i\bigr)\right|.

For the parallelogram above the four terms are 0, 24, 24, 0, the sum is 48 and the area is 24=6·4, base times height, as it must be. For A(0,0), B(6,0), C(6,4), D(2,6) the terms are 0,24,28,0, so the area is 26; check by splitting into triangles ABC (12) and ACD (14).

The sign of the sum tells the direction: positive for counterclockwise vertices, negative for clockwise, which is why we take the absolute value. The order matters. A bow-tie such as (0,0),(4,4),(4,0),(0,4) gives a sum of exactly 0, because its two lobes cancel: the formula is only an area for a simple polygon.

How do you classify a quadrilateral in code?

Write the three tests as small functions on integer or Fraction coordinates, and the classifier is a few comparisons. The code assumes a, b, c, d are the vertices in order round a simple convex quadrilateral.

from fractions import Fraction as F

def sub(p, q): return (p[0] - q[0], p[1] - q[1])
def cross(u, v): return u[0] * v[1] - u[1] * v[0]
def dot(u, v): return u[0] * v[0] + u[1] * v[1]
def d2(p, q): return dot(sub(p, q), sub(p, q))     # squared length

def kind(a, b, c, d):
    ab, bc, cd, da = sub(b, a), sub(c, b), sub(d, c), sub(a, d)
    p1, p2 = cross(ab, cd) == 0, cross(bc, da) == 0   # AB || CD, BC || DA
    s = [d2(a, b), d2(b, c), d2(c, d), d2(d, a)]
    if p1 and p2:
        rhombus, right = len(set(s)) == 1, dot(ab, bc) == 0
        if rhombus and right: return "square"
        if right: return "rectangle"
        return "rhombus" if rhombus else "parallelogram"
    if p1 or p2: return "trapezoid"
    if (s[0] == s[1] and s[2] == s[3]) or (s[0] == s[3] and s[1] == s[2]): return "kite"
    return "none of these"

def area(*pts):
    s = sum(p[0] * q[1] - q[0] * p[1] for p, q in zip(pts, pts[1:] + pts[:1]))
    return F(abs(s), 2)

>>> kind((0, 0), (6, 0), (8, 4), (2, 4))
'parallelogram'
>>> kind((4, 0), (8, 3), (4, 6), (0, 3))
'rhombus'
>>> kind((0, 0), (3, 2), (0, 6), (-3, 2))
'kite'
>>> area((0, 0), (6, 0), (6, 4), (2, 6))
Fraction(26, 1)
>>> area((F(1, 2), 0), (4, 0), (4, F(7, 3)), (0, 2))     # fractions stay exact
Fraction(49, 6)
>>> 0.7 * 0.3 - 0.1 * 2.1                                 # should be 0 ...
-2.7755575615628914e-17
>>> F(7, 10) * F(3, 10) - F(1, 10) * F(21, 10)            # ... and with fractions it is
Fraction(0, 1)

const cross = (u, v) => u[0] * v[1] - u[1] * v[0]
cross([6, 0], [-6, 0])                    // 0: parallel (exact for small integers)
cross([0.7, 0.1], [2.1, 0.3])             // -2.7755575615628914e-17, not 0
Math.abs(cross([0.7, 0.1], [2.1, 0.3])) < 1e-9   // true: with floats, compare with a tolerance

const shoelace = (pts) =>
  Math.abs(pts.reduce((s, [x, y], i) => { const [u, v] = pts[(i + 1) % pts.length]; return s + x * v - u * y }, 0)) / 2
shoelace([[0, 0], [6, 0], [6, 4], [2, 6]])   // 26

#include <iostream>
#include <cstdio>
using namespace std;

int cross(int ux, int uy, int vx, int vy) { return ux * vy - uy * vx; }
int dot(int ux, int uy, int vx, int vy) { return ux * vx + uy * vy; }
int d2(int px, int py, int qx, int qy) { return (px - qx) * (px - qx) + (py - qy) * (py - qy); }

const char* kind(int ax, int ay, int bx, int by, int cx, int cy, int dx, int dy) {
  bool p1 = cross(bx - ax, by - ay, dx - cx, dy - cy) == 0;      // AB parallel to CD
  bool p2 = cross(cx - bx, cy - by, ax - dx, ay - dy) == 0;      // BC parallel to DA
  int s0 = d2(ax, ay, bx, by), s1 = d2(bx, by, cx, cy), s2 = d2(cx, cy, dx, dy), s3 = d2(dx, dy, ax, ay);
  if (p1 && p2) {
    bool rhombus = s0 == s1 && s1 == s2 && s2 == s3;
    bool right = dot(bx - ax, by - ay, cx - bx, cy - by) == 0;
    if (rhombus && right) return "square";
    if (right) return "rectangle";
    return rhombus ? "rhombus" : "parallelogram";
  }
  if (p1 || p2) return "trapezoid";
  if ((s0 == s1 && s2 == s3) || (s0 == s3 && s1 == s2)) return "kite";
  return "none of these";
}

int twiceArea(int x[], int y[], int n) {                          // the shoelace formula, doubled: whole numbers
  int s = 0;
  for (int i = 0; i < n; i++) s += x[i] * y[(i + 1) % n] - x[(i + 1) % n] * y[i];
  return s < 0 ? -s : s;
}

int main() {
  cout << kind(0, 0, 6, 0, 8, 4, 2, 4) << endl;                   // parallelogram
  cout << kind(4, 0, 8, 3, 4, 6, 0, 3) << endl;                   // rhombus
  cout << kind(0, 0, 3, 2, 0, 6, -3, 2) << endl;                  // kite
  int x[4] = {0, 6, 6, 2}, y[4] = {0, 0, 4, 6};
  cout << twiceArea(x, y, 4) / 2 << endl;                         // 26
  printf("%.20f\n", 0.7 * 0.3 - 0.1 * 2.1);                       // should be 0 ...
  cout << 7 * 3 - 1 * 21 << endl;                                 // ... and with whole numbers (tenths) it is
  return 0;
}

/* output:
parallelogram
rhombus
kite
26
-0.00000000000000002776
0
*/

The pitfalls, in order of how often they bite:

Where do the quadrilaterals come from?

