Solid Shapes
📥  Solid Geometric Shapes 
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 + 2πr^{2} = 2πr(h+r) sq. Units
Curved Surface Area of a Cylinder = 2πrh sq.units
Volume of a cylinder = πr^{2}h 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 m^{2}. Find the total cost of painting.
Answer:
Total Surface Area = Curved Surface Area + area of top and bottom faces
= 2πrh + 2πr^{2 }= 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 + πr^{2}
= πrl + πr^{2 }= 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 × πr^{2}h
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 = 4πr^{2} sq.units
Volume of a sphere = 4/3 × πr^{3} cubic units
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 = 4πr^{2} 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
= 2πr^{2} + πr^{2} = 3πr^{2}
Curved Surface Area of a Hemisphere = 2πr^{2}
Volume of a hemisphere = 2/3 × πr^{3} cubic units.
Example
Find the volume of a hemisphere cup of radius 9 m.
Answer:
Volume of hemisphere solid = 2/3 × (πr^{3}) 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’
= a^{2} + a^{2} + a^{2} + a^{2} + a^{2} + a^{2} = 6 a^{2} sq. units.
Volume of a cube = area of one side × length (height) of the cube
= a^{2} × a = a^{3} cubic units.
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 = 6a^{2}
= 1350sq.m., ‘a’ being edge length of each face.
a^{2} = 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
Example
Find the total surface area of a cuboidshaped 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 a^{2} 
6a^{2} 
a^{3} 
Cylinder 
2 πrh 
2 πr(r+h) 
πr^{2}h 
Cone 
πrl 
πr(r+l) 
1/3 πr2h 
Sphere 
4 πr^{2} 
4 πr^{2} 
4/3 πr^{3} 
Hemisphere 
2 πr^{2} 
3 πr^{2} 
2/3 πr^{3} 
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 3dimensions. 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 + 2πr^{2 }
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 × πr^{2}h
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 = πr^{2}h
What is the Formula for Volume for the Different Solid shapes
 Volume of Cylinder = πr^{2}h
 Volume of Cone = 1/3 × πr^{2}h
 Volume of Sphere = 4/3 × πr^{3}
 Volume of Hemisphere = 2/3 × πr^{3}
 Volume of Cube = a^{3}
 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 = 4a^{2}
 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 πr^{2}
 Curved Surface Area of Hemisphere = 2 πr^{2}
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 πr^{2}
 Total Surface Area of Hemisphere = 3 πr^{2}
 Total Surface Area of Cube = 6a^{2}
 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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