 A quadrilateral is a polygon with 4 edges, corners, and interior angles. The main shapes are square, rectangle, rhombus, kite, parallelogram, and trapezoid.

In geometry, a quadrilateral is a two-dimensional closed shape or polygon that has four straight sides, four corners or vertices, and four interior angles. The sum of the interior angles is 360 degrees. The word “quadrilateral” comes from the Latin words quadri, meaning “four”, and latus, meaning “side.” A less common name for the shape is a tetragon, which comes from the Greek words tetra, meaning “four”, and gon, meaning “corner or angle.”

Quadrilaterals are important not only in geometry, but for understanding complex geometric shapes and for their wide practical applications.

There are several common types of quadrilaterals. The terminology is mostly the same in both American and British English, except for a trapezoid (American) which is often referred to as a trapezium in British English.

1. Square: A square is a quadrilateral with all sides of equal length and all internal angles of 90 degrees.
2. Rectangle: A rectangle is a quadrilateral with opposite sides of equal length and all internal angles of 90 degrees.
3. Rhombus (Rhomb or Diamond): A rhombus is a quadrilateral with all sides of equal length, opposite angles of equal measure, but not necessarily angles of 90 degrees.
4. Parallelogram: A parallelogram is a quadrilateral with opposite sides of equal length and opposite angles of equal measure. Adjacent angles are supplementary (they add up to 180 degrees).
5. Trapezoid (American) / Trapezium (British): A trapezoid is a quadrilateral with at least one pair of parallel sides. In American usage, it refers to a quadrilateral with exactly one pair of parallel sides, while the British usage typically includes shapes with at least one pair of parallel sides.
6. Trapezium (American) / Irregular Quadrilateral (British): In American usage, a trapezium refers to a quadrilateral with no parallel sides. The British often refer to this as an irregular quadrilateral.
7. Kite: A kite is a quadrilateral with two pairs of adjacent sides of equal length. This implies that a kite has a pair of equal angles.

Remember, all these figures are quadrilaterals, meaning they all have four sides and the sum of their internal angles equals 360 degrees. The specific names (like square, rectangle, etc.) just give more information about the properties of the sides and angles of the quadrilateral.

Some of the quadrilateral shapes are types of other shapes. For example:

• A square is also a rectangle and a rhombus.
• However, a rectangle and rhombus are not square.
• A square, rectangle, and rhombus are all types of parallelograms.
• A parallelogram is a trapezoid (American) or trapezium (British). However, a parallelogram is not an American trapezium.
• Similarly, a British irregular quadrilateral is not a parallelogram.
• A kite is not necessarily a parallelogram. However, a rhombus is a type of kite and is also a parallelogram.
• Both a square and a rhombus are types of quadrilaterals that have four congruent sides.

### Perimeter and Area Formulas

Each quadrilateral shape has its own perimeter and area formula:

1. Square:
• Perimeter = 4a (where a = length of a side)
• Area = a² (where a = length of a side)
2. Rectangle:
• Perimeter = 2(l + w) (where l = length and w = width)
• Area = l * w (where l = length and w = width)
3. Rhombus (Rhomb or Diamond):
• Perimeter = 4a (where a = length of a side)
• Area = d₁d₂ / 2 (where d₁ and d₂ are the lengths of the diagonals)
4. Parallelogram:
• Perimeter = 2(l + w) (where l = length and w = width)
• Area = b * h (where b = base and h = height)
5. Trapezoid (American) / Trapezium (British):
• Perimeter = a + b + c + d (where a, b, c, and d are the lengths of the sides)
• Area = (a + b) / 2 * h (where a and b are the lengths of the parallel sides and h is the height)
6. Trapezium (American) / Irregular Quadrilateral (British):
• Perimeter = a + b + c + d (where a, b, c, and d are the lengths of the sides)
• Area: Depending on the information available, there are different methods for calculating area. One common method for irregular quadrilaterals is dividing them into triangles and adding the areas of those triangles.
7. Kite:
• Perimeter = 2(a + b) (where a and b are the lengths of the different sides)
• Area = d₁d₂ / 2 (where d₁ and d₂ are the lengths of the diagonals)

The distinction between convex and concave quadrilaterals lies in their interior angles and the relative positioning of their vertices.

