Introduction

 

What is the Perimeter of a Triangle?

 

The perimeter of a triangle is the total length of its boundary, calculated by adding up the lengths of its three sides. It’s a simple yet vital concept in mensuration and geometry that lays the foundation for more advanced topics.

 

Introduction

 

The concept of perimeter is fundamental in geometry and mensuration. Whether you are planning a DIY project, fencing your garden, or helping your child with math homework, understanding of how to calculate the perimeter is an essential skill. Here, in this article, we will break down the process of finding the perimeter, cover formulas for various shapes, and provide practical examples to help you get a solid grasp on this concept.

Pyramid

Pyramids are three dimensional figure like a prism. It is a solid with polygonal base and triangular lateral faces meeting at a common point called vertex or apex. This figure has fascinated human beings from the ancient times. Pyramid of Egypt are one of then seven wonders of the world. These pyramids were built during the period 3000 - 2000 B.C.

Volume of a Cone

Volume of a cone is given by the one third of the product of its area of base and height i.e. volume of cone =  πr²h/3. This formula can be understood and derived by the following activity:

1.     Take a hollow cylindrical jar of radius r and height h, whose volume is πr²h.

2.     Take a hollow cone which has the same height h and same radius r.

3.     Fill the cone with water and pour into the jar. And repeat it.

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In this activity, the cylindrical jar will be full filled with water when 3 cones of full water is pour into it. It shows that,

3 × Volume of cone = Volumeof the cylinder

or,       Volume of cone = 1/3 × Volume of the cylinder

                                     = 1/3 × πr²h

                                     = πr²h/3

∴ Volume of a cone = πr²h/3

Workout Examples

Example 1: Find the volume of a cone whose radius = 4 cm and height = 21 cm.

Solution: Here,

                           Radius (r) = 4 cm

                           Height (h) = 21 cm

                We know,

                           Volume of a cone = πr²h/3

                                                          = 1/3 × 22/7 × 4² × 21

                                                          = 352 cm³

                Thus, the volume of the cone is 352 cm³.

Example 2: The vertical height of a right circular cone is three times its diameter and its volume is 54π cm³. Find its height.

Solution: Here,

                           Let, the diameter of cone be d.   ∴  d = 2r

                           Then, height (h) = 3d = 3 × 2r = 6r

                           Given, volume of cone = 54π cm³

                i.e.      πr²h/3 = 54π

                or,      r²h/3 = 54

                or,      r² × 6r = 162

                or,      6r³ = 162

                or,      r³ = 162/6

                or,      r³ = 27

                or,      r³ = 3³

                or,      r = 3 cm

                Thus, height (h) = 6r = 6 × 3 cm = 18 cm.

Example 3: Calculate the volume of adjoining solid given in the figure.

Solution: Here,

                           The given figure is a combined solid of two cones.

                           Radius of both the cones (r) = 42 cm

                           Total height of the solid = 180 cm

                           Let the height of upper cone be h₁ and the height of lower cone be h₂.

                           So, h₁ + h₂ = 180 cm

                We know,

                           Volume of the solid = V(upper cone) + V(lower cone)

                                                             = πr²h₁/3 + πr²h₂/3

                                                             = πr²(h₁ + h₂)/3

                                                             = 1/3 × 22/7 × 42² × 180     [∵  h₁ + h₂ = 180 cm]

                                                             = 332640 cm³

                Thus, the volume of the solid is 332640 cm³.

You can comment your questions or problems regarding the volume of a cone here.

Surface Area of a Cone

 

A cone is a pyramid with a circular base. A solid cone has two types of surfaces, one is the curved surface around the cone and another is the plane surface of its circular base.

Cone

A cone is a pyramid with circular base. Take a circular piece of paper with centre O. cut off the sector APB and join the edges AO and BO. In this way a pyramid with a circular base is formed which is called a circular pyramid or a cone.

A cone has a curved surface and a circular base. In the figure given below, O is the centre of the circular base, OA is the radius (r), OP is the vertical height (h) of the cone and PA is the slant height (l).

In right angled triangle AOP, using Pythagoras Theorem,

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Surface Area and Volume of Cone

A cone is formed from the sector of a circle. So its curved surface area is the surface of the sector.

1.  The curved surface area of a cone = Area of the sector = πrl, where r is the radius of the circular base and l is the slant height.

2.  The total surface area of a cone = curved surface area + area of circular base

                                                        = πrl + πr²

                                                        = πr (r + l)

3.  Volume of the cone = volume of the circular pyramid

                                        = 1/3 of area of circular base × height

                                        = 1/3 × πr² × h

                                        = πr²h/3

Workout Examples

Example 1: Calculate the curved sirface area, total surface area and volume of the given cone.

Solution: Here,

                                Height of cone (h) = 8 cm

                                Slant height (l) = 10 cm 

                Now,

                                Curved surface area (CSA) = πrl

                                                                               = 22/7 × 6 × 10

                                                                               = 188.57 cm²

                                Total surface area (TSA) = πr(r + l)

                                                                           = 22/7 × 6 (6 + 10)

                                                                           = 301.71 cm²

                                Volume of cone (V) = πr²h/3

                      = 1/3 × 22/7 × 6² × 8

                      = 22/21 × 36 × 8

                      = 301.71 cm³

Example 2: If the total surface area of a cone is 704 cm² and radius of its base is 7 cm, find the volume of the cone.

