The surface area of \u200b\u200bthe right quadrangular pyramid formula. Side surface area of \u200b\u200bthe right quadrangular pyramid: Formulas and examples of tasks. Square side surface of the pyramid
Definition. Side - This is a triangle, in whom one corner lies in the top of the pyramid, and the party opposed to him coincides with the base side (polygon).
Definition. Side edges - These are common side of the side faces. The pyramid has so many ribs how many corners have a polygon.
Definition. Height of the pyramid - This is a perpendicular, lowered from the top to the base of the pyramid.
Definition. Apothem - This is a perpendicular of the side face of the pyramid, lowered from the top of the pyramid to the side of the base.
Definition. Diagonal section - This is a cross section of a pyramid with a plane passing through the top of the pyramid and the base diagonal.
Definition. Right pyramid - This is a pyramid in which the basis is the right polygon, and the height falls into the center of the base.
The volume and surface area of \u200b\u200bthe pyramid
Formula. Pyramid volume Through the base area and height:
Properties of the pyramid
If all the side ribs are equal, then around the base of the pyramid can be described, and the center of the base coincides with the center of the circle. Also, perpendicular, lowered from the top passes through the center of the base (circle).
If all side ribs are equal, then they are tilted to the base plane at the same angles.
The side ribs are equal when they form with the plane of the base equal angles or if the circle can be described around the base of the pyramid.
If the side faces are tilted to the base plane at one angle, then in the base of the pyramid you can enter the circle, and the peak of the pyramid is designed to its center.
If the side faces are tilted to the base plane at one angle, then the apophems of the side faces are equal.
Properties of the right pyramid
1. The vertex of the pyramid is equidistant from all corners of the base.
2. All side ribs are equal.
3. All side edges are tilted under the same corners to the base.
4. Apofims of all side faces are equal.
5. Areas of all side faces are equal.
6. All faces have the same dihedral (flat) angles.
7. Around the pyramid you can describe the sphere. The center of the described sphere is the intersection point of perpendiculars, which pass through the middle of the ribs.
8. In the pyramid you can enter the sphere. The center of the inscribed sphere will be the intersection point of bisector emanating from the corner between the edge and the base.
9. If the center of the inscribed sphere coincides with the center of the described sphere, then the sum of flat corners at the top is equal to π or vice versa, one angle is π / n, where N is the number of angles at the base of the pyramid.
Pyramid connection with sphere
Around the pyramid, you can describe the sphere when at the base of the pyramid lies a polyhedron around which you can describe the circle (necessary and sufficient condition). The center of the sphere is the intersection point of the planes passing perpendicular through the middle of the side ribs of the pyramids.
Around any triangular or correct pyramid can always be described by the sphere.
In the pyramid, you can enter the sphere if the biss-sector planes of the internal dwarfrani corners of the pyramids intersect at one point (necessary and sufficient condition). This point will be the center of the sphere.
Pyramid connection with cone
The cone is called inscribed in the pyramid if their vertices coincide, and the base of the cone is inscribed in the base of the pyramid.
The cone can be entered into the pyramid if the apophems of the pyramids are equal to each other.
The cone is called the pyramid described around, if their vertices coincide, and the base of the cone is described around the base of the pyramid.
The cone can be described around the pyramid if all the side ribs of the pyramid are equal to each other.
Pyramid connection with cylinder
The pyramid is called inscribed in the cylinder if the top of the pyramid lies on one basis of the cylinder, and the base of the pyramid is written to another base of the cylinder.
The cylinder can be described around the pyramid if around the base of the pyramid you can describe the circle.
The four-edged four faces and four vertices and six ribs, where any two ribs do not have common vertices but not come into contact.
Each peak consists of three faces and ribs that form three corner.
The segment connecting the vertex of the tetrahedron with the center of the opposite face is called median tetrahedron (GM).
Bimedian It is called a segment connecting the mid-opposite ribs that do not come into contact (KL).
All bimedians and medians of the tetrahedral intersect at one point (s). At the same time, the bimedians are divided by half, and medians in respect of 3: 1 starting from the vertex.
Definition. Acreditated pyramid - This is a pyramid in which the apophem is more than half the length of the base side of the base.
