GRE Geometry | Quadrilaterals

A **quadrilateral** is a 2 – D shape which contain 4 sides. It consist of four sides, four angles and four vertices.

**Different type of quadrilateral and their properties:**

**Rectangle:**

The rectangle, like the square, is one of the most commonly known quadrilaterals. It is defined as having all four interior angles 90° (right angles).- All four angles in a rectangle are 90°.
- Opposite side are parallel and of same length(or congruent).
- Diagonals of rectangle bisect each other and divide a the rectangle into two congruent triangles.
- Area of rectangle = length * breadth.
- Perimeter of rectangle = 2*(length + breadth)

**Square:**

In geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, or (100-gradian angles or right angles). It can also be defined as a rectangle in which two adjacent sides have equal length.- All four angles in a rectangle are 90°.
- All sides are equal and opposite sides are parallel(or congruent).
- Diagonals of square bisect each other at 90° and divide the square into two congruent triangle.
- Area of square = side * side
- Perimeter of square = 4 * side

**Rhombus:**

A Rhombus is a flat shape with 4 equal straight sides. A rhombus looks like a diamond. All sides have equal length. Opposite sides are parallel, and opposite angles are equal (it is a Parallelogram).- All sides are equal in a rhombus.
- Opposite angles are equal in rhombus.
- The diagonals of rhombus intersect each other at equal angles.
- Area of rhombus = (diagonal
_{1}* diagonal_{2}) / 2 - Perimeter of rhombus = 4 * side

**Trapezium:**

In Euclidean geometry, a convex quadrilateral with at least one pair of parallel sides is referred to as a trapezoid.- Atleast one pair of opposite side is parallel to each other.
- Area of trapezium = (sum of length of parallel side(or bases)) * height / 2.
- Perimeter of trapezium = sum of all sides.

**Parallelogram:**

In Euclidean geometry, a parallelogram is a simple quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure.- Opposite sides are parallel and congurent to each other.
- Opposite angles are congurent to each other.
- Sum of adjacent angles is 180°
- Diagonals of parallelogram bisect each other and divide the parallelogram into two congurent triangles.
- Area of parallelogram = length * breadth
- Perimeter of parallelogram = 2(length + breadth)

**Some examples on quadrilateral:**

What will be the area and perimeter of given rectangle if length of rectangle is 5cm and breadth of rectangle is 7cm?**Solution:**

Area of rectangle = length * breadth. Area of rectangle = 5 * 7 = 35 cm^{2}Perimeter of rectangle = 2*(length + breadth) Perimeter of rectangle = 2*(5 + 7) = 24cm

What will be the are of given rhombus if diagonal of rhombus are 4cm and 6cm?

Area of rhombus = (diagonal_{1}* diagonal_{2}) / 2 Area of rhombus = (4 * 6) / 2 = 12 cm^{2}

What will be area and perimeter of given trapezium if AB is parallel to CD?

Area of trapezium = (sum of length of parallel side(or bases)) * height / 2. Area of trapezium = (3 + 4. 5) * 3 / 2 = 11.25 cm^{2}Perimeter of trapezium = Sum of all sides = 3 + 3 + 3.5 + 4.5 = 14 cm