1. Convex Quadrilaterals: These are quadrilaterals in which all the interior angles are less than 180°. Another key characteristic is that for any two points within the shape, the line segment that connects them is also be entirely within the shape. All the types of quadrilaterals we discussed earlier (square, rectangle, rhombus, parallelogram, trapezoid/trapezium, kite) are examples of convex quadrilaterals.
2. Concave Quadrilaterals: These are quadrilaterals in which at least one interior angle is more than 180°. This forms a ‘dent’ or ‘cave’ in the shape (which is why it’s called ‘concave’). For some pairs of points within the shape, the line segment that connects them is not entirely within the shape. Concave quadrilaterals are also known as re-entrant quadrilaterals.

It’s important to note that the sum of interior angles in both convex and concave quadrilaterals is always 360° since they both have four sides. The distinction lies in the measure of individual angles and how their vertices are arranged.

Quadrilaterals, four-sided polygons, are an important concept in geometry due to their variety and ubiquity. They serve as a bridge between simpler shapes, like triangles, and more complex polygons. Here’s a detailed explanation of their importance:

1. Basic geometry understanding: Understanding the properties of quadrilaterals is a key part of learning about two-dimensional shapes. This includes understanding their angles, sides, diagonals, and area.
2. Variety of types: There are several types of quadrilaterals, each with their own unique properties. For example, rectangles have four right angles, parallelograms have opposite sides that are equal in length, and trapezoids have one pair of parallel sides. Understanding these varieties enriches one’s comprehension of geometrical shapes and their properties.
3. Foundational to complex concepts: The principles learned from quadrilaterals apply to more complex shapes and principles. For example, any polygon divides into triangles, but quadrilaterals provide a simpler step up in complexity from triangles that prepares students for dealing with polygons that have even more sides.
4. Practical applications: Quadrilaterals are common in everyday life and various fields such as architecture, design, engineering, and computer graphics. For instance, rectangles are important in the design of buildings and furniture. In computer graphics, meshes consisting of quadrilaterals (usually rectangles) model complex shapes.
5. Analytical skills: Studying the properties of quadrilaterals also develops deductive reasoning and problem-solving skills. For instance, if a student knows that the opposite angles of a parallelogram are equal, they deduce the measure of missing angles in a given problem.

1. Problem: A rectangle has a length of 12 cm and a width of 5 cm. What is the area and the perimeter of the rectangle
Solution:
• The area of a rectangle is found by multiplying the length by the width, so area = length x width = 12 cm x 5 cm = 60 cm².
• The perimeter of a rectangle is found by adding up all its sides, so perimeter = 2(length + width) = 2(12 cm + 5 cm) = 2(17 cm) = 34 cm.
2. Problem: A parallelogram has a base of 8 cm and a height of 6 cm. What is the area of the parallelogram?
Solution: The area of a parallelogram is the base multiplied by the height, so area = base x height = 8 cm x 6 cm = 48 cm².
3. Problem: A rhombus has diagonals of lengths 10 cm and 6 cm. What is the area of the rhombus?
Solution: Find the area of a rhombus by multiplying the lengths of the diagonals and then dividing by 2, so area = (d1 x d2) / 2 = (10 cm x 6 cm) / 2 = 30 cm².
4. Problem: The three angles of a quadrilateral are 85°, 95°, and 100°. Find the measure of the fourth angle.
Solution: In any quadrilateral, the sum of all interior angles is 360°. To find the fourth angle, we subtract the sum of the known angles from 360°. fourth angle = 360° – (85° + 95° + 100°) = 360° – 280° = 80°.
5. Problem: In a square, the length of one side is 7 cm. Find the perimeter of the square.
Solution: In a square, all sides are equal. Therefore, the perimeter is four times the length of one side. perimeter = 4 * side = 4 * 7 cm = 28 cm.
6. Problem: One angle in a parallelogram is 120°. Find the measure of the adjacent and opposite angles.
Solution: In a parallelogram, consecutive angles are supplementary (add up to 180°) and opposite angles are equal.
• The measure of the adjacent angle = 180° – 120° = 60° (because consecutive angles are supplementary).
• The measure of the opposite angle = 120° (because opposite angles are equal).

### References

• Alsina, Claudi; Nelsen, Roger (2010). Charming Proofs : A Journey Into Elegant Mathematics. Mathematical Association of America. ISBN 978-0-88385-348-1.
• Beauregard, R. A. (2009). “Diametric Quadrilaterals with Two Equal Sides”. College Mathematics Journal. 40 (1): 17–21. doi:10.1080/07468342.2009.11922331
• Hartshorne, R. (2005). Geometry: Euclid and Beyond. Springer. ISBN 978-1-4419-3145-0.
• Jobbings, A. K. (1997). “Quadric Quadrilaterals”. The Mathematical Gazette. 81 (491): 220–224. doi:10.2307/3619199
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