Definition. Stupid pyramid - This is a pyramid in which the apophem is less than half the length of the base side.
Definition. Right tetrahedron - A tetrahedron who has all four faces - equilateral triangles. It is one of the five right polygons. In the right tetrahedron, all dumarted angles (between the edges) and triangular angles (at the top) are equal.
Definition. Rectangular tetrahedron A tetrahedron is called a straight angle between three ribs at the top (ribs perpendicular). Three faces form rectangular triangular corner And the faces are rectangular triangles, and the basis of an arbitrary triangle. Apothem of any face is equal to half the side of the foundation that the apophem falls.
Definition. A washing tetrahedron The tetrahedron is called the lateral facets are equal to each other, and the base is the right triangle. Such a tetrahedron serves an isolated triangles.
Definition. Orthocentric tetrahedron A tetrahedron is called all heights (perpendicular), which is omitted from the top to the opposite face, intersect at one point.
Definition. Star pyramid The polyhedron is called the base is the star.
Typical geometric tasks on the plane and in three-dimensional space are the problems of determining the areas of surfaces of different figures. In this article, we give the formula of the side surface area of \u200b\u200bthe correct pyramid of quadrangular.
What is the pyramid?
Let us give a strict geometric definition of the pyramid. Suppose there is some polygon with N sides and with n angles. Choose an arbitrary point of space that will not be in the plane of the specified N-carbon, and connect it from each pin of the polygon. We will get a figure having some volume called an N-coal pyramid. For example, we will show in the figure below how the pentagonal pyramid looks like.
Two important elements of any pyramid is its base (N-square) and therapy. These elements are connected to each other n triangles, which are generally not equal to each other. Perpendicular, lowered from the top to the base, is called the height of the figure. If it crosses the base in the geometric center (coincides with the center of the masses of the polygon), then such a pyramid is called straight. If, in addition to this condition, the base is the right polygon, then the entire pyramid is called proper. The figure below shows how the right pyramids with triangular, quadrangular, pentagonal and hexagonal base look like.
The surface of the pyramid
Before switching to the question about the side surface area of \u200b\u200bthe right pyramid of quadrangular, it is necessary to dwell on the concept of the surface itself.
As mentioned above and shown in the drawings, any pyramid is formed by a set of faces or sides. One side is the basis, and N of the parties are triangles. The surface of the whole figure is the sum of the area of \u200b\u200beach side.
The surface is convenient to study on the example of the scan of the figure. The scan for the correct quadrangular pyramid is shown in the drawings below.
We see that its surface area is equal to the sum of the four squares of the same inaccessible triangles and square square.
The total area of \u200b\u200ball triangles, which form the side sides of the figure, is customary to be called the side surface area. Next, we show how to calculate it for the quadrangular pyramid correctly.
Side surface of the quadrangular correct pyramid
To calculate the side surface area of \u200b\u200bthe specified figure, turn back to the above scan. Suppose we know the side of the square foundation. Denote it with a symbol a. It can be seen that each of the four identical triangles has a base of length a. To calculate their total area, you need to know this value for one triangle. From the course of geometry, it is known that the triangle SA T is equal to the product of the base to the height, which should be divided by half. I.e:
Where H b is the height of an inaccessible triangle, conducted to the base a. For the pyramid, this height is appotable. Now it remains to multiply the obtained expression on 4 to obtain the area S B side surface for the pyramid under consideration:
S B \u003d 4 * S T \u003d 2 * H b * a.
This formula contains two parameters: apotheme and side of the base. If the latter is known in most conditions, then the first one has to calculate, knowing other values. We give formulas for calculating Apotheme H B for two cases:
- when the length of the side edge is known;
- when the height of the pyramid is known.
If you designate the length of the side of the side (side of an equilibried triangle) symbol L, then Apotheme H B is to determine the formula:
h B \u003d √ (L 2 - A 2/4).
This expression is the result of the use of the Pythagorean theorem for the side surface triangle.
If the height h of the pyramid is known, then the H B is designed as follows:
h B \u003d √ (H 2 + A 2/4).