Solution: Here,

                           Radius of cone (r) = 7 cm

                           Total surface area of cone = 704 cm²

                i.e.      πr (r + l) = 704

                or,      22/7 × 7 (7 + l) = 704

                or,      22 (7 + l) = 704

                or,      154 + 22l = 704

                or,      22l = 704 – 154

                or,      l = 550/22

                or,      l = 25 cm

Now,

                Volume of cone (V) = πr²h/3

       = 1/3 × 22/7 × 7² × 24

       = 1232 cm³

Example 3: If the volume of the given cone is 1848 cm³, and the radius of its base is 14 cm, find its curved surface area.

Solution: Here,

                           Radius of cone (r) = 14 cm

                           Volume of cone = 1848 cm³

                i.e.      πr²h/3  = 1848

                or,      1/3 × 22/7 × 14² × h = 1848

                or,      22/21 × 196 × h = 1848

                or,      h = 1848 × 21/4312

                or,      h = 9 cm               

Now,

                Curved surface area of cone (CSA) = πrl

                                                                             = 22/7 × 14 × 16.64

                                                                             = 732.16 cm²

You can comment your questions or problems regarding the surface area and volume of cone here.

Hemisphere

Hemisphere is half of a sphere. When a sphere is divided equally into two parts, each part is called hemisphere. It has two types of surfaces, one is circular face (great circle) and another is curved surface. The radius of hemisphere is equal to the radius of sphere.

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Total Surface Area of a Hemisphere

Total surface area of a hemisphere is the sum of its area of curved surface and the area of circular face (great circle). Area of curved surface of a hemisphere is the half of the total surface area of a sphere.

So, area of curved surface of a hemisphere = ½ of total surface area of the sphere

                                                                      = ½ × 4πr²

                                                                      = 2πr²

And, area of circular face (great circle) = πr²

Now,

Total surface area of hemisphere = area of curved surface + area of circular face

                                                      = 2πr² + πr²

                                                      = 3πr²

∴   Total surface of hemisphere = 3πr²

Volume of a Hemisphere

To find the volume of a hemisphere, we can do the following activity:

-    Take a cylindrical vessel circumscribing a plastic spherical ball as given in the figure above.

-    Take out the spherical ball from the cylinder and cut it into two equal parts. In this way we will get two hemispheres of plastic ball.

-    Fill a hemisphere with rice or sand and pour it into the cylinder. And repeat it again.

-    Then we will find that the cylinder will be full filled by 3 hemispheres.

Thus, the volume of 3 hemisphere = volume of 1 cylinder  

                                                        = πr²h    [∵ volume of cylinder = πr²h]

                                                        = πr² × d    [∵ h = d]

                                                        = πr² × 2r    [∵ d = 2r]

                                                        = 2πr³

          ∴   Volume of 1 hemisphere = 2πr³/3

Thus, the volume of a hemisphere = 2πr³/3

Workout Examples

Example 1: Find the total surface area and the volume of a hemisphere of radius 3.5cm.

Solution: Here, radius of hemisphere (r) = 3.5cm

                Now,

                         Total surface area of hemisphere = 3πr²

                                                                                  = 3 × 22/7 × (3.5)²

                                                                                  = 115.50cm²

                         Volume of hemisphere = 2πr³/3

                                                                 = 2/3 × 22/7 × (3.5)³

                                                                 = 89.83cm³

                Thus, total surface area is 115.50cm² and the volume is 89.83cm³.

Example 2: The circumference of the edge of a hemispherical bowl is 132cm. Find the capacity of the bowl.

Solution: Here, circumference = 132cm

                i.e.      2πr = 132

                or,      2 × 22/7 × r = 132

                or,      r = 132 × 7/44

                or,      r = 21cm

                Now,

                          The capacity of the bowl = 2πr³/3

                                                                     = 2/3 × 22/7 × (21)³

                                                                     = 19404cm³  

         

                Thus, the capacity of the bawl is 19404cm³.

Example 3: Find the total surface area and the volume of the given combined solid figure.

Solution: Here, the given combined solid figure is a cylinder and a hemisphere,

    radius (r) = 7cm

    height (h) = 10cm

                Now,

                         Total surface area of solid = 2πr² + 2πrh + πr²

                                                                       = 3πr² + 2πrh

                                                                       = 3 × 22/7 × (7)² + 2 × 22/7 × 7 × 10

                                                                       = 462 + 440

                                                                       = 902cm²

                         Volume of solid = πr²h + 2πr³/3

                                                     = 22/7 × 7² × 10 + 2/3 × 22/7 × 7³

                                                     = 1540 + 718.67

                                                     = 2258.67cm³

                Thus, total surface area is 902cm² and the volume is 2258.67cm³.

You can comment your questions or problems regarding the surface area and volume of hemisphere here.