Rectangular pyramid formula. Pyramid and her elements

We continue to consider the tasks of the Mathematics in the exam. We have already investigated the tasks where the condition is given and you want to find the distance between the two dots or angle.

The pyramid is a polyhedron, the base of which is a polygon, the rest of the face - triangles, with what they have a common vertex.

The correct pyramid is a pyramid at the base of which the correct polygon lies, and its peak is projected into the center of the base.

The correct quadrangular pyramid - again the square is the square. The pyramid is designed to the point of intersection of the base diagonals (square).

ML - apophem
∠mlo - dihedral angle at the base of the pyramid
∠mco - angle between the side edge and the plane of the base of the pyramid

In this article, we will consider the tasks of solving the right pyramid. It is required to find any element, a side surface area, volume, height. Of course, you need to know the theorem of the Pyphagora, the formula of the side surface of the pyramid, the formula for finding the volume of the pyramid.

In the article "" Are presented formulas that are necessary for solving tasks for stereometry. So, the tasks:

Sabcd. point O. - center base,S. vertex, SO. = 51, AC \u003d 136. Find the side edgeSC.

In this case, there is a square at the base. This means that the AC and BD diagonals are equal, they intersect and the intersection point is divided by half. It should be noted that in the right pyramid, the height of its vertices passes through the center of the base of the pyramid. Thus, SO is a height, and a triangleSOC. rectangular. Then, on the Pythagora theorem:

How to extract the root of a large number.

Answer: 85.

Solve yourself:

In the right quadrangular pyramid Sabcd. point O. - center base, S. vertex, SO. = 4, AC \u003d 6. Find the side edge SC.

In the right quadrangular pyramid Sabcd. point O. - center base, S. vertex, SC = 5, AC \u003d 6. Locate the length of the segment SO..

In the right quadrangular pyramid Sabcd. point O. - center base, S. vertex, SO. = 4, SC \u003d 5. Locate the length of the segment AC.

Sabc. R. - Mid-rib BC., S. - Top. It is known that AB \u003d 7, and Sr. \u003d 16. Find the side surface area.

The side surface area of \u200b\u200bthe correct triangular pyramid is equal to half the work of the perimeter of the base on an apothemy (apophem is the height of the side face of the right pyramid, conducted from its vertex):

Or it can be said: the area of \u200b\u200bthe side surface of the pyramid is equal to the sum of the area of \u200b\u200bthree side faces. Sidelights in the correct triangular pyramid are equal in the area of \u200b\u200btriangles. In this case:

Answer: 168.

Solve yourself:

In the right triangular pyramid Sabc. R. - Mid-rib BC., S. - Top. It is known that AB \u003d 1, and Sr. \u003d 2. Locate the side surface area.

In the right triangular pyramid Sabc. R. - Mid-rib BC., S. - Top. It is known that AB \u003d 1, and the side surface area is equal to 3. Locate the length of the segment Sr..

In the right triangular pyramid Sabc. L. - Mid-rib BC., S. - Top. It is known that SL. \u003d 2, and the side surface area is equal to 3. Locate the length of the segment AB.

In the right triangular pyramid Sabc. M.. Area of \u200b\u200ba triangle ABC equal to 25, the pyramid volume is 100. Locate the length of the segment MS..

Pyramid base - equilateral triangle. therefore M. is the center of the base, andMS. - the height of the right pyramidSabc.. Volume of pyramid Sabc. equal: to inspect the decision

In the right triangular pyramid Sabc. Medians grounds intersect at point M.. Area of \u200b\u200ba triangle ABC equal to 3, MS. \u003d 1. Find the volume of the pyramid.

In the right triangular pyramid Sabc. Medians grounds intersect at point M.. The volume of the pyramid is 1, MS. \u003d 1. Find the triangle area ABC.

On this finish. As you can see, the tasks are solved in one or two actions. In the future, we consider with you other tasks from this part, where the bodies of rotation are given, do not miss!

I wish you success!

Sincerely, Alexander Krutitsky.

P.S: I will be grateful if you tell about the site on social networks.