It is also not difficult to obtain this expression, if you consider inside the pyramid, a rectangular triangle formed by Cates H and A / 2 and hypotenuse H b.
We show how to apply these formulas by deciding two interesting tasks.
Task with a well-known surface area
It is known that the side surface area of \u200b\u200bthe quadrangular is 108 cm 2. It is necessary to calculate the value of its length of its Apotheme H B, if the height of the pyramid is 7 cm.
We write the surface formula s b surface of the side through the height. We have:
S B \u003d 2 * √ (H 2 + A 2/4) * a.
Here we simply substituted the appropriate formula of Apotheme into an expression for s b. Erected both parts of equality in the square:
S b 2 \u003d 4 * A 2 * H 2 + A 4.
To find the value A, we will replace the variables:
t 2 + 4 * H 2 * T - S b 2 \u003d 0.
Now we substitute the known values \u200b\u200band solve the square equation:
t 2 + 196 * T - 11664 \u003d 0.
We prescribed only a positive root of this equation. Then the bases of the base of the pyramid will be equal to:
a \u003d √t \u003d √47,8355 ≈ 6,916 cm.
To get the length of Apotheme, it is enough to use the formula:
h B \u003d √ (H 2 + A 2/4) \u003d √ (7 2 + 6.916 2/4) ≈ 7.808 cm.
Side Surface of Heops Pyramid
We define the importance of the side for the largest Egyptian pyramid. It is known that in its foundation there is a square on the side of 230,363 meters. The height of the structure was initially 146.5 meters. We substitute these numbers into the appropriate formula for s b, we get:
S b \u003d 2 * √ (H 2 + A 2/4) * a \u003d 2 * √ (146.5 2 +230.363 2/4) * 230,363 ≈ 85860 m 2.
The value found is a bit more square 17 football fields.
Before learning questions about this geometric shape and its properties, it should be understood in some terms. When a person hears about the pyramid, he is preserving the tremendous buildings in Egypt. So looks like the simplest one. But they come in different types and forms, which means that the calculation formula for geometric shapes will be different.
Types of Figure
Pyramid - Geometric Figuredenoting and representing a few faces. In fact, it is the same polyhedron, at the base of which a polygon lies, and the sides are the triangles connecting at one point - the top. The figure is two main species:
- proper;
- truncated.
In the first case, at the bottom there is a regular polygon. Here all side surfaces are equal Between themselves, the figure itself will delight the eye of the perfectionist.
In the second case, the bases are two are large at the very bottom and small between the vertex, repeating the form of the main one. In other words, the truncated pyramid is a polyhedron with a cross section formed in parallel base.
Terms and notation
Major terms:
- Correct (equilateral) triangle - Figure with three identical corners and equal parties. In this case, all angles have 60 degrees. The figure is the simplest of the right polyhedra. If this figure lies at the base, then such a polyhedron will be called proper triangular. If there is a square, the pyramid will be referred to as the correct four-graded pyramid.
- Vertex - The top point where the edges are converged. The height of the vertex is formed by a straight line emanating from the vertex to the base of the pyramid.
- Face - One of the planes of the polygon. It can be in the form of a triangle in the case of a triangular pyramid or in the form of a trapezium for a truncated pyramid.
- Section - Flat figure formed as a result of dissection. It is not necessary to be confused with a cut, as the incision shows what is in cross section.
- Apothem - Cut, conducted from the top of the pyramid to its base. It is also the height of that edge where the second height point is located. This definition is valid only with respect to the correct polyhedron. For example, if this is not a truncated pyramid, the edge will be a triangle. In this case, the height of this triangle will become apophey.
Formulas Square
Find the sideways of the pyramid Any type can be in several ways. If the figure is not symmetric and is a polygon with different sides, then in this case it is easier to calculate the total surface area through the totality of all surfaces. In other words - it is necessary to calculate the area of \u200b\u200beach face and fold them together.
Depending on which parameters are known, formulas for calculating the square, trapezoids, an arbitrary four-broth, etc. may be required. Formulas themselves in different cases Also will have differences.
In the case of the correct figure, find the area is much easier. It is enough to know only a few key parameters. In most cases, calculations are required for such figures. Therefore, the corresponding formulas will be given further. Otherwise, I would have to paint everything into several pages, which will only confuse and dismiss.
