A quadrilateral is a polygon with four straight sides, four vertices and four angles. Its four interior angles always add to 360 degrees. Squares, rectangles, rhombuses, parallelograms, trapezoids and kites are special quadrilaterals, each named for extra properties of its sides or angles.
One diagonal cuts any quadrilateral into two triangles, so the angles add to 2 × 180° = 360°. In the tree, every shape is a special case of the shapes above it.
Formal definition and notation
A quadrilateral is a polygon ABCD formed by four line segmentsAB, BC, CD and DA joining four points in a plane, no three of which are collinear, with sides meeting only at their endpoints. Its interior angles satisfy ∠ A + ∠ B + ∠ C + ∠ D = 360°.
Symbol
Read it as
Meaning
ABCD
quadrilateral A B C D
vertices listed in order around the shape, so AC and BD are diagonals
Key facts
Every quadrilateral has 4 sides, 4 vertices, 4 angles and 2 diagonals.
The angles add to 360°, because one diagonal splits the shape into two triangles.
Parallelogram: two pairs of parallel sides. Trapezoid: exactly one pair (US school definition). Kite: two pairs of equal sides that are next to each other.
Rectangle: four right angles. Rhombus: four equal sides. Square: four right angles and four equal sides, so it is both a rectangle and a rhombus.
Properties flow down the family tree: a square has every property of a rectangle, a rhombus and a parallelogram.
Quadrilateral angle sum
Quadrilateral angle sum
∠ A + ∠ B + ∠ C + ∠ D = 360°
∠ A, ∠ B, ∠ C, ∠ D
the four interior angles
360°
two triangles’ worth of angles: 2 × 180°
To find a missing angle, subtract the other three from 360. Angles of 85°, 110° and 70° leave 360 − 265 = 95°.
Quadrilateral examples, step by step
Example 1: Find the missing angle
Warm-up
Three angles of a quadrilateral measure 85°, 110° and 70°. Find the fourth angle.
Add the known angles: 85 + 110 + 70 = 265.
Subtract from 360: 360 − 265 = 95.
Answer: The fourth angle is 95°.
Example 2: Angles in a ratio
Core
The angles of a quadrilateral are in the ratio 1 : 2 : 3 : 4. Find each angle.
Write the angles as k, 2k, 3k and 4k. They add to 360: 10k = 360.
So k = 36.
The angles are 36°, 72°, 108° and 144°. Check: 36 + 72 + 108 + 144 = 360.
Answer: The angles are 36°, 72°, 108° and 144°.
Example 3: Solve for x
Test-style
A quadrilateral has angles x, 2x, x + 20 and 100degrees. Find x and the largest angle.
Sum of the angles: x + 2x + (x + 20) + 100 = 360.
Combine: 4x + 120 = 360.
Solve: 4x = 240, so x = 60.
The angles are 60°, 120°, 80° and 100°. The largest is 2x = 120°.
Answer: x = 60, and the largest angle is 120°.
Common mistakes
✗ MistakeAngles of 90°, 90° and 120° leave 180 − 300 for the fourth angle, which is impossible.
Every parallelogram is a quadrilateral, but a quadrilateral needs only four sides. A kite or a trapezoid is a quadrilateral that is not a parallelogram.
Quadrilateralvs.Rectangle
A rectangle is one kind of quadrilateral, with four right angles. Most quadrilaterals have no right angles at all.
Quick check: 3 questions
Question 1 of 3
A quadrilateral has angles of 90°, 90° and 120°. What is the fourth angle?
Show answer and explanation
Answer: A.360 − (90 + 90 + 120) = 60. Choice B uses 330 as the total; C and D just repeat one of the given angles.
Question 2 of 3
Which shape is a quadrilateral but not a parallelogram?
Show answer and explanation
Answer: D. A trapezoid has exactly one pair of parallel sides, so it is not a parallelogram. A square, a rhombus and a rectangle all have two pairs of parallel sides.
Question 3 of 3
A quadrilateral has four equal sides and no right angles. What is it?
Show answer and explanation
Answer: B. Four equal sides without right angles is a rhombus. A square needs right angles, a rectangle needs right angles but not equal sides, and a trapezoid has only one pair of parallel sides.
Quadrilateral: frequently asked questions
Do the angles of every quadrilateral add to 360 degrees?
Yes. Draw one diagonal and the quadrilateral splits into two triangles. Each triangle has angles that add to 180 degrees, so the quadrilateral has 2 times 180, or 360 degrees. This holds for squares, kites, trapezoids and irregular four-sided shapes alike, including concave ones with a dent.
Is a square a rectangle, a rhombus, or both?
