Geometry and Mensuration - Important Concepts & Formulas

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Solid Geometry

Areas and Volumes of solids

Basically solid figures are three dimensional objects which have surface area and volume too. As per the required testing area of the competitive exams, here we are dealing with three well defined categories of three dimensional objects such as Right Prisms, Pyramids and Spheres.

Right Prisms

A right prism is solid object which has two identical and parallel faces (named base and top) connected by a lateral surface. The number of lateral faces is equal to the number of sides of the base polygon and each lateral face is in the shape of a rectangle.

The distance between the base and top faces of a prism is known as the length or height of the prism.

The name of a prism is based on the shape of the base polygon. i.e. if the base of a prism is in the shape of circle then it is a circular prism or cylinder or the base is a triangle then it is a triangular prism and so on.

Above diagrams are examples for right prisms.

General results for any right prisms:

Lateral surface area = Base perimeter * Height of the prism

Total surface area = Lateral surface area + 2 (Base area)

Volume = Base area * Height of the prism.

Classification of prisms

Cuboids:

A right prism whose base is a rectangle is called cuboid (rectangular prism)

Base area = Length * Breadth = lb

Base perimeter = 2(l + b)

Lateral surface area = 2(l + b) * h

Total surface area = 2(lh + bh + lb)

Volume = lbh

Longest diagonal of a cuboid = √(l2 + b2 + h2)

Cube:

A right prism whose all faces are identical squares is called a cube. i.e. the height of cube is equal to the side of base.

Base area = a2

Base perimeter = 4a

Lateral surface area = 4a2

Total surface area = 6a2

Volume = a3

Longest diagonal = a√3

Cylinder:

A right prism whose base is a circle is known as cylinder.

Base area = πr2

Base perimeter = πr

Lateral surface area = πrh

Total surface area = 2πr(h+r)

Volume = πr2h

Pyramids

Pyramids are solid objects (three dimensional figures) whose base is a polygon and lateral faces are triangles. The lateral faces have a common vertex and this vertex is not co-planar with the base.

Line segments from the common vertex to the vertices of the base are called the lateral edges (e).

Pyramids are named according to the shape of its base. i.e. if the base is a triangle, then it is a triangular pyramid. If the base is a square, then it is a square pyramid, and so on.

If the base has a center, then the line joining the common vertex to the center of base is called axis.

A pyramid that has the axis perpendicular to the base is a right pyramid; otherwise it is an oblique pyramid.

If the base is a regular polygon and the pyramid is a right pyramid, then it is called a regular pyramid.

The altitude (h) of a pyramid is the perpendicular distance from the vertex to the base.

The slant height (l) of a pyramid is the altitude of the lateral face.

General results for pyramids:

Lateral surface area =1/2 * Permieter of the base *slant height

Total surface area = Lateral surface area + Area of the base.

Volume = 13 * Area of base * Height

Cone:

Cone is a right pyramid which has its base as a circle.

Radius of base → r

Height of the cone → h

Slant height of the cone → l

l2 = r2 + h2

Volume = 13πr2h

Curved surface area = πrl

Total surface area = πr(l+r)

Cone cut in to two parts:

In the diagram, a cone is cut into two parts by a plane parallel to the base of the cone. There are two similar cones forms, one is the larger cone and the other is the smaller cone formed by cutting the lager cone.

Measures of the larger cone:

Radius = R

Height = H

Slant Height = L

Measures of the smaller cone:

Radius = r

Height = h

Slant Height = l

Following relations can furnish on the measures of both the cones;

r/R = h/H = l/L = 2πr/2πR = √(πr2)/√(πR2) = ∛(πr2 h)/∛(πR2 H) = √(πrl)/√(πRL)

Frustum of a cone:

When cutting a cone into two parts by a plane parallel to the base of the cone, the portion that contains the base is called the frustum of the cone.

Bottom radius = R

Top radius = r

Height = h

Slant height = l

l2 = (R-r)2 + h2

Volume = 1/3 * πh (R2 + Rr+r2)

Lateral surface area = πl ( R+r)

Total surface area = π(R2+ r2+ Rl+rl)

Frustum of any pyramid:

If a pyramid cut into two portions by a plane parallel to the base, then the part that containing the base is called the frustum of a pyramid.

If,

Area of the base = A

Area of the top = a

Height of the pyramid = h

Slant height = l

Then;

Volume = 1/3 h (a + A + √(a * A))

Lateral surface area = 1/2 * (sum of the perimeters of base and top)*slant height

Total surface area = Lateral surface area + a + A

Sphere:

Sphere is a collection of all the points which are equidistant a particular point in a space. This particular point is called the center of the sphere and the distance between from the center to all the points in a sphere is the radius of the sphere.

Volume = 4/3 * π r3

Surface area = 4πr2

Hemisphere:

This is exactly half portion of a sphere, or the portion of a sphere formed by cutting a sphere through its center by a plane in any direction.

A hemisphere has two faces, one curved face and another circular plane face.

Volume = 2/3 * πr3

Curved surface area = 2πr2

Total surface area = 3πr2

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