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#### Define Radius: Height and Slant Height of Cone

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A cone is a solid figure that is composed of an interior base and a circle along with the point that is given on the plain of circle which is known as vertex along with all the segments on the point of the circular base. A cone is a geometrical shape having three dimension that tapers from the flat base to the point that is called vertex and apex. Therefore, it is the surface that is traced by the moving straight line passing through the fixed point that is known as the vertex. The cone generatix is assumed to have the infinite length as it extends in both the direction from the vertex. There are basically two types of cones that is oblique cone and right circular cone. Right circular cone is the cone where the axis to the base from the cone vertex through the centre of the base of circle. The centre of the circular base is below the vertex of the cone. The cone is called as the right circular cone because the axis forms a ninety degree angle that is perpendicular to the circular base of the cone. In geometry, this is the most commonly used cone. Oblique cone on other hand is a cone having circular base, however, the axis of the cone is not perpendicular with the base. In the oblique cone, there is no vertex directly over the centre of base of circus. There are various properties of a cone that is listed below:

ü There is only one vertex or the apex point of the cone

ü There is only one face of the cone which do not have vertices or edge that is, the face of the cone is its circular base.

ü The formula for deriving the total surface area of the cone is πr (l + r)

ü The formula for deriving the volume of the cone is 1/3πr2h

ü The formula for deriving the slant height of the cone is

l = √ h2 + r2

The area and volume of a cone can be found using the formula that is derived based on the radius, height and slant height of the cone.  That is the cone is formed using the line or the line segments connecting to the common point of the base that is circle. Cone can also be defined as the pyramid and it can also be stated as circular cone having cross section. Cone has got three dimension and they are as follows:

2. Altitude or the vertical height that is the perpendicular distance drawn from the top of the cone to the base.
3. The last dimension is the slant height and such height is the distance from the down and top the side to the point on the base of circumference.

All the three dimensions are related and for defining the cone, it is required to have only two dimension. The third dimension is the slant height that can be defined by forming the right triangle and the slant height is viewed as the hypotenuse and the hypotenuse can be found using the Pythagorean Theorem. A radius of a cone is the distance from the middle of the circle that is the base of a cone to the perimeter that is the circumference of circle. Therefore, the radius of a cone is the radius of the base of circle of the cone. The radius of the base of the circle of the cone can be found through its height and volume. Moreover, the radius forms a part of the triangle depicted in the figure above and given the slant height and height of the cone, the radius of the cone can be ascertained using the Pythagorean Theorem.

Radius of the cone is often represented by the symbol “r” and this r is the distance from the centre of the circular base of the cone to the side of the base of the cone. Altitude is the perpendicular drawn to the base of the circle from the vertex. Cone axis on other hand is the segment whose endpoints are the centre of the base and vertex. In addition to this, cone is considered to be the right cone if the axis is perpendicular to the circular base cone, or else it is known as oblique cone.

Height of the cone is represented by the distance from the base to the vertex or from the vertex to the base. There is a measured value of the cone which is indicated by the radius. Slant height is the segment length drawn to the circle of the base from the vertex of the cone.

Slant height is usually denoted by l which represents the distance of the apex to the point on the circular base boundary. Radius of a cone is represented by r and height of the cone represented by h. For finding out the slant height of the cone, the application of Pythagoras theorem is done. Therefore, using the theorem, slant height is represented by the following formula.

l2 = h2 + r2

l = √ h2 + r2

The above formula is the required formula for ascertaining the slant height.

In addition to this, the truncated cone is obtained cutting the cone’s upper portion. Cylinder with the tapered radius looks like a truncated cone. The slant height of the truncated con e can be found by using the following formula

l2 = h2 + (R – r)2

l2 = √ h2 + (R – r)2

It can be observed that given the slant height and radius of the circular cone, it is easy to obtain the length of height of cone using the formula mentioned above.

l2 = h2 + (R – r)2

h2 = l2 - (R – r)2

h = √ l2 - (R – r)2

Using this formula, it is possible to ascertain the height, slant height and radius of the cone.

Given the height and radius of the cone, the slant height can be found.

Say, if the value of r is 5 cm and the height of cone is 8 cm. Then, the slant height can be obtained using the formula.

l2 = √ h2 + r2

= √ 72 + 52

=√49+25= √89 =√74= 8.60 cm.

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