Geometric Shapes

Geometrical Shapes are also categorized by the points that they join. The geometrical shapes that consist of three sides are referred to as triangles, whereas the geometrical shapes that consist of four sides are referred to as quadrilaterals. Quadrilaterals consist of many shapes like squares, rectangles, rhombus, trapezium, etc. Circles are shapes that don’t consist of any sides or vertices. 

Polygon is referred to any geometric shape which has straight sides and is completely closed and that shape has to be two-dimensional. A circle is not a polygon because it doesn’t consist of any sides. Polygons also consist of Geometric Shapes with many sides. For instance, a five-sided polygon is called a Pentagon. Similarly, six-sided, seven-sided, and eight-sided geometric shapes are known as Hexagon, Heptagon, and octagon respectively.

In this topic, we will look at multiple geometric shapes and how to calculate their physical attributes like area, perimeter, etc.

 Circle

As we already saw, a circle is a two-dimensional closed geometric shape that doesn’t have any sides or vertices.

R is known as the radius of the circle. It is simply the distance from its centre to any point which lies on the circumference of the circle.

D is the diameter of the circle.

D=2R

Area=πR2= ((π*D2)/4)

The perimeter (P) of a circle and the Circumference of a Circle(C) are the same things.

C=2πR=πD

Q. Find the Area and Perimeter of a circle with a radius of 2.

Soln. 

A=πR2=π*(22) = π*4 = 12.566

C=2πR=2*π*2=π*4 = 12.566

 Triangle

Triangle is a two-dimensional closed Geometrical Shape that consists of three vertices joined by three sides. The total sum of all the angles which the sides make with each other is equal to 180°.

To find the area of the triangle, we need to drop a perpendicular from one vertex onto the base. 

In this case Area (A) is given by:-

A= (1/2)*b*h

If we are given such a figure, there is another way to find out the Area of the Triangle.

A=√(s(s-a) (s-b) (s-c))

This formula is known as The Heron’s Formula

a= length of AB

b= length of BC

c= length of AC

s= (a+b+c)/2

The perimeter of the Triangle (P) is given by:-

P= (a+b+c)

Q. What is the Area and Perimeter of a triangle with sides 5,3 and 7.

Soln.

Using the Heron’s formula to find out the Area,

A=√(s(s-a) (s-b) (s-c))

s= (5+3+7)/2=15/2=7.5

So using this value to calculate the Area,

A=√ (7.5*(7.5-7)*(7.5-5)*(7.5-3))

A=√ (7.5*(0.5)*(2.5)*(4.5))

A=√ 42.188

A= 6.495

P=a+b+c=7+5+3=15 

Square

A Square is a two-dimensional Geometric Shape with four vertices joined by four sides. These four sides are all equal in length. Also, all the four angles which the sides make with each other are 90°.  That means the total sum of the angles which the sides make with each other is equal to 360°.  The Square belongs to the Quadrilateral Family.

The area of the square in the above figure will be given by:-

A= a * a

The perimeter of the square in the above figure will be given by:-

P= a+a+a+a = 4*a

 Q. Find the Area and Perimeter of a Square with a side equal to 10.

Soln.

Given a=10,

Area A= 10*10=100

Perimeter P= 4*10=40

 Rectangle

A rectangle like the square is also a two-dimensional closed geometric shape with four vertices joined by four sides.  But this is where the similarities end. Unlike the square (in which all the four sides are equal), two sides of a rectangle are equal in length to each other and the other two sides are equal in length to each other. These two pairs of sides are different to each other in length. All the angles which the sides make with each other are equal to 90°. That means that the sum of all the angles in a rectangle is equal to 360°.

Perimeter of a rectangle (P) is given by:-

P= l+b+l+b-2*l+2*b=2*(l+b)

Area of a rectangle (A) is given by:-

A= l*b 

Q. Find the Area and Perimeter of a Rectangle in which the length and breadth are equal to 5 and 2 respectively.

Soln.

P=2*(l+b) =2*(5+2) =2*7=14

A=l*b=5*2=10

Now we will look at some three-dimensional closed Geometric Shapes.

 Cube

A cube is a three-dimensional closed Geometric Shape consisting of eight vertices joined by twelve edges. A cube has six square faces in the three-dimension. All the angles which the edges make with each other are 90°

In three-dimensional figures, the physical attributes are different from that of two-dimensional figures. For instance, in three-dimensional figures we have Lateral Surface Area (LSA), Total Surface Area (TSA) and Volume (V).

For the cube in the above figure,

LSA=4*a*a=4*a2=4a2

TSA=6*a*a=6*a2=6a2

V=a3 

Q. For a cube with a side equal to 6, find the Lateral Surface Area(LSA), Total Surface Area(TSA) and Volume(V).

Soln.

LSA=4a2=4*(6)2=4*36=144

TSA=6a2=6*(6)2=6*36=216

V=63=216

 Conclusion

In the chapter, first, we have defined Geometric Shapes in brief. Then we moved into the Introduction part where we looked into categorising the Geometric Shapes. Then we looked into a few Geometric Shapes. We learnt how to calculate the Perimeter and Area of such shapes and we solved a few problems related to these shapes. We also looked at a special Geometric Shape called The Cube which is a three-dimesnsional figure.