Table of Contents

1. Introduction
2. Solid Shapes with Curved Faces
3. Solid Shapes with only Flat Faces
4. Formulas
5. Summary
6. Frequently Asked Questions (FAQs)

September 8, 2020,     

Reading Time: 5 mins       

Introduction

Objects, occupying space in the three dimensions, are known as having solid shapes. Cylinder, cone, sphere,prism, and hemisphere are examples of solids, which have curved shapes with or without a flat top and bottom or both. The cube, pyramid, and cuboid are also solids, which have only flat sides around and top and bottom.

The outer surfaces of these solids are exposed to the atmosphere. The total such area of the solid is the total of all its surface, both the curved and flat areas in solid having curved shapes. In the case of solids with flat sides, the total exposed area of a surface is the total of areas of all its sides including top and bottom. The area of each solid is calculated in square units. 

The inner space of the solids may be solid as a cube or as a road- roller or empty as a pot, measuring jar, drum, gas cylinder. Cube, cuboid, cone, etc. The empty interior space of these solids facilitates the use of them as containers. Thus, they have volumes. The volumes of individual solids can be calculated in cubic units according to their shapes. 

Solid Shapes

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📥 Solid Geometric Shapes

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Shapes

Faces

Vertex

Edges

2

1 1
 

1 0 1
 

6 8 12

6 8 12

3 0 2

5 5 8

5 6 9

Solid Shapes with Curved Faces

Cylinder

It has a round shape with flat faces on top and bottom.

Example – Pen holder

Consider a cylinder of radius r and height h. It can be considered to have a surface of rectangular height ‘h’ and width 2πr which is the circumference of the circular top and bottom end faces, as shown when it is opened and spread. 
Thus, the Total Surface Area of a cylinder = Area of curved surface + Areas of 2 circular faces
                 = 2πrh + 2πr2 = 2πr(h+r) sq. Units

Curved Surface Area of a Cylinder = 2πrh sq.units
Volume of a cylinder = πr2h cubic units.      

 Example
1. A cylindrical tank of radius 3.5m and height 6m is to be painted for the outer surface. The cost of painting is Rs. 5 per m2. Find the total cost of painting.

Answer:

Total Surface Area = Curved Surface Area + area of top and bottom faces
                 = 2πrh + 2πr2 = 2πr (r + h)
                 = 2 × 22/7 × 3.5 × (3.5 + 6)
                 = 22 × 9.5 = 209 sq.m.
Cost of painting at Rs.5 per sq.m = 209 × 5 = Rs.1045.

 

Cone

Cone has a circular top side and a curved surface starting from all points on the circumference of the circular flat face and converging uniformly to a point as the vertex with a constant length and slope around, as shown in the figure:

This can be considered as an area spread across the circumference of the top side, which will appear as below.

The Total Surface Area of a Cone= curved surface area + circular face area
                 = ½ 2πrl + πr2
                 = πrl + πr2 = πr (l + r) sq.units where l = length of slanting surface and r = radius of the top side.

Curved Surface Area of a Cone = πrl
Volume of a Cone = 1/3 × πr2h

Transparent cone isolated on white background Premium Vector

Example
Find the surface area of a cone of radius 14cms and height 15cms.

Answer:

Surface area of cone = curved area + area of top face (circular)
                 = πr(r + l)
r = 14cms, h = l = 15cms. 
Therefore, SA = 22/7 × 14 × (14 +15) = 44 × 29 = 1376 sq.cm.          

 

Sphere

  • A Sphere is a solid shape. It has only a curved surface throughout and no flat side.
  • All the points on the surface of the sphere are at equal distance from the center of the sphere. It is the radius of the sphere. 
  • Example - Fruits like orange, balls, glass balls, and the earth.

Total Surface Area of a sphere = 4πr2 sq.units
Volume of a sphere = 4/3 × πr3 cubic units

Collection of various designs of wireframe globes Free Vector

Example
Find the cost of painting the surface of a spherical solid of radius 4.2m.  if the cost of painting 1 sq.m is Rs.5.

Answer:

Surface area of the spherical solid = 4πr2 sq.mts
                 = 4 × 22/7 × (4.2) × 2= 221.76 sq.m = Area to be painted 
Total Painting cost at Rs. 5 per sq.m = 221.76 × 5 = Rs. 1108.80

 

Hemisphere

Hemisphere is a solid shape exactly equal to half of a sphere, but with a top flat circular face.

Example - Watermelon

Total Surface Area of a Hemisphere = Area of curved space + Area of circular face
                 = 2πr2 + πr2 = 3πr2

Curved Surface Area of a Hemisphere = 2πr2
Volume of a hemisphere = 2/3 × πr3 cubic units.

Concrete hemispheres parking limiter Premium Photo

Example
Find the volume of a hemisphere cup of radius 9 m.

Answer:

Volume of hemisphere solid = 2/3 × (πr3) cubic units
                 = 2/3 × 22/7 × 93= 44/7 × 81 × 3 = 1527 × 3/7cu.m.


Solid Shapes with only Flat Sides

Cube

  • A Cube is a solid shaped object which has six flat sides of a square shape, each side meeting the adjacent sides at the edge and vertices.
  • Each side is at 90⁰ to the adjacent sides.
  • Each side has equal length and forms a square face.

