Surface Area of a Cone and Its Notion
Cone (in Ancient Greek «konos») literally means a pine cone. The cone is known to people since ancient times. In 1906, the book «On the Method» written by Archimedes was found. This book provides a solution to the problem of the total volume of the intersecting cylinders, including surface area of a cone. Archimedes said that this discovery belonged to the ancient Greek philosopher Democritus, who with the help of this principle attained the formula for calculating surface area of a cone and a pyramid.
The most complete works dedicated to the conic sections and surface area of a cone were «Conic Sections» by Apollonius of Perga.
Apollonius of Perga, the Greek mathematician and astronomer, is known as «the great geometer». Most of his works are based on the foundations laid by Euclid. Only one of his books retained: «On the conic sections» has some mentions of the surface area of a cone. In the book, he showed that the ellipse, parabola, and hyperbole can be obtained by cutting the cone at different angles. In astronomy, he described the movement of the planets, using the concept of epicycles. Epicycle is the way, made by a point moving in a circle, which, in turn, moves in such a way that its center lies on the circumference of the other, larger circle. His works became the basis of the Ptolemaic system, which was used until the discoveries of Copernicus.
In classical mechanics, the trajectory of the free movement of spherical objects in a vacuum is subject to one of the applications of the inverse square law – the law of gravity, and therefore is one of the conical curves – parabola, hyperbola, ellipse, or line. The orbits of the planets are ellipses, the trajectory of comets is hyperbole, and the flight trajectory of a cannonball, net of the effect of the air, is a parabola.
Cone is the body in Euclidean space, which was formed as a result of a union of all rays emanating from one point (the apex) and passing through a flat surface. Sometimes cone is referred to as such a portion of the body that has a limited volume and was formed as a result of a union of all segments connecting the top and the points of the flat surface (the latter in this case is called the base of the cone and such cone is called a cone relying on this surface). If the base of the cone is a polygon, then a cone is a pyramid.
Related definitions
 The segment connecting the top and bottom border, called the generatrix cone.
 The unification of generatrixes forming the cone is called a generator (or side) of the cone surface. The generatrix surface of the cone is a conical surface.
 The segment dropped perpendicular from the top to the ground plane (as well as the length of such a segment), is called the height of the cone.
 Cone angle is the angle between two opposing generators (apex angle of a cone within a cone).
 If the base of the cone has a center of symmetry (for example, a circle or ellipse) and the orthogonal projection of the cone’s apex to its base coincides with the center, then the cone is called direct. In this case the line connecting the top and center of the base, called the axis of the cone.
 Oblique cone is a cone, whose orthogonal projection of the top on the base does not coincide with its center of symmetry.
 The circular cone is a cone, whose base is a circle.
 Right circular cone (often referred to simply as a cone) can be obtained by rotating a right triangle around a straight line containing the leg (this line is the axis of the cone).
 Cone based on an ellipse, parabola, or hyperbole is called respectively elliptical, parabolic, or hyperbolic cone (the last two have an infinite surface area of a cone).
 The part of the cone lying between the base and a plane, which is parallel to the base and located between the top and base, is called a truncated cone or tapered layer.
Sweep
Right circular cone as a body of revolution is formed by right triangle, rotating around one of the legs, where h is height of the cone from the bottom center to the top, is the leg of a right triangle around which rotation occurs. The second leg of a right triangle r is radius of the base of the cone. The hypotenuse of a right triangle is the l, which is generatrix of a cone.
In creating the sweep of the cone it is possible to use only two values – r and l. The radius of the base r determines the unfolding circle of the cone base, and the generatrix l defines the sector of a side surface, which is the radius of the side surface of the sector. The sector angle in the sweep of conical surface defined by the formula: 360^{0}*(r/l).
We know what the cone is, but what about surface area of a cone? Why do we need to solve problems using the surface area of a cone? For example, you need to know how much dough you need to cook a waffle cone, or how many bricks you will need to lay down a brick roof of the castle. That’s where the surface area of a cone comes in handy.