Both. A rectangle is a quadrilateral with four right angles, and a rhombus is one with four equal sides. A square has four right angles and four equal sides, so it belongs to both groups. It is also a parallelogram and a quadrilateral, since it has two pairs of parallel sides and four sides.
What are the 6 types of quadrilaterals?
The six usually taught are the parallelogram, rectangle, rhombus, square, trapezoid and kite. They overlap: rectangles, rhombuses and squares are all parallelograms, and a square is both a rectangle and a rhombus. Any four-sided polygon that fits none of these names is still a quadrilateral, just an irregular one.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 3.G.A.1, CCSS 5.G.B.4
Cite this entry
Effortless Math Team. “Quadrilateral.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/quadrilateral/
Flashcard
TermQuadrilateral4 sides, angles sum to 360°
DefinitionA polygon with four sides and four angles that add to 360°. Angles 85°, 110° and 70° leave 95° for the fourth.
PolygonA polygon is a flat, closed shape made of straight line segments, called sides, that meet only at their endpoints. Each corner is a vertex. Polygons are named by their number of sides: a triangle has 3, a quadrilateral 4, a pentagon 5, a hexagon 6 and an octagon 8.Geometry · Grade 2, Grade 3, Grade 5Line SegmentA line segment is a straight part of a line with two endpoints. It starts at one point, stops at another, and includes every point between them. A segment with endpoints A and B is written AB. Unlike a line, which goes on forever, a segment has a definite length that you can measure.Geometry · Grade 4, Grade 5, Grade 6SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2Interior AngleAn interior angle is an angle inside a polygon, formed at a vertex by two sides that meet there. A polygon with n sides has n interior angles, and they add up to (n − 2) × 180 degrees. A triangle’s interior angles total 180 degrees, and a rectangle’s four right angles total 360 degrees.Geometry · Grade 7, Grade 8, GeometryTriangleA triangle is a polygon with three sides and three angles. Its three corners are called vertices, and the angle measures in any triangle add to 180 degrees. Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse).Geometry · Grade 3, Grade 4, Grade 5ParallelogramA parallelogram is a four-sided shape whose opposite sides are parallel. Its opposite sides are also equal in length, its opposite angles are equal, and its diagonals cut each other in half. Rectangles, rhombuses and squares are special parallelograms. Its area is base times height: A = bh.Geometry · Grade 3, Grade 5, Grade 6TrapezoidA trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are the bases, and the other two sides are the legs. Its area is the average of the bases times the height: A = b1 + b22 × h. In British English it is called a trapezium.Geometry · Grade 3, Grade 5, Grade 6Right AngleA right angle is an angle that measures exactly 90 degrees, a quarter of a full turn. It looks like the corner of a square or a sheet of paper, and it is marked with a small square at the vertex. Two lines that meet at a right angle are perpendicular.Geometry · Grade 2, Grade 3, Grade 4RhombusA rhombus is a quadrilateral with four sides of equal length. It is a special parallelogram, so its opposite sides are parallel and its opposite angles are equal. Its diagonals cross at right angles and cut each other in half, and its area is half the product of the diagonals.Geometry · Grade 3, Grade 4, Grade 5SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3RatioA ratio compares two quantities by division. Ratios can be written with words, a colon, or a fraction: 3 boys to 5 girls is 3 to 5, 3 : 5, or 35. Order matters, and a ratio can compare part to part or part to whole. Multiplying both numbers by the same amount gives an equivalent ratio.Ratios and Proportions · Grade 6, Grade 7, Grade 8DegreeA degree is the standard unit for measuring an angle, written with the symbol °. One degree is 1360 of a full turn, so a full turn is 360°, a straight angle is 180° and a right angle is 90°. An angle that turns through 25 one-degree angles measures 25°.Geometry · Grade 4, Grade 5, Grade 7LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3AngleAn angle is formed by two rays that share the same endpoint, called the vertex. It is measured in degrees by how far one ray is turned from the other. A right angle measures 90 degrees, a straight angle measures 180 degrees, and one full turn is 360 degrees.Geometry · Grade 4, Grade 5, Grade 6Polygon Angle SumThe polygon angle sum is the total of the interior angles of a polygon. For a polygon with n sides it equals (n − 2) × 180°, because diagonals from one vertex cut the polygon into n − 2 triangles. A pentagon’s angles add to 540°, and a hexagon’s add to 720°.Geometry · Grade 8, GeometryPerimeterPerimeter is the total distance around a two-dimensional shape. To find it, add the lengths of all the sides. A rectangle with sides 5 and 3 has perimeter 5 + 3 + 5 + 3 = 16. Perimeter is a length, so it uses length units such as inches or meters, never square units.Geometry · Grade 3, Grade 4, Grade 5