The study of the properties of spatial figures plays an important role in solving practical problems. The science that deals with figures in space is called stereometry. In this article, from the point of view of stereometry, we consider the cone and show how to find the area of ββthe cone.
Round base cone
In the general case, a cone is a surface constructed on some plane curve, all points of which are connected by segments with one point in space. The latter is called the top of the cone.
From the above definition it is clear that the curve can have an arbitrary shape, for example, parabolic, hyperbolic, elliptic and so on. Nevertheless, in practice and in problems in geometry, it is often a round cone that occurs. It is shown below in the figure.
Here, the symbol r denotes the radius of the circle located at the base of the figure, h is the perpendicular to the plane of the circle, which is drawn from the top of the figure. It is called height. The quantity s is the generator of the cone, or its generator.
It can be seen that the segments r, h and s form a right triangle. If it is rotated around the leg h, the hypotenuse s will describe the conical surface, and the leg r forms the round base of the figure. For this reason, the cone is considered a rotation figure. The three named linear parameters are interconnected by equality:
s 2 = r 2 + h 2
Note that the above equality is valid only for a round straight cone. A straight figure is only if its height falls exactly in the center of the circle of the base. If this condition is not satisfied, then the figure is called oblique. The difference between the straight and inclined cones is shown in the figure below.
Shape Sweep
The study of the surface area of ββthe cone is conveniently carried out by examining it on a plane. This method of representing the surface of figures in space is called their development. For a cone, this scan can be obtained as follows: you need to take a figure made, for example, of paper. Then, with scissors cut the round base around the circumference. After that, along the generatrix, make a cut of the conical surface and deploy it on a plane. The result of these simple operations will be a cone scan, as shown in the figure below.
As you can see, the surface of the cone can really be represented on the plane. It consists of two parts:
- a circle of radius r representing the base of the figure;
- circular sector of radius g, which is a conical surface.
The cone area formula assumes finding the areas of both unfolded surfaces.
Calculation of the surface area of ββa figure
We divide the task into two stages. First we find the area of ββthe base of the cone, then the area of ββthe conical surface.
The first part of the problem is easy to solve. Since the radius r is given, to calculate the area of ββthe base, it suffices to recall the corresponding expression for the area of ββthe circle. We write it down:
S o = pi Γ r 2
If the radius is not known, then you should first find it, using the formula of the relationship between it, height and the generatrix.
The second part of the problem of finding the area of ββthe cone is somewhat more complicated. Note that the circular sector is built on the radius g of the generator and is bounded by an arc whose length is equal to the circumference of the circle. This fact allows you to record the proportion and find the angle of the sector in question. Denote it by the Greek letter Ο. This angle will be equal to:
2 Γ pi => 2 Γ pi Γ g;
Ο => 2 Γ pi Γ r;
Ο = 2 Γ pi Γ r / g
Knowing the central angle Ο of the circular sector, it is possible to find its area using the appropriate proportion. Denote it by S b . It will be equal to:
2 Γ pi => pi Γ g 2 ;
Ο => S b ;
S b = pi Γ g 2 Γ Ο / (2 Γ pi) = pi Γ r Γ g
That is, the area of ββthe conical surface corresponds to the product of the generator g, the radius of the base r and the number Pi.
Knowing what the areas of both surfaces are equal, we can write the final formula for the area of ββthe cone:
S = S o + S b = pi Γ r 2 + pi Γ r Γ g = pi Γ r Γ (r + g)
The written expression assumes for calculating S the knowledge of two linear parameters of the cone. If g or r is unknown, then they can be found through the height h.
The task of calculating the area of ββthe cone
It is known that the height of a round straight cone is equal to its diameter. It is necessary to calculate the area of ββthe figure, knowing that the area of ββits base is 50 cm 2 .
Knowing the area of ββthe circle, you can find the radius of the figure. We have:
S o = pi Γ r 2 =>
r = β (S o / pi)
Now we find the generator g through h and r. According to the condition, the height h of the figure is equal to two radii r, then:
h = 2 Γ r;
g 2 = (2 Γ r) 2 + r 2 =>
g = β5 Γ r = β (5 Γ S o / pi)
The formulas found for g and r should be substituted into the expression for the entire area of ββthe cone. We get:
S = S o + pi Γ β (S o / pi) Γ β (5 Γ S o / pi) = S o Γ (1 + β5)
In the resulting expression, substitute the base area S o and write the answer: S β 161.8 cm 2 .