Shapes

Shapes are everywhere we look; they are the basis of everything we see and touch in everyday life. From the spherical wheels that propel our cars to the square screens we communicate with, shapes determine form and space. In mathematics, the study of shapes is more than just their identification—it's delving into their properties, their dimensions, and their relationships. So let's see all these in this article.

1.0Shapes Definition

In mathematics, a shape is the boundary of a figure, object, or surface and is the edge that distinguishes the figure from exterior space. Shapes may be regular when all sides and angles are equal, or irregular when the angles and sides are not equal. Shapes may be divided into various types, each type having some unique properties and features.

Naming and describing shapes help students become proficient in mathematical problem-solving and investigating real-world objects. As students learn about shapes, they are acquiring skills to identify edges, locate an area, measure a perimeter, and understand how real-world objects maintain their shape, and whether the shape is stable or unstable.

2.0Shapes Types

In mathematics, shapes are categorised together based on their dimensions and properties. 

A. Open Shape:

A shape or figure whose line segments and/or curves do not meet is known as an open shape in geometry. They don't begin and finish at the same time.

B. Closed Shape:  A closed shape in geometry is an enclosed figure or shape with connected or meeting line segments and/or curves. They begin and finish at the same location.

Two-dimensional shapes and three-dimensional shapes are the two main categories into which closed geometric shapes can be further divided.

C. Two-Dimensional (2D) Shapes:

These are two-dimensional shapes, or shapes that have length and breadth but not thickness. These shapes exist on a plane, and we tend to discuss them in terms of perimeter (the circumference around the outside of the shape) and area (the space contained in the shape). Some of the most common 2D shapes are:

1. Triangle – A Triangle is a polygon with three sides and three angles. Triangles can be equilateral (all sides equal), isosceles (two sides equal), scalene (all sides are different), and right-angled (one angle is 90). Triangles are applied in bridges, roofs, and other structures to make them stronger.

• Perimeter = All the sides added up.
• Area = ½ × base × height

2. Square – A Square is a four-sided polygon (quadrilateral) in which all sides are equal and have right angles. You see squares in tiles, chessboards, and windows.  

• Perimeter = 4 × side
• Area = side2

3. Rectangle – It is a four-sided polygon (quadrilateral) that has opposite sides that are equal and has right angles. You see books, doors, and screens in the shape of rectangles.

• Perimeter = 2 × (length + breadth)
• Area = length × breadth

4. Circle – a curved, rounded shape in which all the points on the shape are the same distance from the centre of the circle. Circles appear in wheels, clocks, and coins.

• Circumference = 2πr
• Area = πr2

5. Trapezium (or Trapezoid) – A four-sided figure with one set of parallel sides. Trapeziums appear in standard examples of ramps, bridge construction, and other architectural ornamentation details.

• Perimeter = sum of the sides
• Area = ½ × (sum of parallel sides) × height

6. Parallelogram – A four-sided figure with opposite sides equal in length and parallel. Parallelograms appear in slanted windows, roof panels, and certain tile flooring.

• Perimeter = 2 × (base + side)
• Area = base × height

D. Three-Dimensional (3D) Shapes:

Three-dimensional figures are shapes having length, breadth, and height, due to which they occupy space. In contrast to 2D shapes, 3D shapes can hold volume, and you can grasp them, touch them, and observe them from various sides. Some of the common 3D shapes include:

1. Cube – A 3D shape that consists of six square faces with 12 edges and 8 vertices/Corners. Cubes occur in most everyday household or game objects, such as dice, ice cubes, and building blocks.

• Surface Area = 6a²
• Volume = a³

2. Cuboid – A rectangular box shape that consists of six rectangular faces (2 sets of opposite rectangular faces). Bricks or books/ paper are some examples of cuboids.

• Surface Area = 2(lb + bh + hl)
• Volume = l × b × h

3. Sphere – A solid shape that is completely round with no edges or vertices; a part involved in a sphere-type solid shape is a football, globe, or marble.

• Surface Area = 4πr²
• Volume = (4/3)πr³

4. Cylinder – A solid that has two parallel circular bases joined together by a curved surface. Examples of cylinders include Can-type objects. Pipes and tanks, for instance, are cylinders too.

• Surface Area = 2πr(h + r)
• Volume = πr²h

5. Cone – A solid with a circular base tapering smoothly to a single point known as the apex. Ice cream cones, traffic cones, and pine cones (natural cones) are examples.

• Surface Area = πr(l + r), l = slant height
• Volume = (1/3)πr²h

3.0Transforming Shapes

In mathematics, transforming shapes is considered a change in position, size, or motion of that shape, but it does not affect the nature of the shape. Transformations are an essential aspect of geometry since understanding how shapes will move and scale is important.

Types of Transformations

• Translation: Moving a shape from one location to another without rotating or resizing it.
• Reflection: Flipping a shape over a line (a mirror line) to produce a mirror image.
• Rotation: Rotating a shape around a fixed point, which is usually measured in degrees.
• Dilation (Enlargement/Reduction): Resizing a shape to be bigger or smaller. (the proportions remain the same)