Basic formula for calculating The side surface area of \u200b\u200bthe right pyramid will have the following form:
S \u003d ½ PA (P is the perimeter of the base, and - apophem)
Consider one of the examples. The polyhedron has a base with segments A1, A2, A3, A4, A5, and all of them are equal to 10 cm. Appeham Let it be 5 cm. To begin with, it is necessary to find a perimeter. Since all five faces of the base are the same, it is possible to find this: p \u003d 5 * 10 \u003d 50 cm. Next, we use the basic formula: S \u003d ½ * 50 * 5 \u003d 125 cm in a square.
Side surface area of \u200b\u200bthe correct triangular pyramid Calculate the easiest. The formula has the following form:
S \u003d ½ * AB * 3, where a is apophem, B is a base of the base. The tripler multiplier here means the number of bases of the base, and the first part is the side surface area. Consider an example. The figure with an apophural 5 cm and the base of the base is 8 cm. Calculate: S \u003d 1/2 * 5 * 8 * 3 \u003d 60 cm in square.
Side side surface of a truncated pyramid Calculate a little more difficult. The formula looks like this: S \u003d 1/2 * (p _01 + p _02) * A, where p_01 and p_02 are the perimeters of the bases, and - apophem. Consider an example. Suppose the sizes of the bases of the base 3 and 6 cm are given for a quadriginal figure, the apophem is 4 cm.
Here, for a start, it is necessary to find the perimeters of the base: p_01 \u003d 3 * 4 \u003d 12 cm; p_02 \u003d 6 * 4 \u003d 24 cm. It remains to substitute the values \u200b\u200bto the main formula and we obtain: S \u003d 1/2 * (12 + 24) * 4 \u003d 0.5 * 36 * 4 \u003d 72 cm in the square.
Thus, you can find the side surface area of \u200b\u200bthe correct pyramid of any complexity. Should be attentive and not confused These calculations with the total area of \u200b\u200bthe entire polyhedron. And if it still needs to be done - it is enough to calculate the area of \u200b\u200bthe largest base of the polyhedron and add it to the side surface area of \u200b\u200bthe polyhedron.
Video
Secure information on how to find the side surface area of \u200b\u200bdifferent pyramids, this video will help you.
What kind of figure we call the pyramid? First, this is a polyhedron. Secondly, at the base of this polyhedron there is an arbitrary polygon, and the sides of the pyramid (side faces) necessarily have the form of triangles converging in one total vertex. Now, having understood with the term, find out how to find the surface area of \u200b\u200bthe pyramid.
It is clear that the surface area of \u200b\u200bsuch a geometric body will be made up of the sum of the base area and its entire side surface.
Calculation of the base area of \u200b\u200bthe pyramid
The choice of the calculated formula depends on the form of a polygon underlying in the foundation of our pyramid. It can be correct, that is, with the sides of the same length, or incorrect. Consider both options.
Based on the right polygon
From the school course it is known:
- the square of the square will be equal to the length of its side, erected into the square;
- the equilateral triangle area is equal to the square of it, shared on 4 and multiplied on the square root of three.
But there is a general formula for calculating the area of \u200b\u200bany correct polygon (SN): it is necessary to multiply the perimeter value of this polygon (P) to the radius inscribed in it (R), and then split the resulting result into two: Sn \u003d 1 / 2p * R .
Based on the wrong polygon
The scheme of finding its area is to first divide the entire polygon on triangles, calculate the area of \u200b\u200beach of them by the formula: 1 / 2a * h (where A is the base of the triangle, H is the height-lowered to this base), fold all the results.
Square side surface of the pyramid
Now we calculate the area of \u200b\u200bthe side surface of the pyramid, i.e. The sum of the squares of all its side. There are also 2 options here.
- Let us have an arbitrary pyramid, i.e. Such, at the base of which - an irregular polygon. Then the area of \u200b\u200beach face should be calculated and folded the results. Since only triangles can be the side sides of the pyramid, then the calculation is based on the formula mentioned above: S \u003d 1 / 2a * h.
