The cylinder is geometric body, bounded by two parallel planes and a cylindrical surface. In the article we will talk about how to find the area of a cylinder and, using the formula, we will solve several problems as an example.
A cylinder has three surfaces: a top, a base, and a side surface.
The top and base of a cylinder are circles and are easy to identify.
It is known that the area of a circle is equal to πr 2. Therefore, the formula for the area of two circles (the top and base of the cylinder) will be πr 2 + πr 2 = 2πr 2.
The third, side surface of the cylinder, is the curved wall of the cylinder. In order to better imagine this surface, let's try to transform it to get a recognizable shape. Imagine that the cylinder is an ordinary tin can that does not have a top lid or bottom. Let's make a vertical cut on the side wall from the top to the bottom of the can (Step 1 in the figure) and try to open (straighten) the resulting figure as much as possible (Step 2).
After the resulting jar is fully opened, we will see a familiar figure (Step 3), this is a rectangle. The area of a rectangle is easy to calculate. But before that, let's return for a moment to the original cylinder. The vertex of the original cylinder is a circle, and we know that the circumference is calculated by the formula: L = 2πr. It is marked in red in the figure.
When the side wall of the cylinder is fully opened, we see that the circumference becomes the length of the resulting rectangle. The sides of this rectangle will be the circumference (L = 2πr) and the height of the cylinder (h). The area of a rectangle is equal to the product of its sides - S = length x width = L x h = 2πr x h = 2πrh. As a result, we received a formula for calculating the area of the lateral surface of the cylinder.
Formula for the lateral surface area of a cylinder
S side = 2πrh
Total surface area of a cylinder
Finally, if we add the area of all three surfaces, we get the formula for the total surface area of a cylinder. The surface area of a cylinder is equal to the area of the top of the cylinder + the area of the base of the cylinder + the area of the side surface of the cylinder or S = πr 2 + πr 2 + 2πrh = 2πr 2 + 2πrh. Sometimes this expression is written identical to the formula 2πr (r + h).
Formula for the total surface area of a cylinder
S = 2πr 2 + 2πrh = 2πr(r + h)
r – radius of the cylinder, h – height of the cylinder
Examples of calculating the surface area of a cylinder
To understand the above formulas, let’s try to calculate the surface area of a cylinder using examples.
1. The radius of the base of the cylinder is 2, the height is 3. Determine the area of the lateral surface of the cylinder.
The total surface area is calculated using the formula: S side. = 2πrh
S side = 2 * 3.14 * 2 * 34.6. Total ratings received: 990.
Instructions
First of all, it is worth understanding that the lateral surface of the pyramid is represented by several triangles, the areas of which can be found using a variety of formulas, depending on the known data:
S = (a*h)/2, where h is the height lowered to side a;
S = a*b*sinβ, where a, b are the sides of the triangle, and β is the angle between these sides;
S = (r*(a + b + c))/2, where a, b, c are the sides of the triangle, and r is the radius of the circle inscribed in this triangle;
S = (a*b*c)/4*R, where R is the radius of the triangle circumscribed around the circle;
S = (a*b)/2 = r² + 2*r*R (if the triangle is right-angled);
S = S = (a²*√3)/4 (if the triangle is equilateral).
In fact, these are only the most basic known formulas for finding the area of a triangle.
Having calculated the areas of all triangles that are the faces of the pyramid using the above formulas, you can begin to calculate the area of this pyramid. This is done extremely simply: you need to add up the areas of all the triangles that form the side surface of the pyramid. This can be expressed by the formula:
Sp = ΣSi, where Sp is the area of the lateral surface, Si is the area of the i-th triangle, which is part of its lateral surface.
For greater clarity, we can consider a small example: given a regular pyramid, the side faces of which are formed by equilateral triangles, and at its base lies a square. The length of the edge of this pyramid is 17 cm. It is required to find the area of the lateral surface of this pyramid.
Solution: the length of the edge of this pyramid is known, it is known that its faces are equilateral triangles. Thus, we can say that all sides of all triangles on the lateral surface are equal to 17 cm. Therefore, in order to calculate the area of any of these triangles, you will need to apply the formula:
S = (17²*√3)/4 = (289*1.732)/4 = 125.137 cm²
It is known that at the base of the pyramid lies a square. Thus, it is clear that there are four given equilateral triangles. Then the area of the lateral surface of the pyramid is calculated as follows:
125.137 cm² * 4 = 500.548 cm²
Answer: The lateral surface area of the pyramid is 500.548 cm²
First, let's calculate the area of the lateral surface of the pyramid. The lateral surface is the sum of the areas of all lateral faces. If you are dealing with a regular pyramid (that is, one that has a regular polygon at its base, and the vertex is projected into the center of this polygon), then to calculate the entire lateral surface it is enough to multiply the perimeter of the base (that is, the sum of the lengths of all sides of the polygon lying at the base pyramid) by the height of the side face (otherwise called the apothem) and divide the resulting value by 2: Sb = 1/2P*h, where Sb is the area of the side surface, P is the perimeter of the base, h is the height of the side face (apothem).
If you have an arbitrary pyramid in front of you, you will have to separately calculate the areas of all the faces and then add them up. Since the side faces of the pyramid are triangles, use the formula for the area of a triangle: S=1/2b*h, where b is the base of the triangle, and h is the height. When the areas of all the faces have been calculated, all that remains is to add them up to get the area of the lateral surface of the pyramid.
Then you need to calculate the area of the base of the pyramid. The choice of formula for calculation depends on which polygon lies at the base of the pyramid: regular (that is, one with all sides of the same length) or irregular. Square regular polygon can be calculated by multiplying the perimeter by the radius of the circle inscribed in the polygon and dividing the resulting value by 2: Sn = 1/2P*r, where Sn is the area of the polygon, P is the perimeter, and r is the radius of the circle inscribed in the polygon.
