Surface Area of a Cone Formula

Surface Area of a Cone Formula

Surface Area of a Cone

A right circular cone is a solid shape (3D shape) with one circular face and a curved face. It has one circular edge and one vertex. The circular face is called the base of the cone. The radius of the cone is called the base radius. The perpendicular distance from the vertex to the centre of the circular base is called the height of the cone.

The real-life examples of the cone include ice-cream cone, joker’s cap, pylon, etc.

A cone has two surfaces, one is curved and other is circular plane. The area of the curved surface is called the curved surface area or lateral surface area and the area of curved surface and the circular place surface together is called the total surface area of the cone.

Let us derive the formula to calculate the surface area of a cone.

Take a cone made up of paper. Now, cut the cone along its slant height and spread it on the ground as shown in the given figure. Let the slant height of the cone be l.

Now, divide the figure in smaller triangles, such as Ob1b2, Ob2b3, and so on.

The area of all these triangles gives the curved surface area of the cone.

Thus, curved surface area of the cone = Area of triangles (Ob1b2 + Ob2b3 + …….)

                                                             = ½ × b1b2 × l + ½ × b2b3 × l + ……….

                                                             = ½ × (b1b2 + b2b3 + ……….) × l

But, b1b2 + b2b3 + ………. = Circumference of the circle = 2πr

Therefore, the curved surface area of the cone = ½ × 2πr × l

                                                                           = πrl

Curved surface area of a cone = πrl

Where l is the slant height of the cone.

We can calculate the slant height of the cone if its height and base radius are given.

Using Pythagoras theorem, l2 = h2 + r2

Or, l = (h2 + r2)

Hence, the curved surface area of a cone = πr × (h2 + r2)


Now, the total surface area of the cone = Curved surface area + Area of the circular base

                                                               = πrl + πr2

                                                               = πr(l + r)

Total surface area of a cone = πr(l + r)

 

Surface Area of a Cone Formula

Curved surface area of a cone = πrl

Total surface area of a cone = πr(l + r)

 

Surface Area of a Cone Example

Example 1: Find the curved surface area of a cone whose base radius is 14 cm and slant height is 10 cm.

Solution: Given: r = 14 cm and l = 10 cm

Curved surface area of a cone = πrl

                                                 = 22/7 × 14 × 10

                                                 = 440 sq. cm

 

Example 2: Find the total surface area of a cone whose base diameter is 14 cm and height is 8 cm.

Solution: Given: Diameter = 14 cm, then, r = 14/2 = 7 cm

Height, h = 8

l2 = h2 + r2

l2 = (8)2 + (7)2

l2 = 64 + 49 = 113

l = 113

l = 10.63 cm

Curved surface area of a cone = πrl

                                                 = 22/7 × 7 × 10.63

                                                 = 233.86 sq. cm

 

Example 3: Find the height of a cone whose base radius is 21 cm and curved surface area is 1650 sq. cm.

Solution: Given: r = 21 cm and curved surface area of = 1650 sq. cm

Curved surface area of a cone = πrl

                                        1650 = 22/7 × 21 × l

                                        1650 = 66 × l

                                               l = 1650/66

                                               l = 25 cm

Using, l2 = h2 + r2

h2 = l2 – r2

    = (25)2 – (21)2

    = 625 – 441

    = 184

h = 184

h = 13.56 cm

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