A right prism whose base is a square and height is equal to the side of the base is called a cube. .

Volume = a^{3} where a is the edge of
the cube. .

Lateral Surface Area = 4a^{2}

Total Surface Area = 6a^{2}

The longest diagonal of the cube (i.e., the line joining one vertex on the top face to the diagonally opposite vertex on the bottom face) is called the diagonal of the cube. The length of the diagonal of the cube is a.

A cylinder is equivalent to a right prism whose base is a circle. A cylinder has a single curved surface as its lateral faces. If r is the radius of the base and h is the height of the cylinder, .

Volume = p r^{2}h

Curved Surface Area = 2p rh

Total Surface Area = 2 p rh + 2p r^{2} =
2p r(h + r)

A hollow cylinder has a cross-section of a ring. .

Volume of the material contained in a hollow
cylindrical ring = p(R^{2} -
r^{2})h where R is the outer radius, r is the inner radius and h, the
height. .

A solid whose base is a polygon and whose faces are triangles is called a pyramid. The triangular faces meet at a common point called vertex. The perpendicular from the vertex to the base is called the vertical height of the pyramid. .

A pyramid whose base is a regular polygon and the foot of the perpendicular from the vertex to the base coincides with the centre of the base, is called a right pyramid.

The length of the perpendicular from the vertex to any side of the base (please note that this side will be the base of one of the triangular lateral faces of the prism) along the slant lateral surface is called the slant height of the prism. .

Volume of a pyramid

= 1/3 x Area of base x Vertical Height Lateral Surface area

= 1/2 x Perimeter of the base x Slant height

Total Surface Area

= Lateral Surface Area + Area of the base. .

**Fig****. 5.41**.

A cone is equivalent to a right pyramid whose base is a circle. The lateral surface of a cone does not consist of triangles like in a right pyramid but is a single curved surface. .

If r is the radius of the base of the cone, h is height of the cone and’ 1 is the slant height of the cone, then we have the relationship (Fig. 5.41) .

l^{2} = r^{2} + h^{2}

Volume = pr^{2}h

Curved Surface Area = p r.1 .

Total Surface Arca = p rl + pr^{2} = pr(1 + r)

A cone can be formed by taking the sector of a circle and joining together its straight edges. If the radius of the sector is R and the angle of the sector is q°, then we have the following relationships between the length of the are and area of the sector on the one hand and base perimeter of the cone and curved surface area of the cone on the other hand. .

Radius of the sector = Slant height of the cone i.e., R = 1 .

Length of the are of the sector = Circumference of the base of the cone i.e., x 2 x pR = 2pr ⇒ r = x R and Area of the sector = Curved surface area

(Actually, from this last equation, substituting the values from the first two equations, we can get the curved surface area of the cone, which is what is given previously as equal to prl)

**Fig****. 5.42**.

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

If r is the top radius; R, the radius of the base; h the height and 1 the slant height of a frustum of a cone (Fig. 5.42), then, .

Lateral Surface Area of the frustum of a cone = p(R + r)

Total Surface Area = p (R^{2} + r^{2} + R.l + r.l) .

Volume = 1/3 ph(R^{2} + Rr + r^{2})

l^{2} = (R - r)^{2} + h^{2}

If H is the height of the complete cone from which the frustum is cut, then from similar triangles, we can write the following relationship. .

A bucket that is normally used in a house is a good example of the frustum of a cone. The bucket is actually the inverted form of the frustum that is shown in the figure above. .

A pyramid left after cutting of a portion at the top by a plane parallel to the base is called a frustum of a pyramid. .

If A_{1} is the area of the base; A^{2}
the area of the top and h, the height of the frustum,

Volume of frustum = 1/3 x h x (A_{1}
+ A_{2} + )

Lateral Surface Area = ½ x (Sum of perimeters of base and top) x Slant height

Total Surface Area = Lateral Surface Area + A_{1}
+ A_{2}

**Fig****. 5.43**.

A torus is a three-dimensional figure produced by the revolution of a circle about an axis lying in its plane but not intersecting the circle. The shape of the rubber tube in a bicycle (when it is inflated fully) is an example of a torus. If r is the radius of the circle that rotates and a is the distance between the centre of the circle and the axis of revolution, .

Surface Area of the torus = 4p^{2}ra

Volume of the torus = 2p^{2}r^{2}a

A torus is also referred to as a solid ring. (rig. 5.43) .

Any point on the surface of a sphere is equidistant from the centre of the sphere. This distance is the radius of the sphere. .

Surface Area of a sphere = 4pr^{2}

Volume of a sphere = (4/3)pr^{3}

The curved surface area of a hemisphere is
equal to half the surface area of a sphere, i.e., 2pr^{2}. .

{Note: Among the solids discussed above, Pyramid, Frustum of a Pyramid and Torus are not important from the point of view of the entrance exams and hence can be ignored if you so wish. Similarly, among the plane figures, Ellipse may be ignored if you so wish.} .

The following examples cover various properties/theorems discussed in Geometry as well as areas and volumes discussed in Mensuration. You should learn all the properties of triangles, quadrilaterals and circles as well as areas/volumes of plane figures and solids thoroughly before starting with the worked out examples and the exercise that follows the worked out examples. .