Imagine that waffle cone wrapped in a cloth. In order to find the area of the piece of fabric (surface area of a cone) you need to wrap the cone with, you need to cut the fabric and put it on the table. This way, you will get a flat shape, which surface area of a cone we can find.
The same applies to the cone. Cut its lateral surface along any generatrix. Then «unwind» the lateral surface onto a plane. There will be the sector. The center of this sector is the vertex of a cone. The sector radius is equal to the generatrix of the cone. The length of its arc coincides with the length of the circumference of the cone base. This sector is called the sweep of the cone. Now, let’s see how to calculate the surface area of a cone and use other cone’s formulas.
Formulas for Calculating the Surface Area of a Cone and the Volume of a Cone
The sweep of a side surface area of a cone is the circular sector whose radius is equal to the generatrix of the cone, and the length of the sector’s arc is the length of the circumference of the base of the cone.
The surface area of a cone consists of the side surface area of a cone and the base area (circle).
The side surface area of a cone is calculated as follows: S (side) = πRl, where R is the radius of the cone, l is the forming of a cone.
The area of the cone base is calculated by the formula: S = S (side) + S (circle) = S (circle) = πR2.
The surface area of a cone as a whole is calculated according to the formula: S = S (side) + S (circle) = πRl + πR2.
The side surface area of a cone is an area of its sweep. The sweep of the side surface of the cone is a circular sector.
The volume of a circular cone is calculated by the following formula: V = 1/3 πR2H, where the R is the radius of the base, H is the height of the cone.
The side surface area of a cone that is truncated can be found by the formula: S side = π (R + r) l, where R is the low base radius, r is the radius of the upper base, and l is length of the generator.
The total surface area of a cone that is truncated can be found by the formula: S = πR2 + πr2 + π (R + r) l, where R is the low base radius, r is the radius of the upper base, and l is the length of the generator.
The volume of a truncated cone can be found as follows: V = 1/3 πH (R2 + Rr + r2), where R is the radius of the lower base, r is the radius of the upper base, and H is the height of the cone.
Additional Information on the Cone
There is the «debris cone» notion in geology. It is a form of relief, formed by the accumulation of clastic rocks (pebbles, gravel, and sand), carried to the piedmont plain or to a broader surface by mountain rivers.
In biology, there is the notion of a «cone of origin». This is the tip of the shoot and plant root, which comprises of cells of a formative tissue.
Cone is also related to the family of marine mollusks of the front branchial subclass. Conical shell (216 cm) is brightly colored. There are more than 500 species of cones. They live in the tropics and subtropics. They are predators and have a poisonous gland. The bite of cones is very painful. There were death cases. The shells are used as decorations and souvenirs.
According to the statistics, every year from lightning strikes 6 people in 1000000 die around the world, mostly inhabitants of the southern countries. This would not have happened if there were lightning rods everywhere as this way the safety cone is formed. The higher the lightning rod, the greater the volume of the cone. In order to build the lightning rod of the right size we need to use the surface area of a cone. Some people try to hide from lightning under a tree, but trees are not conductors, as they accumulate the charges and can be a source of voltage.
In physics, there is a concept of «bodily cone». This is a cone angle, which is carved inside a sphere. The unit of measure of the bodily cone angle is one steradian. One steradian is a bodily angle, the square of the radius of which is equal to the area of the sphere it cuts. If we put a light source in this angle, which is equal to 1 candela, then we will obtain the luminous flux of 1 lumen. The light from a movie camera or a spotlight is distributed in the form of a cone.
Online Calculators for Finding the Surface Area of a Cone
In order to find the surface area of a cone using online services you need to input the radius of the base circle r and the length of the cone.
To find the side surface area of a cone or the total surface area of a cone input the values of the radius of the base and the value of the length of the cone’s generatrix, and press «Calculate» button.
The program will determine the side surface area of a cone or the total surface area of a cone.
Initial data and results of calculations can be copied to the clipboard for further use in other applications.
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