The pyramid is a polyhedron, at the base of which the polygon is lying. All facets, in turn, form triangles that converge in one vertex. Pyramids are triangular, quadrangular and so on. In order to determine which pyramid in front of you is enough to calculate the number of corners in its base. The definition of the "pyramid height" is very often found in the geometry tasks in the school program. In the article, try to consider different ways to find it.

Pieces of pyramid

Each pyramid consists of the following elements:

• side faces that have three angle and converge in the top;
• apofhem is a height that sinks from its vertices;
• the top of the pyramid is a point that connects the side ribs, but does not lie in the base plane;
• the base is a polygon on which the top does not lie;
• the height of the pyramid is a segment that crosses the peak of the pyramid and forms a straight angle with its base.

How to find the height of the pyramid if its volume is known

Through the formula V \u003d (S * H) / 3 (in the formula V - volume, S - the base area, H is the height of the pyramid) we find that H \u003d (3 * V) / s. To secure the material, let's immediately solve the task. In the triangular base is 50 cm 2, whereas its volume is 125 cm 3. The height of the triangular pyramid, which we need to find is unknown. Here everything is simple: insert the data into our formula. We obtain H \u003d (3 * 125) / 50 \u003d 7.5 cm.

How to find the height of the pyramid, if the length of the diagonal and its ribs are known

As we remember, the height of the pyramid forms with its base a straight angle. And this means that the height, edge and half of the diagonal together form many, of course, remember the theorem of Pythagora. Knowing two dimensions, the third magnitude will be easy to find. Recall the famous A² \u003d B² + C² theorem, where a is hypotenuse, and in our case the edge of the pyramids; B - the first catt or half is diagonally and C - respectively, the second catat, or the height of the pyramid. From this formula C² \u003d A² - B².

Now the task: in the correct pyramid, the diagonal is 20 cm, when as the length of the ribs is 30 cm. It is necessary to find a height. We decide: C² \u003d 30² - 20² \u003d 900-400 \u003d 500. Hence C \u003d √ 500 \u003d about 22.4.

How to find a height of a truncated pyramid

It is a polygon, which has a cross section parallel to its base. The height of the truncated pyramid is a segment that connects two bases. The height can be found in the right pyramid if the length of the diagonals of both bases will be known, as well as the edge of the pyramids. Let the larger base diagonal equal to D1, while the diagonal of a smaller base is d2, and the rib has a length - l. To find the height, you can lower the height on its base from the two upper opposite points of the diagram. We see that we have two rectangular triangles, it remains to find the lengths of their cathets. For this, from the larger diagonal, we subtract smaller and divide on 2. So we will find one roll: a \u003d (d1-d2) / 2. After that, according to the Pythagora theorem, we can only find the second catt, which is the height of the pyramid.

Now consider the whole thing in practice. Before us the task. The truncated pyramid has a square at the base, the length of the larger base diagonal is 10 cm, while smaller - 6 cm, and the edge is 4 cm. It is required to find height. To begin, we find one or \u003d (10-6) / 2 \u003d 2 cm. One catat is 2 cm, and hypotenuse - 4 cm. It turns out that the second catat or height will be 16-4 \u003d 12, that is, H \u003d √12 \u003d about 3.5 cm.

This video tutorial will help users get an idea of \u200b\u200bthe theme of the pyramid. Proper pyramid. In this lesson, we will get acquainted with the concept of the pyramid, let her define. Consider what the right pyramid and what properties it possesses. Then we prove the theorem about the side surface of the right pyramid.

In this lesson, we will get acquainted with the concept of the pyramid, let her define.

Consider a polygon A 1 and 2...A N.which lies in the plane α and the point P.which does not lie in the plane α (Fig. 1). Connect the point P. With vertices A 1, and 2, and 3, … A N.. Receive n. Triangles: A 1 A 2 p, A 2 A 3 p etc.

Definition. Polyhedron Ra 1 A 2 ... A Ncomposed of n.-Golfish A 1 and 2...A N. and n.triangles Ra 1 and 2, 3 a 3 …Ra n a n -1, called n.-Gal pyramid. Fig. one.

Fig. one

Consider a quadrangular pyramid Pabcd. (Fig. 2).

R - Top of the pyramid.

Abcd. - The base of the pyramid.

R. - Side edge.

AU - Rib foundation.

