What are sides in math?

In geometry, a side is one of the line segments that makes up the border of a two-dimensional figure such as a triangle, square, rectangle, or other polygon. A side is made up of two endpoints joined by a straight or curved line. Some key points about sides in math:

Table of Contents

Types of Sides

There are a few main types of sides:

Straight sides – These are sides made up of straight line segments. Squares, rectangles, and triangles have straight sides.
Curved sides – These are sides made up of curved line segments. Some polygons like octagons can have curved sides.
Line segments – These are sides made up of an undefined length. Line segments have two endpoints but the length between them is not specified.

Side lengths – These refer to the specific length of a side. For example, one side of a rectangle may have a side length of 5 cm.

Sides and Shapes

Different shapes have different numbers and configurations of sides:

Triangles – Triangles have 3 straight sides

Quadrilaterals – Quadrilaterals (four-sided polygons) have 4 straight sides. These include squares, rectangles, rhombuses, and trapezoids.
Pentagons – Pentagons have 5 straight sides
Hexagons – Hexagons have 6 straight sides

Heptagons – Heptagons have 7 straight sides
Octagons – Octagons have 8 straight sides
Nonagons – Nonagons have 9 straight sides

Decagons – Decagons have 10 straight sides

In general, an n-gon is a polygon with n sides. As the number of sides increases, polygons begin to look more circular.

Parts of a Side

The main parts of a side are:

Endpoints – These are the two points at either end of a side where two line segments meet. Every side has two endpoints.
Length – The length of a side is the distance between the endpoints. This can be measured in units.
Midpoint – This is the point on the side that divides it into two equal halves. It is halfway between the endpoints.

Side Relationships

There are some important relationships between the sides of polygons:

Adjacent sides – These are sides that share a common endpoint in a polygon. For example, in a rectangle, each side has one adjacent side.
Opposite sides – These are sides positioned across from one another in a polygon. Opposite sides are equal in length and parallel in rectangles, parallelograms, and rhombuses.

Perpendicular sides – These are sides that intersect at right (90 degree) angles. In rectangles and squares, opposite sides are perpendicular.
Parallel sides – These are sides that never intersect because they are always equidistant. Opposite sides in parallelograms and trapezoids are parallel.

Side Measurements

Some ways to measure the sides of polygons include:

Length – Using a ruler to directly measure the distance between endpoints of a side.
Coordinates – Plotting the endpoints on a coordinate plane and using the distance formula to calculate length.
Trigonometry – Using trigonometric ratios like sine, cosine, and tangent to find side lengths.

Area/Perimeter formulas – Using formulas for perimeter and area that include the side lengths of a shape.

Sides in 3D Shapes

Three-dimensional shapes like cubes, prisms, pyramids, cylinders, cones, and spheres do not technically have sides. However, their two-dimensional faces can be considered analogous to sides:

Cubes – Cubes have 6 square faces that could be considered sides

Rectangular Prisms – Rectangular prisms have 6 rectangular faces that could be sides
Triangular Prism – Triangular prisms have 5 triangular faces that could be sides
Pyramids – Pyramids have triangular faces meeting at a vertex that could be considered sides

Cylinders – Cylinders have two circular bases and a curved lateral surface that could be considered sides
Cones – Cones have a circular base and curved lateral surface that could be considered sides
Spheres – Spheres do not have flat sides, but great circles on spheres could be considered analogous to sides

Properties of Sides

Some important properties and facts about the sides of polygons:

A polygon’s sides always join together so that no gaps exist between them
The sum of the measures of the interior angles of a polygon depends on the number of sides

The sum of the side lengths of any quadrilateral equals the sum of the lengths of its diagonals
Equilateral polygons have all equal side lengths
Isosceles polygons have at least two equal side lengths

In rectangles and parallelograms, opposite sides have equal lengths
In kites and isosceles trapezoids, adjacent sides have equal lengths
In regular polygons, all sides and angles are equal

Formulas Involving Sides

There are many important geometry formulas that involve the sides of polygons:

Perimeter of a polygon = Sum of the side lengths
Area of a rectangle = Length x Width (side lengths)

Area of a parallelogram = Base x Height
Area of a triangle = 1/2 (Base x Height)
Area of a trapezoid = 1/2 (a + b) x h (side lengths)

Distance formula = √(x2 – x1)2 + (y2 – y1)2 (uses side endpoints)
Pythagorean Theorem = a2 + b2 = c2 (relates triangle side lengths)
Trig ratios like sine, cosine, tangent use side lengths

These are just a few examples of how side lengths are incorporated into area, perimeter, and other geometry formulas.

Sides and Angles

There is an important relationship between the sides and angles of polygons:

As the number of sides increases in a regular polygon, each interior angle decreases

The sum of a polygon’s interior angles depends on the number of sides (Triangles = 180°, Quadrilaterals = 360°, Pentagons = 540°, etc)
In an equiangular polygon like a regular polygon, the sides and interior angles are equal
In a right triangle, the side opposite the right angle is the longest side (hypotenuse)

Isosceles triangles have two equal sides and two equal base angles

So the number of sides, side lengths, and angle measures are all closely related in geometry.

Using Sides in Proofs and Theorems

Sides play an important role in geometry proofs and theorems like:

Triangle sum theorem – The sum of the measures of the interior angles of a triangle equals 180°
Isosceles triangle theorem – If two sides of a triangle are equal, then the angles opposite those sides are also equal
Third angle theorem – If two angles of a triangle are known, the measure of the third angle can be determined

Pythagorean Theorem – In a right triangle, a^2 + b^2 = c^2, where a and b are the legs and c is the hypotenuse
Parallelogram side properties – Opposite sides are equal and parallel. Consecutive sides are equal.

Side relationships and lengths are often used as givens or to derive conclusions in geometric proofs.

Conclusion

In summary, the main points about sides in geometry are:

Sides are line segments that form the boundaries of polygons
Triangles have 3 sides, quadrilaterals 4 sides, pentagons 5 sides, etc

Side length is the distance between the endpoints of a side
Adjacent and opposite sides have key relationships in polygons
Side lengths are used in formulas for area, perimeter, distance, and trigonometry

Sides and angles are interconnected, based on the properties of the polygon
Understanding sides is key for geometric proofs and theorems

Sides are a fundamental building block of geometric shapes and analysis. Understanding the properties and behavior of sides is essential for working with and proving relationships in geometry.