Quadrilaterals are among the oldest shapes in mathematics, because fields, walls and tiles are four-sided, and the names we use come mostly from Greek.

• c. 1550 BCE. The Rhind Papyrus, problem 52, finds the area of a trapezoid, which the Egyptian scribe describes as a triangle with its tip cut off.
• c. 300 BCE. Euclid's Elements, Book I, definition 22, names the square, the oblong (our rectangle), the rhombus and the rhomboid (our parallelogram), and calls every other quadrilateral a trapezium. Proposition I.34 proves that the opposite sides and angles of a parallelogram are equal and that a diagonal bisects it. Later Greek writers narrowed trapezium to a quadrilateral with a pair of parallel sides.
• c. 150 CE. Ptolemy's theorem: in a quadrilateral whose vertices lie on a circle, the product of the diagonals equals the sum of the products of the opposite sides.
• 628 CE. Brahmagupta gives the area of a quadrilateral inscribed in a circle, √((s-a)(s-b)(s-c)(s-d)), where s is half the perimeter. It generalizes Heron's formula for a triangle.
• 1769. Albrecht Meister describes the shoelace formula for the area of a polygon from its coordinates; it is now also known by Gauss's name, and as the surveyor's formula.

The names themselves drifted. Over time British English kept trapezium for a quadrilateral with one pair of parallel sides, while American English moved to trapezoid, which is why one shape has two names today.

What are the common mistakes with quadrilaterals?

The most common mistakes with quadrilaterals are the nine below, each with the correct statement.

Practice: test your understanding

These questions mix everything above. A wrong answer costs nothing here: read the hint and try again.

Summary: quadrilaterals at a glance

• Definition: four sides, four vertices, two diagonals; the angles of a simple quadrilateral add up to 360^\circ.
• Family tree: square ⊂ rectangle, rhombus ⊂ parallelogram; rhombus ⊂ kite; a trapezoid has exactly one parallel pair.
• Parallelogram: opposite sides parallel and equal, opposite angles equal, diagonals bisect each other.
• Rectangle / rhombus / square: equal diagonals / perpendicular diagonals / both.
• Area: lw, bh, ((a+b)h)/(2), and (d_1d_2)/(2) for a rhombus or kite; perimeter is the sum of the sides.
• Coordinates: cross product 0 is parallel, dot product 0 is perpendicular, squared distances for equal length; area is \frac12\left|\sum(x_iy_i+1-x_i+1y_i)\right|.
• Code: use integers or Fraction, never float equality; order the vertices round the shape.

Glossary: key terms

Quadrilateral

A polygon with four sides, four vertices and four interior angles that add up to 360 degrees.

Parallelogram

A quadrilateral whose two pairs of opposite sides are parallel, which makes opposite sides and opposite angles equal.

Rectangle

A quadrilateral with four right angles, which is a parallelogram with equal diagonals.

Rhombus

A quadrilateral with four equal sides, which is a parallelogram whose diagonals are perpendicular.

Square

A quadrilateral with four equal sides and four right angles, so it is both a rectangle and a rhombus.

Kite

A quadrilateral with two pairs of equal adjacent sides, whose diagonals are perpendicular.

Trapezoid

A quadrilateral with exactly one pair of parallel sides, called the bases, in the school definition used here.

Diagonal

A segment that joins two opposite vertices of a quadrilateral.

Convex quadrilateral

A quadrilateral in which every interior angle is less than 180 degrees and both diagonals lie inside.

Shoelace formula

A formula giving the area of a simple polygon as half the absolute sum of x sub i times y sub i plus 1 minus x sub i plus 1 times y sub i.

Line of symmetry

A line that divides a shape into two mirror-image halves, so that reflecting the shape in it leaves it unchanged.

Frequently asked questions

References

1. Thomas L. Heath (translator) (1908). The Thirteen Books of Euclid's Elements (2nd ed.), Book I, definition 22 and propositions 33–36. Cambridge University Press.
2. Henry Thomas Colebrooke (translator) (1817). Algebra, with Arithmetic and Mensuration, from the Sanskrit of Brahmegupta and Bhascara. John Murray.
3. Arnold Buffum Chace (1927). The Rhind Mathematical Papyrus. Mathematical Association of America.
4. H. S. M. Coxeter and Samuel L. Greitzer (1967). Geometry Revisited, chapter 2 (cyclic quadrilaterals). Mathematical Association of America.
5. Bart Braden (1986). The Surveyor's Area Formula. The College Mathematics Journal, 17(4), 326–337.
6. Python Software Foundation. The Python Standard Library: fractions, rational numbers. docs.python.org. https://docs.python.org/3/library/fractions.html.

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