Frustum of a Cone-Slant Height

Cone is a three-dimensional geometric shape that has a circular base and the circular base narrows towards the vertex of that cone. Slant height can be introduced as the line drawn from the vertex to the extreme point of a radius of the circular base of the right cone. A frustum of a cone has two circular faces.  One circular face is at the top and one is at the bottom of the cone.

Frustum of a Cone Slant Height

The frustum of a cone is the segment of a cone that can be produced after cutting the cone through the middle portion. According to the Latin language frustum means “piece cut off”. 

The upper part of the cone after cutting through the middle looks the same as the cone. The bottom part that has two circular surfaces is called the frustum of a cone. Both two parts have different volumes and surface areas. The formulas to determine the volumes and surface areas are also different for these two parts.

The slant height is the line drawn from the vertex to the periphery. Several geometrical shapes have slant height such as the square pyramid, cone, frustum of a cone, etc. According to Kern and Bland, the slant height is the altitude of a triangle that has lateral faces. 

What is the slant height of a frustum of a cone?

The determination of the slant height of an object can help to determine the actual height of the object. Slant height and radius are required to determine the height of a cone. Calculating the height of a pyramid is not an easy job. The slant height of a pyramid helps to calculate the height of a pyramid with the help of the Pythagorean Theorem.    

The slant height of a frustum of a cone can be introduced as the line drawn from the circumference of one circular face to the circumference of the other circular face. The line that is known as slant height should be orthogonal to the radii of two circular faces.  On the other hand, the slant height of a frustum of a cone can be expressed in different ways also. The line that is joining the outermost points of two parallel radii of two circular plates of the cone is known as slant height. Several items in real life look the same as a frustum of a cone such as buckets, glasses, and typical lampshades. Street cones also can be introduced as examples of a frustum of cones.

Frustum of a cone slant height formula

The formula that can help to determine the slant height of a frustum of a cone is given below:

“l =√ (h2+(R-r) 2)”

In this equation, l denotes the slant height of the frustum of a cone, h denotes the height of the frustum of a cone. Also, R and r denote the radius of the bottom circular plate and a top circular plate of the frustum of a cone respectively. Effortlessly, the description of this formula can be described. Two parallel radii of the frustum of a cone are taken. These two radii should be taken from different circular bases of the frustum of a cone. Now, the combination of two radii, slant height, and actual height of the frustum of a cone is like a trapezoid. This trapezoid has two parallel sides and the other two sides are not parallel. One angle of this trapezoid is a right angle. Now, the actual height of the frustum of a cone can be expressed as the height of the trapezoid. A triangle can be cut from the trapezoid such that the height of the trapezoid should be the height of the triangle and the triangle is a right-angle triangle. The slant height of the frustum of the cone is the hypotenuse of this right-angle triangle. Now, with the application of the Pythagorean Theorem the length of the hypotenuse of the right angle triangle can be calculated. It has mentioned below:

“l2= r2+h2”

In this equation l denotes the hypotenuse of the right angle; r and h denote the radius and height of the right-angle triangle. This equation also can be expressed in a different way like: 

“l=√ (r2+h2)”

Here l which denotes the hypotenuse of the right angle triangle is the same as the slant height of the frustum of the cone. Here r which denotes the radius of the right angle triangle is equal to the difference between the two parallel radii of the circular plates of the frustum of the cone. The height of this right-angle triangle and the height of the frustum of the cone is the same.

Conclusion

It can be concluded that the frustum of a cone has two different circular plates. These two plates are parallel to each other. The lengths of the radii of these two plates are different. The slant height of the frustum of a cone can be used to calculate the actual height of the object. The slant height of an object is always greater than the actual height of that object.