Download Frustum: When a plane section is taken of a right prism parallel to its

Frustum:
When a plane section is taken of a right prism parallel to its end (i.e.,
perpendicular to its axis), the section is known as a cross-section of the prism
and the two positions of the prism are still prisms. If, however, the plane
section taken is not parallel to the ends, the portion of the prism between the
plane section and the base is called frustum.
Volume of a frustum of a Prism:
In figure ABCEFGHI represents a frustum of a prism whose cutting
plane EFGH is inclined at an angle to the horizontal. In this case, the
frustum can be taken as a prism with base ABEF and height BC.
(i)
Volume of the frustum
ABEF is a trapezium whose area is
Volume of frustum
i.e.
Lateral Surface Area of a Prism:
If the cutting plane is inclined at an angle
figure we have
to the horizontal then from
or
or
or
Hence
Total surface area = area of the base + area of the section + lateral
surface area
Note: Lateral surface area of the frustum is the combination of rectangle and
trapeziums whose area can be calculated separately.
Example:
A hexagonal right prism, whose base is inscribed in a circle of radius
2m, is cut by a plane inclined at an angle of
to the horizontal. Find the
volume of the frustum and the area of the section when the heights of the
frustum are 8m and 6m respectively.
Solution:
Area of cross-section
Here
Area of base
Volume of frustum
Area of the section
Frustum of a Right Circular Cylinder
If
(i)
is the radius and
is the height of the frustum, then
Volume of the frustum of the circular cylinder
i.e.
(ii)
Curved surface area of the frustum
(iii)
Total surface area = curved surface area + area of the ends
Example:
A circular cylinder, having a radius 2.3m, is cut in the shape of a frustum with
volume and the lateral surface area.
. Find the
Solution:
(a)
(b)
Curved surface area
Frustum of a Pyramid
If a pyramid is cut through by a plane parallel to its base portion of the pyramid between that
plane and the base is called frustum of the pyramid.
Volume of Frustum of a Pyramid:
A general formula for the volume of any pyramid can be derived in terms of the areas of the two
bases and the height of the frustum.
Consider any frustum of a pyramid
altitude . Complete the pyramid
in figure with the lower base
of which the frustum
Denote by , the volume of the small pyramid
of
is
Let
and
, upper base
and the
is a part.
, whose altitude is . Then the altitude
.
respectively, represents the volume of the frustum
From the figure it is easily seen that
we may write.
and the pyramid
.
. Expressing this equality in terms of the dimensions,
…...... (1)
The pyramid
may be considered as cut by the two parallel planes
and
. Hence
& nbsp;
(as if a pyramid is cut by two parallel planes, the areas of the sections are proportional to the squares of their
distances from the vertex).
Taking the square root of both sides, we have
or
Transposing
to the L.H.S. of this equation and factorizing,
Substituting the value of
in (1), we have
i.e. The volume of a frustum of a pyramid is equal to the one-third the product of the altitude and the sum of
the upper base, the lower base and the square root of the product of two bases.
Lateral Surface Area of Frustum of a Pyramid:
Each of the faces such as CDML, of the frustum of a pyramid is a trapezium and the area of each
trapezium will be half the sum of the parallel sides, CD and ML, multiplied by the slant distance between
them.
In the frustum of pyramid on a square base, let
length of each side of the other end,
denote the length of each side of the base,
the
the height of the frustum.
Each face CDML is a trapezium, the lengths of the parallel sides
and .
Area
i.e.
Example:
A frustum of a pyramid has rectangular ends, the sides of the base being 25dm and 36dm. If the
area of the top face is 784sq.dm and the height of the frustum is 60dm, find its volume.
Solution:
Here
Frustum of a Cone
If a cone is cut by a plane parallel to its base, the portion of a solid between this plane and
the base is known as frustum of a cone.
The volume denoted by ABCD in figure is a frustum of the cone ABE.
Volume of Frustum of a Cone:
Since, we know that cone is a limit of a pyramid therefore; frustum of a cone will be the limit of
frustum of a pyramid. But volume of a pyramid is
Where
Example:
A cone 12cm high is cut 8cm from the vertex to form a frustum with a volume of 190cu.cm. Find
the radius of the cone.
Solution:
Given that:
Height of cone
Height of frustum
Volume of frustum
Now volume of frustum cone
or
or
Hence required radius of cone
Curved Surface Area of a Frustum of a Cone:
Since, a cone is the limiting case of a pyramid, therefore the lateral surface of frustum of a cone can
be deduced from the slant surface of frustum of a pyramid, i.e., curved (lateral) surface of frustum of cone.
, being the slant height of frustum,
and
being two radius of bases.
Note:
(1)
Total surface area of frustum of a cone
(2)
To find the slant height of the cone, use Pythagorean theorem.
Example:
A material handling bucket is in the shape of the frustum of a right circular cone as shown in figure.
Find the volume and the total surface area of the bucket.
Solution:
Slant height
Lateral surface area
Base areas
Total surface area
Volume
Zone or Frustum of a Sphere
The portion of a sphere intercepted between two parallel planes is called a zone (i.e.
frustum).
1. The volume of the zone (or frustum) of a sphere may be found by taking the difference between
segment EBD and the segment ABC (see figure) that is
Where
is the altitude,
and
are respectively the radii of the small circle (bases).
(ii)
The surface area of the zone or frustum is equal to the circumference of the great circle of the
sphere times the altitude of the same.
i.e.
Where
(iii)
Total surface area of a zone
Example:
The sphere of radius 8cm is cut by two parallel planes, one passing 2cm from the center and
other 6cm from the center. Find the area of the zone and the volume of the segment between two planes if
both planes are on the same side of the center.
Solution:
Surface Area of the zone
Now
Example:
A stone was rolled into a hemispherical basin 8cm diameter having
depth of water in it, when the
water immediately rose to the tip of the basin. What was the cubic content of the stone?
Solution:
Let the figure represents a vertical mid-section of the basin and stone. Let DE and ABindicate the
level of the water before and after the rolling of the stone. Then
If
, then
Now, the cubical content of the stone
nearly.
Segment of a Sphere:
For a special segment of one base, the radius of the lower base
is equal to zero. Therefore,
In this case, the total surface area of the segment
Example:
Find the volume of a segment of a sphere whose height is
is 8cm.
Solution:
Given that:
Volume of the segment
and the diameter of whose base