Quadrilateral

Quadrilateral
Quadrangle

Hierarchy diagram of Special Quadrilaterals: Parallelogram (→Rhombus→Square, →Rectangle→Square), Kite, Trapezoid (→Isosceles...

Key Formula

S = (n - 2) \times 180^\circ = (4 - 2) \times 180^\circ = 360^\circ

Where:

$S$ = Sum of the interior angles of the quadrilateral
$n$ = Number of sides (always 4 for a quadrilateral)

Worked Example

Problem: A quadrilateral has three interior angles measuring 90°, 85°, and 110°. Find the fourth angle.

Step 1: Recall that the sum of interior angles in any quadrilateral is 360°.

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

Step 2: Substitute the three known angles into the equation.

90^\circ + 85^\circ + 110^\circ + \angle D = 360^\circ

Step 3: Add the three known angles together.

285^\circ + \angle D = 360^\circ

Step 4: Subtract 285° from both sides to find the missing angle.

\angle D = 360^\circ - 285^\circ = 75^\circ

Answer: The fourth angle measures 75°.

Another Example

This example differs by finding the area of a general quadrilateral using coordinates, rather than finding a missing angle. It also introduces the Shoelace Formula, a useful technique for coordinate geometry problems.

Problem: Find the area of a quadrilateral with vertices at A(0, 0), B(6, 0), C(5, 4), and D(1, 4).

Step 1: Use the Shoelace Formula for the area of a polygon given its vertices listed in order. The formula is:

\text{Area} = \frac{1}{2} |x_A(y_B - y_D) + x_B(y_C - y_A) + x_C(y_D - y_B) + x_D(y_A - y_C)|

Step 2: Substitute the coordinates of each vertex.

\text{Area} = \frac{1}{2} |0(0 - 4) + 6(4 - 0) + 5(4 - 0) + 1(0 - 4)|

Step 3: Evaluate each term inside the absolute value.

= \frac{1}{2} |0 + 24 + 20 + (-4)| = \frac{1}{2} |40|

Step 4: Compute the final area.

\text{Area} = \frac{1}{2} \times 40 = 20 \text{ square units}

Answer: The area of the quadrilateral is 20 square units.

Frequently Asked Questions

What is the sum of the interior angles of a quadrilateral?

The interior angles of every quadrilateral add up to 360°. This follows from the general polygon angle sum formula: (n − 2) × 180°, where n = 4. You can also see this by drawing one diagonal, which splits the quadrilateral into two triangles, each with angles summing to 180°.

What are the different types of quadrilaterals?

The main types are the trapezoid (exactly one pair of parallel sides), parallelogram (two pairs of parallel sides), rectangle (parallelogram with four right angles), rhombus (parallelogram with four equal sides), square (both a rectangle and a rhombus), and kite (two pairs of consecutive equal sides). Every square is a rectangle, every rectangle is a parallelogram, and every parallelogram is a trapezoid (under the inclusive definition).

Is a square a quadrilateral?

Yes. A square is a special type of quadrilateral that has four equal sides and four right angles. All squares are quadrilaterals, but not all quadrilaterals are squares. The square sits at the most specialized end of the quadrilateral classification hierarchy.

Quadrilateral vs. Triangle

	Quadrilateral	Triangle
Number of sides	4	3
Angle sum	360°	180°
Number of diagonals	2	0
Minimum sides to define a polygon	No (triangle is the minimum)	Yes (simplest polygon)
Common area formulas	Depends on type; Shoelace Formula for coordinates	½ × base × height

Why It Matters

Quadrilaterals appear constantly in geometry courses, from basic angle problems through coordinate geometry and proofs. Understanding the hierarchy of quadrilateral types (trapezoid → parallelogram → rectangle/rhombus → square) is essential for classifying shapes and applying the correct area or perimeter formula. Real-world applications include architecture, tiling patterns, and computer graphics, where surfaces are often broken into quadrilateral meshes.

Common Mistakes

Mistake: Assuming the angle sum of a quadrilateral is 180° (confusing it with a triangle).

Correction: A quadrilateral's interior angles sum to 360°, not 180°. Remember: each time you add a side to a polygon, the angle sum increases by 180°. A triangle has 180°, so a quadrilateral (one more side) has 360°.

Mistake: Thinking that a quadrilateral must have parallel sides or equal-length sides.

Correction: A quadrilateral only needs to be a four-sided polygon. It can be completely irregular, with no parallel sides and no sides of equal length. Parallel or equal sides are additional properties that define special types like parallelograms or rhombuses.

Related Terms

Polygon — General category that includes quadrilaterals
Side of a Polygon — A quadrilateral has exactly four sides
Trapezoid — Quadrilateral with at least one pair of parallel sides
Parallelogram — Quadrilateral with two pairs of parallel sides
Rectangle — Parallelogram with four right angles
Rhombus — Parallelogram with four equal sides
Square — Both a rectangle and a rhombus
Kite — Quadrilateral with two pairs of consecutive equal sides