Introduction
A quadrilateral is a plane closed geometric figure with four sides and four angles. A quadrilateral can be constructed using a compass and a ruler. There are several real-life examples of a quadrilateral: kites, squares, rectangles, rhombus, etc. The word quadrilateral is derived from the word “quad” which means four. Hence, there are four adjoining sides of a quadrilateral at all times.
Properties
A quadrilateral always has four sides and four angles.
The length of sides may be the same or different.
The sum of all the internal angles will be 360°.
The sum of all the external angles will also be 360°.
Construction
A quadrilateral can be constructed using a ruler and a compass if any of the following conditions are provided:
Length of four sides and one diagonal are given
Measurement of three sides and two angles are given
Measurements of two sides and three angles are given.
Types
There are a lot of examples of a quadrilateral but the most common geometric figures are square, rectangle, rhombus, parallelogram and trapezium. Except for a trapezium, all the other figures have at least two parallel sides and every quadrilateral has four vertices, four sides and four angles.
Square has all four equal sides and angles. The diagonals of a square bisect each other at 90°. Both the opposite sides of the square are parallel to each other.
Rectangle has equal opposite sides and all four equal angles. The opposite sides of a rectangle are parallel to each other as well.
Rhombus has equal sides, and opposite sides are parallel to each other. Opposite angles are equal and the sum of two adjacent angles is 180°. The diagonals from opposite sides of a rhombus bisect each other and are perpendicular to each other.
A parallelogram has two opposite and equal sides which are also parallel to each other. The opposite angles are equal and the diagonals bisect with each other.
Trapezium has only one opposite parallel and the diagonal bisectors are of the same ratio.
Quadrilaterals can be further classified as convex quadrilateral, concave quadrilateral and intersecting or crossed quadrilaterals based on the diagonals former inside a quadrilateral. In convex quadrilaterals, the diagonals are completely within the boundary of the quadrilateral while in concave quadrilaterals, the diagonals are partially outside the boundary of the quadrilateral.
Formulas
There are two basic formulas of quadrilaterals, namely: area and perimeter.
Area
Area of square: side×side
Area of rectangle: length×breadth
Area of parallelogram: base×height
Area of rhombus: ½× (first diagonal length)×(second diagonal length)
Area of trapezium: ½× (length of first diagonal)×(length of second diagonal)
Perimeter
The perimeter of square: 4×length of side
The perimeter of rectangle: 2×(length of breadth + length of a side)
Perimeter of parallelogram: 2×(base+side)
The perimeter of rhombus: 4×length of side
The perimeter of trapezium: 2×(a+b), a and b are lengths of adjacent sides.
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FAQs on Quadrilateral
1. What is a quadrilateral in geometry?
A quadrilateral is a two-dimensional closed shape in geometry that is formed by joining four non-collinear points. It has four sides, four vertices (corners), and four interior angles. The sum of all its interior angles is always 360 degrees.
2. What are the main types of quadrilaterals based on their properties?
The main types of quadrilaterals are classified based on the properties of their sides and angles. The most common types include:
Trapezium: A quadrilateral with at least one pair of parallel opposite sides.
Parallelogram: A quadrilateral where both pairs of opposite sides are parallel and equal in length.
Rectangle: A parallelogram with four right angles (90°).
Rhombus: A parallelogram with all four sides of equal length.
Square: A parallelogram with four equal sides and four right angles. It is also a special type of rectangle and rhombus.
Kite: A quadrilateral with two pairs of equal-length sides that are adjacent to each other.
3. What are the essential properties that every quadrilateral must have?
Regardless of its specific type, every quadrilateral shares three fundamental properties:
It must have exactly four sides.
It must have exactly four vertices (corners).
The sum of its four interior angles must be equal to 360°. This is known as the Angle Sum Property of a Quadrilateral.
4. How can you tell the difference between a convex and a concave quadrilateral?
The key difference lies in their interior angles and diagonals. In a convex quadrilateral, all interior angles are less than 180°, and both of its diagonals lie completely inside the shape. In a concave quadrilateral, one of the interior angles is a reflex angle (greater than 180°), which causes one of its diagonals to lie partially or entirely outside the shape's boundary.
5. Why is a square considered a special type of both a rectangle and a rhombus?
A square fits the definitions of both a rectangle and a rhombus, making it a special case of both. Here's why:
It is a rectangle because it has four right angles and opposite sides that are parallel and equal.
It is a rhombus because it has four equal sides and opposite sides that are parallel.
Since a square satisfies the more general properties of both shapes, it is considered the most symmetrical type of quadrilateral.
6. What is the primary difference between a parallelogram and a trapezium?
The primary difference is the number of parallel sides. A parallelogram must have two pairs of opposite sides that are parallel. In contrast, a trapezium (also known as a trapezoid) is defined as having only one pair of opposite sides that are parallel. This means that every parallelogram is a special kind of trapezium, but not every trapezium is a parallelogram.
7. Can a quadrilateral have three obtuse angles? Why or why not?
No, a quadrilateral cannot have three obtuse angles. An obtuse angle is greater than 90°. If a quadrilateral had three obtuse angles, the sum of just those three angles would be greater than 270° (e.g., 91° + 91° + 91° = 273°). Since the total sum of all four angles in any convex quadrilateral must be exactly 360°, having three obtuse angles would make it mathematically impossible to form a fourth angle without exceeding this sum.
8. Where can we see examples of quadrilaterals in real-life objects?
Quadrilaterals are found everywhere in our daily surroundings. Some common examples include:
Rectangle: Found in doors, windows, laptop screens, and books.
Square: Seen in chessboard squares, floor tiles, and some traffic signs.
Rhombus: The pattern on a deck of cards (diamonds) and some decorative window panes.
Kite: The classic shape of a flying kite.
Trapezium: The shape of a popcorn box or some lamp shades when viewed from the side.