Example - Cardboard box 

Surface Area of a Cube = areas of six square sides of length ‘a’ 
                 = a2 + a2 + a2 + a2 + a2 + a2 = 6 a2 sq. units.
Volume of a cube = area of one side × length (height) of the cube
                = a2 × a = a3 cubic units.

Box, Cardboard, Cube, Isometric, Storage, Brown Box

Example
The total surface area of a cubical container is 1350 sq.m. Find the edge length of each face of the cube.

Answer:

Total surface area of the cube = 6a2
                 = 1350sq.m., ‘a’ being edge length of each face.
a2 = 1350/6 = 225
Length of each face = a = √225 = 15m.

 

Cuboid

With length ‘a’ and width ‘b’ the area = 2ab for cuboid shape, opposite sides area of flat rectangular faces of equal length and breadth or height.

Each side makes 90⁰ with adjacent sides.

Example - Books, boxes, almirahs, books, fridges are of cuboid shapes.
 

Consider a cuboid of 2 faces with length ‘a’ and width ‘b’ the area = 2ab
2 faces with length ‘b’ and height ‘h’ the area = 2bh
2 faces with length ‘a’ and height ‘h’ the area = 2ah
Surface Area of a Cuboid = 2(ab + ah + bh) sq. units
Volume of a cuboid = a × b × h sq.units. with length ‘a’ and width ‘b’ the area = 2ab

Wooden table with three books on top of each other at daytime Free Photo

Example
Find the total surface area of a cuboid-shaped solid of length 8cms, breadth(width) 6cms, and height 5cms.

Answer:

Given, a = 8cms, b = 6cms, h = 5cms
The total surface area = 2(ab + ah + bh)
                 = 2 × [(8 × 6) + (5 × 6) + (8 × 5)]
                 = 236sq.cms.


Formulas

Name

Curved /lateral surface area

Total surface area (TSA)

volume, V

Cuboid

2 h(l+b)

2 (ab + ah + bh)

l × b × h

Cube

4 a2

6a2

a3

Cylinder

2 πrh

2 πr(r+h)

πr2h

Cone

πrl

πr(r+l)

1/3 πr2h

Sphere

4 πr2

4 πr2

4/3 πr3

Hemisphere

2 πr2

3 πr2

2/3 πr3

Pyramid

½ × (P) × l

½ × (P) × l + B

1/3 Bh

Prism

(P)h

(P)h+2B

Ah

Where (P) is Perimeter, and B is the Base Area.


Summary

Solid shapes have 3-dimensions. They have curved surfaces and flat or lateral sides. Their areas are in square units having a total surface area equal to the sum of the curved surface area and areas of flat sides. They occupy space equal to their volumes cylinder, cone, cuboid, cube, prism, and sphere, etc, are examples of solid shapes.

 

Written by P Reeta Percy Malathi, Cuemath Teacher


Frequently Asked Questions (FAQs)

What is Total Surface Area? Give examples

Total surface area is the sum of the area of the curved surface, or lateral surface, and the area of flat faces of a shape.

For Example, the Total Surface of a Cylinder = Curved Surface Area + Area of Flat faces = 2πrh + 2πr2

What is Curved Surface Area? Give examples

Curved Surface Area is the Area covered by the curved region of a shape.

For Example, the Curved Surface Area of a Cone = πrl

What is the Surface Area and Volume of a Right Circular Cone?

A right circular cylinder is nothing but a regular cone. Hence,

  • Curved Surface Area of a Cone = πrl
  • Total Surface Area of a Cone = πr(r+l)
  • Volume of a Cone = 1/3 × πr2h

What is the Surface Area and Volume of a Right Circular Cylinder?

A right circular cylinder is nothing but a regular cylinder. Hence,

  • Curved Surface Area of a Cylinder = 2 πrh
  • Total Surface Area of a Cylinder = 2 πr(r+h)
  • Volume of a Cylinder = πr2h

What is the Formula for Volume for the Different Solid shapes

  • Volume of Cylinder =  πr2h
  • Volume of Cone = 1/3 × πr2h
  • Volume of Sphere = 4/3 × πr3
  • Volume of Hemisphere = 2/3 × πr3
  • Volume of Cube = a3
  • Volume of Cuboid = l × b × h
  • Volume of Pyramid = 1/3 Bh (B: base area, h: height)

What is the Formula for Lateral Surface Area?

  • Lateral Surface Area of Cube = 4a2
  • Lateral Surface Area of Cuboid = 2 h(l+b)
  • Lateral Surface Area of Pyramid = ½ × (P) × l (P: Perimeter, l: Slant height)

What is the Formula for Curved Surface Area?

  • Curved Surface Area of Cylinder =  2 πrh
  • Curved Surface Area of Cone = πrl
  • Curved Surface Area of Sphere = 4 πr2
  • Curved Surface Area of Hemisphere = 2 πr2

What is the Formula for Total Surface Area?

  • Total Surface Area of Cylinder =  2 πr(r+h)
  • Total Surface Area of Cone = πr(r+l)
  • Total Surface Area of Sphere = 4 πr2
  • Total Surface Area of Hemisphere = 3 πr2
  • Total Surface Area of Cube = 6a2
  • Total Surface Area of Cuboid = 2 (ab + ah + bh)
  • Total Surface Area of Pyramid = ½ × (P) × l + B (B: base area, P: Perimeter, l: Slant height)
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Solid Shapes
  
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