cone (geometry)

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In geometry, a pyramid with a circular base. If the point (vertex) is directly above the centre of the circle, it is known as a right circular cone. The volume (V) of this cone is given by the formula V = 1/3πr²h where h is the perpendicular height and r is the base radius.

A right circular cone is generated by rotating an isosceles triangle about its line of symmetry. The distance from the edge of the base of a cone to the vertex is called the slant height. In a right circular cone of slant height l, the curved surface area is πrl, and the area of the base is πr². Therefore the total surface area A = πrl + πr² = πr(l + r).

For example, to find the volume and surface area of a cone with a perpendicular height of 4 cm and radius of 3 cm:

Volume = 1/3πr²h, so

V = 1/3π × 3² × 4

V = 37.7 cm³ (to 1 decimal place)

Surface area = πr(l + r). First the slant length (l) is calculated:

Using Pythagoras' theorem,

l² = h² + r², so

l² = 4² + 3² = 16 + 9 = 25, so

l = 5 cm

The surface area can now be calculated:

A = πr(l + r), so

A = π × 3(3 + 5) = 3π × 8 = 24π, so

A = 75.4 cm² (to 1 decimal place)

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