Maths

Properties of Polygons

Polygons.

Polygons are closed plane figures bounded by three or more line segments. In the world of geometry, polygons abound. The term refers to a multisided geometric form in the plane. The number of angles in a polygon always equals the number of sides. Polygons are named to indicate the number of their sides or number of noncollinear points present in the polygon .

A square is a special type of polygon, as are triangles, parallelograms, and octagons. The prefix of the term, poly comes from the Greek word for “many,” and the root word gon comes from the Greek word for “angle”.

Classification.

A regular polygon is one whose sides and interior angles are congruent. Regular polygons can be inscribed by a circle such that the circle is tangent to the sides at the centers, and circumscribed by a circle such that the sides form chords of the circle. Regular polygons are named to indicate the number of their sides or number of vertices present in the figure. thus, a hexagon has six sides, while a decagon has ten sides. Examples of regular polygons also include the familiar square and octagon.

Not all polygons are regular or symmetric. Polygons for which all interior angles are less than 180°are called convex. Polygons with one or more interior angles greater than 180° are called concave.

The most common image of a polygon is of a multisided perimeter enclosing a single, uninterrupted area. In reality, the sides of a polygon can intersect to form multiple, distinct areas. Such a polygon is classified as reflex.

Angles.

In a polygon, the line running between nonadja-cent points is known as a diagonal. The diagonals drawn from a single vertex to the remaining vertices in an n-sided polygon will divide the figure into n–2 triangles. The sum of the interior angles of a convex polygon is then just (n–2)×180.

If the side of a polygon is extended past the intersecting adjacent side, it defines the exterior angle of the vertex. Each vertex of a convex polygon has two possible exterior angles, defined by the continuation of each of the sides. These two angles are congruent.

KEY TERMS.

▪️Angle— A geometric figure created by two lines drawn from the same point.

▪️Concave— A polygon whose at least one angle is larger than the straight angle (180°).

▪️Convex— A polygon whose all angles are less than the straight angle (180°).

▪️Diagonal— The line which links-connects any two non-adjacent vertices.

▪️Equiangular— A polygon is equiangular if all of its angles are identical.

▪️Equilateral— A polygon is equilateral if all the sides are equal in length.

▪️Parallelogram— A rectangle with both pair of sides parallel.

▪️Perimeter— The sum of the length of all sides.

▪️Rectangle— A parallelogram in which all angles are right angles.

▪️Reflex polygon— A polygon in which two nonadja-cent sides intersect.

▪️Regular polygon— An equilateral, equiangular polygon.

▪️Rhombus— A parallelogram whose adjacent sides are equal.

▪️Square— A four-sided shape whose sides are equal.

▪️Vertex— The point at which the two sides of an angle meet.

however, so the exterior angle of a polygon is defined as one of the two angles. The sum of the exterior angles of any convex polygon is equal to 360°.

Number of Sides.Polygons are usually defined by the number of sides that they have.

1. Three-Sided Polygons: Triangles.

A three-sided polygon is a triangle. There are several different types of triangle (see diagram), including:

▪️Equilateral – all the sides are equal lengths, and all the internal angles are 60°.

▪️Isosceles – has two equal sides, with the third one a different length. Two of the internal angles are equal.

▪️Scalene – all three sides, and all three internal angles, are different.

Triangles can also be described in terms of their internal angles (see our page on Angles for more about naming angles). The internal angles of a triangle always add up to 180°.

A triangle with only acute internal angles is called an acute (or acute-angled) triangle. One with one obtuse angle and two acute angles is called obtuse (obtuse-angled), and one with a right angle is known as right-angled.

Each of these will also be either equilateral, isosceles or scalene.

2. Four-Sided Polygons – Quadrilaterals.

Four-sided polygons are usually referred to as quadrilaterals, quadrangles or sometimes tetragons. In geometry the term quadrilateral is commonly used.

The term quadrangle is often used to describe a rectangular enclosed outdoor space, for example ‘the freshers assembled in the college quadrangle’. The term tetragon is consistent with polygon, pentagon etc. You may come across it occasionally, but it is not commonly used in practice.The family of quadrilaterals includes the square, rectangle, rhombus and other parallelograms, trapezium/trapezoid and kite.

The internal angles of all quadrilaterals add up to 360°.

▪️Square: Four sides of equal length, four internal right angles.

▪️Rectangle: Four internal right angles, opposite sides of equal length.

▪️Parallelogram: Opposite sides are parallel, opposite sides are equal in length, opposite angles are equal.

▪️Rhombus: A special type of parallelogram in which all four sides are the same length, like a square that has been squashed sideways.

▪️Trapezium (or trapezoid): Two sides are parallel, but the other two sides are not. Side lengths and angles are not equal.

Isosceles Trapezium (or trapezoid): Two sides are parallel and base angles are equal, meaning that non-parallel sides are also equal in length.

▪️Kite: Two pairs of adjacent sides are of equal length; the shape has an axis of symmetry.

▪️Irregular Quadrilateral: a four-sided shape where no sides are equal in length and no internal angles are the same. All internal angles still add up to 360°, as with all other regular quadrilaterals.

3. More than Four Sides.

A five-sided shape is called a pentagon.A six-sided shape is a hexagon, a seven-sided shape a heptagon, while an octagon has eight sides…

There are names for many different types of polygons, and usually the number of sides is more important than the name of the shape.There are two main types of polygon – regular and irregular.

A regular polygon has equal length sides with equal angles between each side. Any other polygon is an irregular polygon, which by definition has unequal length sides and unequal angles between sides.

Circles and shapes that include curves are not polygons – a polygon, by definition, is made up of straight lines.

Angles Between Sides.

The angles between the sides of shapes are important when defining and working with polygons.

▪️The Length of the Sides.

As well as the number of sides and the angles between sides, the length of each side of shapes is also important.

The length of the sides of a plane shape enables you to calculate the shape’s perimeter (the distance around the outside of the shape) and area (the amount of space inside the shape).

If your shape is a regular polygon (such as a square) then it is only necessary to measure one side as, by definition, the other sides of a regular polygon are the same length. It is common to use tick marks to show that all sides are an equal length.

In the example of the rectangle we needed to measure two sides – the two unmeasured sides are equal to the two measured sides.

It is common for some dimensions not to be shown for more complex shapes. In such cases missing dimensions can be calculated.

Bringing All the Information Together.The simplest and most basic polygon for the purposes of calculating area is the quadrilateral. To obtain the area, you simply multiple length by vertical height.

For parallelograms, note that vertical height is NOT the length of the sloping side, but the vertical distance between the two horizontal lines.

This is because a parallelogram is essentially a rectangle with a triangle cut off one end and pasted onto the other.

it onto the other end, the rectangle becomes a parallelogram.The area is length (the top horizontal line) multiplied by height, the vertical distance between the two horizontal lines.

To work out the area of a triangle, you multiple length by vertical height (that is, the vertical height from the bottom line to the top point), and halve it. This is essentially because a triangle is half a rectangle.

To calculate the area of any regular polygon, the easiest way is to divide it into triangles, and use the formula for the area of a triangle.

You can also work out the area of any regular polygon using trigonometry, but that’s rather more complicated.

Reference:

“Polygon”, www.encyclopedia.com

Properties of Polygons”, www.skillsyouneed.com

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