- Let our pyramid be correct, i.e. In its foundation lies the right polygon, and the projection of the peak of the pyramid turns out to be in its center. Then, to calculate the side surface area (SB), it is sufficient to find half the work of the perimeter of the polygon base (P) to the height (H) of the side (the same for all edges): Sb \u003d 1/2 p * h. The perimeter of the polygon is determined by the addition of the lengths of all of its sides.
The total surface area of \u200b\u200bthe right pyramid is due to the summation of the area of \u200b\u200bits base with the area of \u200b\u200bthe entire side surface.
Examples
For example, calculate the algebraically surface area of \u200b\u200bseveral pyramids.
Surface Surface Triangular Pyramid
Based on such a pyramid - a triangle. According to the formula SO \u003d 1 / 2a * h we find the base area. The same formula is used to find the area of \u200b\u200beach face of the pyramid, also having a triangular shape, and we obtain 3 areas: S1, S2 and S3. The area of \u200b\u200bthe side surface of the pyramid is the sum of all areas: Sb \u003d S1 + S2 + S3. After folding the side of the side and base, we obtain the full surface area of \u200b\u200bthe desired pyramid: SP \u003d SO + SB.
The surface area of \u200b\u200bthe quadrangular pyramid
The side surface area is the sum of 4-exharicted: Sb \u003d S1 + S2 + S3 + S4, each of which is calculated by the formula of the triangle area. And the base area will have to search, depending on the form of a quadrangle - correct or incorrect. The area of \u200b\u200bthe full surface of the pyramid will again result in the addition of the base area and the full surface area of \u200b\u200bthe predetermined pyramid.
Triangular pyramid A polyhedron is called, at the base of which the right triangle lies.
In such a pyramid, the edge of the base and edges of the side of the sides are equal to each other. Accordingly, the side of the side faces is located from the sum of the area of \u200b\u200bthe three identical triangles. Find the side surface area of \u200b\u200bthe right pyramid by the formula. And you can make a calculation several times faster. To do this, apply the formula of the side surface area of \u200b\u200bthe triangular pyramid:
where P is the perimeter of the base, in which all parties are equal to b, A - apophem, lowered from the top to this base. Consider an example of calculating the area of \u200b\u200bthe triangular pyramid.
Task: Let the correct pyramid be given. The side of the triangle underlying at the base is B \u003d 4 cm. Apperam pyramid is equal to a \u003d 7 cm. Find the side surface area of \u200b\u200bthe pyramid.
Since, according to the terms of the task, we know the length of all necessary elements, we will find the perimeter. We remember that in the right triangle, all parties are equal, and, therefore, the perimeter is calculated by the formula:
We will substitute the data and find the value:
Now, knowing the perimeter, we can count the side surface area: 
To apply the Formula of the Triangular Pyramid Square to calculate the full value, it is necessary to find the area of \u200b\u200bthe polyhedron base. For this, the formula is used: 
The formula of the base of the triangular pyramid may be the other. It is allowed to apply any calculation of parameters for a given figure, but most often it is not required. Consider an example of calculating the area of \u200b\u200bthe base of the triangular pyramid.
Task: In the right pyramid, the side of the underlying triangle is equal to a \u003d 6 cm. Calculate the base area.
To calculate, we need only the length of the side of the right triangle, located at the base of the pyramid. Substitute data in the formula: 
Quite often, it is required to find the full area of \u200b\u200bthe polyhedron. To do this, it will be necessary to fold the side surface area and base. 
Consider an example of calculating the area of \u200b\u200bthe triangular pyramid.
Task: Let the correct triangular pyramid be given. The base side is equal to B \u003d 4 cm, apophem a \u003d 6 cm. Find the full area of \u200b\u200bthe pyramid.
To begin with, we find the side surface area along the already known formula. Calculate perimeter:
We substitute the data in the formula: 
Now we find the foundation area: 
Knowing the area of \u200b\u200bthe base and side surface, we find the full area of \u200b\u200bthe pyramid: 
When calculating the area of \u200b\u200bthe right pyramid, it is worth not to forget that at the base lies the right triangle and many elements of this polyhedron are equal to each other.
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