When a cone ABC is cut through by a plane parallel to its base, the portion of the cone DECB between the cutting plane and the base is called a frustum of the cone.

**Frustum of a right circular cone**

In olden days a cone shaped buckets [Fig.7.21(a)] filled with sand
/ water were used to extinguish fire during fire accidents. Later, it was
reshaped to a round shaped bottom [Fig.7.21(b)] to increase its volume.

The shape in [Fig.7.21(c)] resembling a inverted bucket is called
as a frustrum of a cone.

The objects which we use in our daily life such as glass, bucket,
street cone are examples of frustum of a cone. (Fig.7.22)

When a cone ABC is cut through by a plane parallel to its base,
the portion of the cone DECB between the cutting plane and the base is called a
frustum of the cone.

Let *R* and *r* be radii of the base and top region of
the frustum *DECB* respectively, *h* is the height and *l* is
the slant height of the same.

Therefore, C.S.A. = ½ (sum of the perimeters of base and top
region) × slant height

= ½ (2*πR *+ 2*πr*)*l*

C.S.A. of a frustum = *π*(*R* + *r*)*l* sq.
units

Where,

T.S.A. = C.S.A. + Area of the bottom circular region* _{ }*+ Area of the top
circular region.

T.S.A. of a frustum = *π(R* + *r*)*l* + *πR*^{2}
+ *πr *^{2} sq. units

Where,

**Example 7.13 **The slant height of a frustum of a cone is** **5** **cm and the radii of its
ends are**
**4 cm and 1 cm. Find its
curved surface area.

** Solution** Let

Given that, *l*= 5 cm, *R* = 4 cm, *r*
= 1 cm_{ }

Now, C.S.A. of the frustum = π (*R* + *r) l* sq.
units

Therefore, C.S.A. = 78.57 cm^{2}

**Example 7.14 **An industrial metallic bucket is in the shape of
the** **frustum of a right
circular cone whose top and bottom diameters are 10 m and 4 m and whose height
is 4 m. Find the curved and total surface area of the bucket.

** Solution **Let

Given that, diameter of the top =10 m; radius of the top *R*
= 5 m.

diameter of the bottom = 4 m; radius of the bottom *r* = 2 m,
height *h*= 4 m

Therefore, C.S.A. = 110 m^{2} and T.S.A. = 201.14 m^{2}

Tags : Definition, Formula, Solved Example Problems | Mensuration | Mathematics , 10th Mathematics : UNIT 7 : Mensuration

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10th Mathematics : UNIT 7 : Mensuration : Surface Area: Frustum of a right circular cone | Definition, Formula, Solved Example Problems | Mensuration | Mathematics

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