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## What is Area and What are Quadrilaterals?

The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m2), which is the area of a square whose sides are one metre long. The two major sides are the Length and the Breadth. A shape with an area of three square meters would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

Area is the quantity that expresses the extent of a two-dimensional figure or shape or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape, or the amount of paint necessary to cover the surface with a single coat. It is the two-dimensional analog of the length of a curve (a one-dimensional concept) or the volume of a solid (a three-dimensional concept).

 Area Common symbols A SI unit Square metre In SI base units 1 m2

The area of a shape can be measured by comparing the shape to squares of a fixed size. In the International System of Units (SI), the standard unit of area is the square metre (written as m2), which is the area of a square whose sides are one metre long. A shape with an area of three square metres would have the same area as three such squares. In mathematics, the unit square is defined to have area one, and the area of any other shape or surface is a dimensionless real number.

There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.

For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area. Formulas for the surface areas of simple shapes were computed by the ancient Greeks, but computing the surface area of a more complicated shape usually requires multivariable calculus.

Area plays an important role in modern mathematics. In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry. In analysis, the area of a subset of the plane is defined using Lebesgue measure, though not every subset is measurable. In general, area in higher mathematics is seen as a special case of volume for two-dimensional regions.

Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers. It can be proved that such a function exists.

There are several well-known formulas for the areas of simple shapes such as triangles, rectangles, and circles. Using these formulas, the area of any polygon can be found by dividing the polygon into triangles. For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus.

In a convex quadrilateral, all interior angles are less than 180°, and the two diagonals both lie inside the quadrilateral.

• Irregular quadrilateral (British English) or trapezium (North American English): no sides are parallel. (In British English, this was once called a trapezoid. For more, see Trapezoid § Trapezium vs Trapezoid)
• Trapezium (UK) or trapezoid (US): at least one pair of opposite sides are parallel. Trapezia (UK) and trapezoids (US) include parallelograms.
• Isosceles trapezium (UK) or isosceles trapezoid (US): one pair of opposite sides are parallel and the base angles are equal in measure. Alternative definitions are a quadrilateral with an axis of symmetry bisecting one pair of opposite sides, or a trapezoid with diagonals of equal length.
• Parallelogram: a quadrilateral with two pairs of parallel sides. Equivalent conditions are that opposite sides are of equal length; that opposite angles are equal; or that the diagonals bisect each other. Parallelograms include rhombi (including those rectangles called squares) and rhomboids (including those rectangles called oblongs). In other words, parallelograms include all rhombi and all rhomboids, and thus also include all rectangles.
• Rhombus, rhomb: all four sides are of equal length. An equivalent condition is that the diagonals perpendicularly bisect each other. Informally: "a pushed-over square" (but strictly including a square, too).
• Rhomboid: a parallelogram in which adjacent sides are of unequal lengths, and some angles are oblique (equiv., having no right angles). Informally: "a pushed-over oblong". Not all references agree, some define a rhomboid as a parallelogram that is not a rhombus.
• Rectangle: all four angles are right angles. An equivalent condition is that the diagonals bisect each other, and are equal in length. Rectangles include squares and oblongs. Informally: "a box or oblong" (including a square).
• Square (regular quadrilateral): all four sides are of equal length (equilateral), and all four angles are right angles. An equivalent condition is that opposite sides are parallel (a square is a parallelogram), and that the diagonals perpendicularly bisect each other and are of equal length. A quadrilateral is a square if and only if it is both a rhombus and a rectangle (i.e., four equal sides and four equal angles).
• Oblong: a term sometimes used to denote a rectangle that has unequal adjacent sides (i.e., a rectangle that is not a square).
• Kite: two pairs of adjacent sides are of equal length. This implies that one diagonal divides the kite into congruent triangles, and so the angles between the two pairs of equal sides are equal in measure. It also implies that the diagonals are perpendicular. Kites include rhombi.
• Tangential quadrilateral: the four sides are tangents to an inscribed circle. A convex quadrilateral is tangential if and only if opposite sides have equal sums.
• Tangential trapezoid: a trapezoid where the four sides are tangents to an inscribed circle.
• Cyclic quadrilateral: the four vertices lie on a circumscribed circle. A convex quadrilateral is cyclic if and only if opposite angles sum to 180°.
• Right kite: a kite with two opposite right angles. It is a type of cyclic quadrilateral.
• Harmonic quadrilateral: the products of the lengths of the opposing sides are equal. It is a type of cyclic quadrilateral.
• Bicentric quadrilateral: it is both tangential and cyclic.
• Orthodiagonal quadrilateral: the diagonals cross at right angles.
• Equidiagonal quadrilateral: the diagonals are of equal length.
• Ex-tangential quadrilateral: the four extensions of the sides are tangent to an excircle.
• An equilic quadrilateral has two opposite equal sides that when extended, meet at 60°.
• A Watt quadrilateral is a quadrilateral with a pair of opposite sides of equal length.
• A quadric quadrilateral is a convex quadrilateral whose four vertices all lie on the perimeter of a square.
• A diametric quadrilateral is a cyclic quadrilateral having one of its sides as a diameter of the circumcircle.
• A Hjelmslev quadrilateral is a quadrilateral with two right angles at opposite vertices.

### Special line Segments

The two diagonals of a convex quadrilateral are the line segments that connect opposite vertices.

The two bimedians of a convex quadrilateral are the line segments that connect the midpoints of opposite sides. They intersect at the "vertex centroid" of the quadrilateral (see § Remarkable points and lines in a convex quadrilateral below).

The four maltitudes of a convex quadrilateral are the perpendiculars to a side—through the midpoint of the opposite side.