A truncated pyramid is a polyhedron that is formed by a pyramid and its cross section parallel to the base. Finding the lateral surface area of the pyramid is not difficult at all. It is very simple: the area is equal to the product of half the sum of the bases by the apothem. Let's consider an example of calculating the lateral surface area of a truncated pyramid. Suppose we are given a regular quadrangular pyramid. The lengths of the base are b = 5 cm, c = 3 cm. Apothem a = 4 cm. To find the area of the lateral surface of the pyramid, you must first find the perimeter of the bases. In a large base it will be equal to p1=4b=4*5=20 cm. In a smaller base the formula will be as follows: p2=4c=4*3=12 cm. Therefore, the area will be equal to: s=1/2(20+12 )*4=32/2*4=64 cm.
A pyramid is a polyhedron, one of whose faces (base) is an arbitrary polygon, and the remaining faces (sides) are triangles having a common vertex. According to the number of angles, the base of the pyramid is triangular (tetrahedron), quadrangular, and so on.
A pyramid is a polyhedron with a base in the form of a polygon, and the remaining faces are triangles with a common vertex. An apothem is the height of the side face of a regular pyramid, which is drawn from its vertex.
The area of the lateral surface of an arbitrary pyramid is equal to the sum of the areas of its lateral faces. It makes sense to give a special formula for expressing this area in the case of a regular pyramid. So, let us be given a regular pyramid, at the base of which lies a regular n-gon with side equal to a. Let h be the height of the side face, also called apothem pyramids. The area of one side face is equal to 1/2ah, and the entire side surface of the pyramid has an area equal to n/2ha. Since na is the perimeter of the base of the pyramid, we can write the found formula in the form:
Lateral surface area of a regular pyramid is equal to the product of its apothem and half the perimeter of the base.
Concerning total surface area, then we simply add the area of the base to the side one.
Inscribed and circumscribed sphere and ball. It should be noted that the center of the sphere inscribed in the pyramid lies at the intersection of the bisector planes of the internal dihedral angles of the pyramid. The center of the sphere described near the pyramid lies at the intersection of planes passing through the midpoints of the edges of the pyramid and perpendicular to them.
Truncated pyramid. If a pyramid is cut by a plane parallel to its base, then the part enclosed between the cutting plane and the base is called truncated pyramid. The figure shows a pyramid; discarding its part lying above the cutting plane, we get a truncated pyramid. It is clear that the small discarded pyramid is homothetic to the large pyramid with the center of homothety at the apex. The similarity coefficient is equal to the ratio of heights: k=h 2 /h 1, or side edges, or other corresponding linear dimensions of both pyramids. We know that the areas of similar figures are related like squares of linear dimensions; so the areas of the bases of both pyramids (i.e. the area of the bases of the truncated pyramid) are related as
Here S 1 is the area of the lower base, and S 2 is the area of the upper base of the truncated pyramid. In the same relation are side surfaces pyramids A similar rule exists for volumes.
Volumes of similar bodies are related like cubes of their linear dimensions; for example, the volumes of pyramids are related as the product of their heights and the area of the bases, from which our rule is immediately obtained. It is of a completely general nature and directly follows from the fact that volume always has a dimension of the third power of length. Using this rule, we derive a formula expressing the volume of a truncated pyramid through the height and area of the bases.
Let a truncated pyramid with height h and base areas S 1 and S 2 be given. If we imagine that it is extended to a full pyramid, then the coefficient of similarity between the full pyramid and the small pyramid can easily be found as the root of the ratio S 2 /S 1 . The height of a truncated pyramid is expressed as h = h 1 - h 2 = h 1 (1 - k). Now we have for the volume of a truncated pyramid (V 1 and V 2 denote the volumes of the full and small pyramids)
formula for the volume of a truncated pyramid
Let us derive the formula for the area S of the lateral surface of a regular truncated pyramid through the perimeters P 1 and P 2 of the bases and the length of the apothem a. We reason in exactly the same way as when deriving the formula for volume. We supplement the pyramid with the upper part, we have P 2 = kP 1, S 2 =k 2 S 1, where k is the similarity coefficient, P 1 and P 2 are the perimeters of the bases, and S 1 and S 2 are the areas of the lateral surfaces of the entire resulting pyramid and its the upper part accordingly. For the lateral surface we find (a 1 and a 2 are apothems of the pyramids, a = a 1 - a 2 = a 1 (1-k))
formula for the lateral surface area of a regular truncated pyramid
In a regular triangular pyramid SABC R- middle of the rib AB, S- top.
It is known that SR = 6, and the lateral surface area is equal to 36
.
Find the length of the segment B.C..
Let's make a drawing. In a regular pyramid, the side faces are isosceles triangles.
Line segment S.R.- the median lowered to the base, and therefore the height of the side face.
The lateral surface area of a regular triangular pyramid is equal to the sum of the areas
three equal side faces S side = 3 S ABS. From here S ABS = 36: 3 = 12- area of the face.
The area of a triangle is equal to half the product of its base and height
S ABS = 0.5 AB SR. Knowing the area and height, we find the side of the base AB = BC.
12 = 0.5 AB 6
12 = 3 AB
AB = 4
Answer: 4
You can approach the problem from the other end. Let the base side AB = BC = a.
Then the area of the face S ABS = 0.5 AB SR = 0.5 a 6 = 3a.
The area of each of the three faces is equal to 3a, the area of the three faces is equal 9a.
According to the conditions of the problem, the area of the lateral surface of the pyramid is 36.
S side = 9a = 36.
From here a = 4.
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