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# 2D Shapes

Mathematics is a vast subject. Geometry is one of the most important and frequently-applied branch of mathematics. Geometry is the study of shapes. It is concerned with the questions related to size, shape, properties and relative position of objects. A geometer is said to be a mathematician who works in geometry. Geometrical shapes are predefined objects each one of which share a certain set of properties belonging to that particular shape. There are different types of geometrical shapes. The most common are two dimensional and three dimensional shapes. Sometimes, even higher dimensional shapes are studied in geometry.

In this article, we are going to learn about two-dimensional geometrical shapes. The 2D shapes are the shapes that have only two dimensions such as length and breadth. So, let us go ahead and get introduced to 2D geometrical shape.

## Definition

The word "2D shape" is referred to "two-dimensional shape". Thus, a 2D shape is a geometrical figure that has two dimensions which are length and width. The two-dimensional shapes have no thickness. The concept of two dimensions can be thought as a flat surface on which one is able to move anywhere. This flat surface is known as a plane. A two dimensional shape is something that always lies on a sheet of paper. A 2D shape has no height, therefore it does not fall above the piece of paper. These shapes are also known as plane shapes or plane figures.
These shapes have areas but no volume.

## List

There are loads of two dimensional shapes around us. We may find 2D shapes almost everywhere we look around. In mathematics, it is practically impossible to list all the 2D shapes because we can name mostly every two dimensional shape we imagine.

The list of most often used 2-D shapes in geometry is given below:
Circular shapes:
1) Circle
2) Semicircle
3) Oval or ellipse

Three-Sided Shape:
Triangle

Quadrilaterals or Four-Sided Shapes:
1) Square
2) Rectangle
3) Parallelogram
4) Rhombus
5) Trapezium
6) Kite

Polygon:
1) Pentagon
2) Hexagon
3) Heptagon
4) Octagon
5) Nonagon
6) Decagon etc

## Properties

The basic properties of some two-dimensional shapes are illustrated below:

Circle:
A circle is a shape each point on which is equidistant from a fixed point called center and this fixed distance is termed as radius.
Area = $\pi r^{2}$
Circumference = 2 $\pi$ r

Semicircle
The half of a circle is termed as a semicircle.
Area = $\frac{1}{2}$ $\pi r^{2}$

Circumference = $\pi$ r
Where, r is radius.

3) Oval or ellipse
If the two perpendicular diameters of a circle are different, i.e. one diameter is larger than its perpendicular diameter, then the oval shape is formed. It may also be called as an ellipse.

4) Triangle
A triangle a plane figure that is bounded by three straight lines.
Area = $\frac{1}{2}$ x base x height
Also,
Area = $\sqrt{s(s-a)(s-b)(s-c)}$

Where, a, b and c are sides of a triangle and s = semi perimeter = $\frac{a+b+c}{2}$

5) Square

A square has four equal sides. It has four equal angles which measure 90$^{\circ}$.
Area = (side)$^{2}$
Perimeter = 4 x side

6) Rectangle
A rectangle is four sided 2-d shape which has two opposite sides equal. It also has four equal angles measuring 90$^{\circ}$.
Area = Length x breadth
Perimeter = 2 x (length + breadth)

7) Parallelogram
A parallelogram does have four sides. The pair of opposite sides are equal and parallel to each other. Opposite angles are also equal.
Area = Base (one side) x Height

8) Rhombus
A four-sided plane figure with all sides equal and opposite sides parallel. Its diagonals bisect each other at right angles
Area = $\frac{1}{2}$ $d_{1} d_{2}$

Where, $d_{1}$ and $d_{2}$ are the length of diagonals.

(5) Trapezium
Trapezium or trapezoid is a plane figure having one pair of opposite sides parallel or other pair is non parallel.
Area = $\frac{1}{2}$ x sum of parallel sides x height(6) Kite
Kite a two-dimensional geometrical figure which has two pairs of adjacent sides equal in length. Its diagonals intersect at right angles. If in case a kite has all four sides equal, then it becomes a rhombus.

## Examples

The examples based on two-dimensional shapes are as follows:
Example 1: Find the area and circumference of a circle whose radius is 7 cm.

Solution : Given that r = 7 cm
Area of circle = $\pi r^{2}$

= $\frac{22}{7}$ $\times$ 7 $\times$ 7
= 22 x 7
= 154

Circumference = 2 $\pi$ r
= 2 $\times$ $\frac{22}{7}$ $\times$ 7
= 2 x 22
= 44
Area and circumference of circle are 154 cm$^{2}$ and 44 cm respectively.

Example 2: What would be the area of a square whose perimeter is 24 mm.

Solution: Perimeter of square = 4 x side
24 = 4 x side
side = 6

Area = side$^{2}$
= 6$^{2}$
= 36
Area of square  is 36 mm$^{2}$

Example 3: Calculate the cost of fencing a rectangular field having dimensions of 15 meter and 10 meter at a rate of $\$$30 per meter. Solution: Fencing is done at the boundary of field. So its perimeter should be calculated. Perimeter of the rectangular field = 2 (length + breadth) = 2(15 + 10) = 50 Cost of fencing per meter = \$$ 30 Cost of fencing 50 meter =$\$$30 x 50 Cost of fencing per meter = \$$ 150

Example 4: Find the area of a triangle whose sides are 3 cm, 4 cm and 5 cm.

Solution: a = 3 cm, b = 4 cm and c = 5 cm

s = $\frac{a+b+c}{2}$

= $\frac{a+b+c}{2}$

= $\frac{3+4+5}{2}$

= $\frac{12}{2}$

= 6
Area = $\sqrt{s(s-a)(s-b)(s-c)}$

= $\sqrt{6(6-3)(6-4)(6-5)}$

= $\sqrt{6(3)(2)(1)}$

= $\sqrt{36}$

= 6
Area of given triangle is 6 cm$^{2}$.
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