A right circular cone is a solid generated by the revolution of a right angled triangle about one of the sides containing the right angle as axis.

**Right Circular Cone**

Observe the given figures in Fig.7.9 and identify which solid
shape they represent?

These objects resemble the shape of a cone.

A right circular cone is a solid generated by the revolution of
a right angled triangle about one of the sides containing the right angle as
axis.

In Fig. 7.10, if the right triangle *ABC* revolves about *AB*
as axis, the hypotenuse *AC* generates the curved surface of the cone
represented in the diagram. The height of the cone is the length of the axis *AB*,
and the slant height is the length of the hypotenuse *AC*.

Suppose the surface area of the cone is cut along the hypotenuse *AC*
and then unrolled on a plane, the surface area will take the form of a sector *ACD*,
of which the radius AC and the arc *CD* are respectively the slant height
and the circumference of the base of the cone.

Here the sector of radius ‘*l* ’ and arc length ‘*s*’
will be similar to a circle of radius *l _{.}*

∴ Curved Surface Area of
the cone = Area of the Sector = *πrl* sq. units.

C.S.A. of a right circular cone = *πrl* sq. units.

**Derivation of slant height ‘ l’**

*ABC *is a right angled triangle, right angled at* B*. The
hypotenuse,* *base and height of the triangle are represented by *l*,
*r* and *h* respectively.

Now, using Pythagoras theorem in Δ*ABC*,

Total surface area of a cone = C.S.A. + base area of the cone

= *πrl *+* πr*^{2}* *(since,
the base is a circle)

T.S.A. of a right circular cone = *πr *(*l* + *r*
) sq. units.

The radius of a conical tent is** **7** **m and the height is** **24** **m. Calculate the** **length of the canvas
used to make the tent if the width of the rectangular canvas is 4 m?

** **Let

Given that, radius *r* =7 m and height *h* = 24 m

Hence,

C.S.A. of the conical tent = *πrl* sq. units

Area of the canvas = (22/7) ×7 ×25 = 550 m^{2}

Now, length of the canvas = Area of the canvas / width =
550/4 = 137.5 m

Therefore, the length of the canvas is 137.5 m

If the total surface area of a cone of radius** **7cm is 704 cm^{2},
then find its** **slant height.

Given that, radius* **r*** **= 7

Now, total surface area of the cone = *πr *(*l* + *r*
) sq. units

T.S.A. = 704 cm^{2}

704 = (22/7) ×7 (*l* +
7)

32 = *l* + 7
implies *l* = 25 cm

Therefore, slant height of the cone is 25 cm.

From a solid cylinder whose height is** **2.4** **cm and** **diameter 1.4 cm, a
conical cavity of the same height and base is hollowed out (Fig.7.13). Find the
total surface area of the remaining solid.

** **Let

Let *l* be the slant height of the cone.

Given that, *h* = 2.4 cm and *d* = 1.4 cm ; *r* =
0.7 cm

1= 2.5 cm

Area of the remaining solid = 2πrh + *πrl* + *πr*^{2}
sq. units

*= πr *(2*h *+* l *+* r *)

= (22/7) × 0.7 × [(2 × 2.4) + 2.5 + 0.7]

= 17.6

Therefore, total surface area of the remaining solid is 17.6 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 of Right Circular Cone | Definition, Formula, Solved Example Problems | Mensuration | Mathematics

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