Quadrilateral

A shape with four sides is a quadrilateral.

These shapes have four corners too. Some shapes are simple. This means the lines do not cross. Other shapes are complex. Their lines cross like a bow-tie.

Some shapes have special rules. A square has four equal sides. It also has four right angles. A rectangle has four right angles. A rhombus has four equal sides.

Lines can connect opposite corners. We call these lines diagonals.

Many different shapes fit this group. You can find them everywhere!

A quadrilateral is a shape with four sides.

The name comes from Latin words. "Quadri" means four. "Latus" means side. You might also hear it called a tetragon. This name comes from Greek words. "Tetra" means four. "Gon" means angle.

Some shapes are simple. This means the lines do not cross each other. In a simple shape, the four inside angles add up to 360 degrees. Simple shapes can be convex or concave. In a convex shape, the corners point out. In a concave shape, one corner points in. A shape that looks like an arrowhead is called a dart.

Other shapes are complex. These are also called crossed quadrilaterals. Their lines cross like a bow-tie.

Many special shapes are quadrilaterals. A parallelogram has two pairs of parallel sides. A rectangle has four right angles. A rhombus has four equal sides. A square is very special. It is both a rhombus and a rectangle. It has four equal sides and four right angles. Lines that connect opposite corners are called diagonals.

A quadrilateral is a shape with four sides and four corners.

The name comes from two Latin words. "Quadri" means four and "latus" means side. You might also hear people call it a tetragon. This name comes from Greek words. "Tetra" means four and "gon" means angle. Some people call it a quadrangle, which means a four-angle. These shapes are very important in geometry. They help us understand how flat surfaces are built.

Shapes can be simple or complex. A simple quadrilateral does not have lines that cross each other. In a simple shape, the four inside angles always add up to 360 degrees. These can be convex or concave. In a convex shape, all corners point outward. In a concave shape, one corner points inward. A shape that looks like an arrowhead is called a dart.

Complex quadrilaterals are different because their lines do cross. These are often called crossed quadrilaterals or butterfly quadrilaterals. They can look like a bow-tie.

There are many special types of simple quadrilaterals. A parallelogram has two pairs of parallel sides. A rectangle is a parallelogram with four right angles. A rhombus has four sides that are all the same length. A square is a very special shape. It is both a rhombus and a rectangle. This means a square has four equal sides and four right angles. A kite is another shape where two pairs of sides next to each other are equal. A trapezoid has at least one pair of parallel sides.

We can also look at how these shapes connect to circles. A cyclic quadrilateral is a shape where all four corners sit on a circle. A tangential quadrilateral is a shape where all four sides touch a circle inside it. If a shape is both, it is called a bicentric quadrilateral. Some shapes have special lines inside them. These lines are called diagonals. They connect the opposite corners of the shape. In a square, the diagonals are equal in length and cross at right angles.

Understanding these shapes helps us see patterns in the world. You can use quadrilaterals to tile a flat floor without any gaps. This works for all non-self-crossing quadrilaterals. You can do this by rotating the shapes around the middle of their edges. Many objects in our homes are quadrilaterals. Windows, doors, and books often use these shapes.

By studying them, we learn how to measure area and space. This makes math feel like a tool for building the world.

In geometry, a quadrilateral is a polygon defined by four edges and four vertices. The term originates from the Latin words "quadri," meaning four, and "latus," meaning side. You may also encounter the term tetragon, which comes from the Greek "tetra" for four and "gon" for angle. Because "gon" refers to an angle, these shapes are sometimes called quadrangles. A quadrilateral can be identified by its vertices, such as points A, B, C, and D. These shapes are fundamental to understanding how flat surfaces and patterns are constructed in mathematics.

Quadrilaterals are categorized by how their sides interact. Simple quadrilaterals are those that do not self-intersect. Within simple quadrilaterals, there are two main types: convex and concave. In a convex quadrilateral, every interior angle is less than 180 degrees. This ensures that both diagonals, the lines connecting opposite vertices, lie entirely inside the shape. In contrast, a concave quadrilateral has one interior angle greater than 180 degrees. This causes one of its diagonals to lie outside the shape. A specific type of concave quadrilateral is called a dart, which has bilateral symmetry.

Complex quadrilaterals, also known as crossed or butterfly quadrilaterals, are self-intersecting. In these shapes, the sides cross over one another. A crossed quadrilateral can take the form of an antiparallelogram, where nonadjacent sides have equal lengths. There are also crossed trapezoids and crossed rectangles. For a crossed quadrilateral, the four interior angles on either side of the intersection point add up to 720 degrees. These complex shapes represent a different way of thinking about how lines and angles can connect in a closed loop.

Simple quadrilaterals can be classified into many specific families based on their symmetry and side properties. A trapezoid (known as a trapezium in British English) has at least one pair of parallel sides. An isosceles trapezoid features equal base angles and equal diagonals. A parallelogram is a more specific type, possessing two pairs of parallel sides. In a parallelogram, opposite sides are equal in length, and opposite angles are also equal. The diagonals of a parallelogram always bisect each other, meaning they cut each other exactly in half.

Within the parallelogram family, we find several highly structured shapes. A rhombus is a parallelogram where all four sides are of equal length. Its diagonals perpendicularly bisect each other. A rectangle is a parallelogram where all four angles are right angles. A square is the most regular quadrilateral because it is both a rhombus and a rectangle. It possesses four equal sides and four right angles. Other shapes include the kite, which has two pairs of equal adjacent sides, and the rhomboid, a parallelogram with unequal adjacent sides and oblique angles.

Mathematical relationships also exist between quadrilaterals and circles. A cyclic quadrilateral is a convex shape where all four vertices lie on a single circumscribed circle. For these shapes, opposite angles always sum to 180 degrees. A tangential quadrilateral is one where all four sides are tangent to an inscribed circle. This occurs if and only if the sums of the opposite sides are equal. If a quadrilateral is both cyclic and tangential, it is called a bicentric quadrilateral. Other specialized types include the orthodiagonal quadrilateral, where diagonals cross at right angles, and the equidiagonal quadrilateral, where diagonals are equal in length.

Calculating the area of a quadrilateral involves various mathematical methods depending on the known information. For a convex quadrilateral, the area can be found using the lengths of the diagonals and the angle between them. Trigonometric formulas, such as Bretschneider's formula, allow for area calculation using the sides and opposite angles. If the quadrilateral is cyclic, Brahmagupta's formula provides a simplified way to find the area using only the side lengths and the semiperimeter. For those using vector calculus, the area is half the magnitude of the cross product of the diagonal vectors.

Beyond individual shapes, quadrilaterals have unique properties in tiling and geometry. All non-self-crossing quadrilaterals can tile a plane. This means they can cover a flat surface without gaps by rotating around the midpoints of their edges. This property is useful in various design and engineering applications. Furthermore, the diagonals and bimedians—lines connecting the midpoints of opposite sides—provide deep insights into the shape's center and balance. Studying these connections helps mathematicians understand the broader systems of Euclidean geometry.