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Basics of Solid Geometry

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Solid geometry deals with the sizes and shapes of three dimensional objects. Some examples of three dimensional objects would be cubes, rectangular solids, prisms, cylinders, spheres, cones and pyramids. We will proceed to discuss the formulae and theorems relating to the above objects in this article.

Definition of solids
Solids are three dimensional objects which in addition to area also have volume. Two different types of areas used for measuring solids can be defined as follows:

(a) Lateral Surface Area: The area of the lateral surfaces of a solid is called as the lateral surface area. If the lateral surface of a solid is curved, like those in cylinders and cones, then it is also referred to as curved surface area.
(b) Total Surface Area: The total surface area of a solid is typically defined as the sum of the area of its lateral surfaces and the area of its top and bottom surfaces.
Total surface area = Lateral surface area + area of top surface + area of bottom surface

Euler’s rule
For any solid in the form of a regular polygon, there is a relationship between the number of vertices, edges and faces. Euler’s law states that the number of vertices plus the number of faces in every regular solid will always equal the number of edges plus two.

Number of faces + Number of vertices = Number of edges + 2

Properties of Cubes


A regular solid body with six equal squares is called a cube. If the length of edge of the cube is ‘a’, then,
Volume = a3
Lateral surface area = 4a2
Total surface area = 6a2

Properties of Cuboid


A regular solid body with six rectangular faces where opposite sides are identical is called a cuboid. If l and b are respectively the length and breadth of the base and h, the height of the cuboid, then,

Volume = lbh
Lateral surface area = 2(l + b) h
Total surface area = 2(lb + bh + hl)

Properties of Prism

A right prism is a solid whose top and bottom faces are parallel to each other and are identical polygons that are parallel. The edges of the top and bottom surfaces are joined by rectangles. Since there is a rectangular face for each side, a right prism with n sided polygonal base will have n rectangular faces. The distance between the base and top is called the length or height of the prism. For any prism,

Volume = area of base x height of the prism
Lateral surface area = perimeter of the base x height of the prism
Total surface area = lateral surface area + (2 x area of base)

Properties of Cylinder



A cylinder is equivalent to a right prism whose base is a circle. If r is the radius of the base and h is the height of the cylinder, then

Volume = πr2h
Curved surface area = 2πrh
Total surface area = 2πr (h + r)

Properties of Pyramid

A solid whose base is a polygon and whose faces are triangles is called a pyramid. The perpendicular from the vertex to the base is called the height of the pyramid. For any pyramid,
Volume = (1/3) x area of base x height
Lateral surface area = (1/2) x perimeter of the base x slant height
Total surface area = lateral surface area + area of the base

Properties of Cone

A cone is equivalent to a right pyramid whose base is a circle. If r is the radius of the base, h is the height of the cone and l is the slant height of the cone, then,
Volume = (1/3) πr2h
Curved surface area = πrl
Total surface area = πr (l + r)

Properties of Sphere

A sphere is the collection of points in three-dimensional space that are equidistant from a fixed point, the centre of the sphere. The distance from the centre of the sphere to any point on the sphere is called the radius of the sphere. If r is the radius of the sphere, then,
Volume = (4/3) πr3
Surface area of a sphere = 4πr2

Properties of Hemisphere


The plane which passes through the centre of the sphere divides the sphere in two equal parts. Each part is called hemi-sphere. If r is the radius of the hemi-sphere, then,
Volume = (2/3) πr3
Lateral surface area = 2πr2
Total surface area = 3πr2


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