of the circle = pr2
where r is the radius of the circle. Circumference = 2pr .
of a circle. Length of arc = x 2pr
Area = x pr2 where q is the angle of the sector in degrees and r
is the radius of the circle.
Area = (1/2)1r; 1 is length
of arc and r is radius. .
Ring is the space enclosed by two concentric circles. Area = pR2 - pr2 = p(R + r) (R - r) where R is the radius of the outer circle and r is
the radius of the inner circle. .
Area = p ab where “a” is semi-major axis and “b” is semi-minor axis.
Perimeter = p (a + b) .
Areas and Volumes of Solids
Solids are three-dimensional objects which,
in addition to areas, have volumes also. For solids, two different types of
areas are defined .
(a) Lateral surface area or curved
surface area and
(b) Total surface area
As the name itself indicates, lateral surface
area is the area of the LATERAL surfaces of the solid. Total surface area
includes the areas of the top and the bottom surfaces also of the solid. Hence,
Total surface area = Lateral surface area + Area of the top face + Area of the
bottom face. .
In solids (like cylinder, cone, sphere) where
the lateral surface is curved, the lateral surface area is usually referred to
as the “curved surface area.” .
For any solid, whose faces are regular
polygons, there is a definite relationship between the number of vertices, the
number of sides and the number of edges of the solid. This relationship is
given by “Euler’s Rule”. .
Number of faces + Number of vertices
= Number of edges + 2 (Euler’s Rule)
A right prism is a solid whose top and bottom
faces (bottom face is called base) are parallel to each other and are identical
polygons (of any number of sides) that are parallel. The faces joining the top
and bottom faces are rectangles and are called lateral faces. There are as many
lateral faces as there are sides in the base. The distance between the base and
the top is called height or length of the right prism. .
In a right prism, if a perpendicular is drawn
from the centre of the top face, it passes through the centre of the base. .
For any prism,
Lateral Surface Area = Perimeter of base x
Height of the prism
Total Surface Area = Lateral Surface Area + 2
x Area of base
Volume = Area of base x Height of the prism
Cuboid or Rectangular Solid
A right prism whose base is a rectangle is
called a rectangular solid or cuboid. If 1 and b are respectively the length
and breadth of the base and h, the height, then .
Volume = lbh
Lateral Surface Area = 2(1 + b) . h .
Total Surface Area = 2(1 + b)h + 2lb = 2(lb +
lh + bh)
Longest diagonal of the cuboid =