From the point R Lower perpendicular PH on the foundation plane Assd.. Perpendicular conducted is the height of the pyramid.

Fig. 2.

The full surface of the pyramid consists of the surface of the side, that is, the area of \u200b\u200ball side faces, and the ground area:

S full \u003d s side + s land

The pyramid is called correct if:

• its foundation is the right polygon;
• the segment connecting the vertex of the pyramid with the center of the base is its height.

Explanation on the example of the correct quadrangular pyramid

Consider the right quadrangular pyramid Pabcd. (Fig. 3).

R - Top of the pyramid. Pyramid base Assd. - The right quadrangle, that is, the square. Point ABOUTThe intersection point of diagonals is the square of the square. It means RO - This is the height of the pyramid.

Fig. 3.

Explanation: In the right one N.The degree of the center inscribed and the center of the circle described coincides. This center is called the center of the polygon. Sometimes they say that the peak is designed to the center.

The height of the side face of the right pyramid, carried out of its vertex, is called apophistician And denotes h a..

1. All side edges of the right pyramid are equal;

2. Side faces are equal to be a feasible triangles.

Proof of these properties Let us give on the example of the correct quadrangular pyramid.

Dano: Ravd. - Proper quadrangular pyramid,

Assd. - Square,

RO - Height of the pyramid.

Prove:

1. RA \u003d PV \u003d PC \u003d PD

2. ΔAvr \u003d ΔVCR \u003d ΔCdr \u003d ΔDap See Fig. four.

Fig. four

Evidence.

RO - Height of the pyramid. That is, straight RO Perpendicular to the plane ABCand therefore direct JSC, in, with and DOlying in her. So triangles Roa, ditch, Ros, RD - rectangular.

Consider the square Assd.. From the properties of the square it follows that JSC \u003d at \u003d with = DO.

Then have rectangular triangles Roa, ditch, Ros, RD cathe RO - General and cathets JSC, in, with and DOequal, it means that these triangles are equal to two categories. From the equality of the triangles, the equality of segments flows, RA \u003d PV \u003d PC \u003d PD.Paragraph 1 is proved.

Segments AUand Sun.equal, as they are the parties to one square, RA \u003d PV \u003d PC. So triangles ABRand VCR -equal and equal in three sides.

Similarly, we get that triangles AVR, VCR, CDR, DAP Equally equally equal, which was required to prove in paragraph 2.

The side surface area of \u200b\u200bthe correct pyramid is equal to half the work of the perimeter of the base on apophem:

To prove, choose the correct triangular pyramid.

Dano: Ravox - Proper triangular pyramid.

AB \u003d Sun \u003d AU.

RO - Height.

Prove: . See fig. five.

Fig. five

Evidence.

Ravox - Proper triangular pyramid. I.e AU= AC \u003d Sun.. Let be ABOUT - Center Triangle ABC, then RO - This is the height of the pyramid. Based on the pyramid lies an equilateral triangle ABC. notice, that .

Triangles Rav, RVC, RSA - equal equal triangles (by property). The triangular pyramid has three side faces: Rav, RVC, RSA. So, the area of \u200b\u200bthe side surface of the pyramid is equal to:

S side \u003d 3S Rav

Theorem is proved.

The radius of the circle inscribed in the base of the correct quadrangular pyramid is 3 m, the pyramid height is 4 m. Find the side surface area of \u200b\u200bthe pyramid.

Dano: Proper quadrangular pyramid Assd.,

Assd. - Square,

r. \u003d 3 m,

RO - the height of the pyramid,

RO \u003d 4 m.

To find: S side. See fig. 6.

Fig. 6.

Decision.

According to the proven theorem,.

Find at first the ground AU. We know that the radius of the circle inscribed in the base of the right quadrangular pyramid is 3 m.

Then, m.

We find the perimeter of Square Assd.with a side of 6 m:

Consider a triangle BCD.. Let be M. - Middle side DC. As ABOUT - Mid. BD.T. (m).

Triangle DPC. - Isol. M. - Mid. DC. I.e, RM. - Median, and hence height in the triangle DPC.. Then RM. - Apperam pyramids.

RO - Height of the pyramid. Then, straight RO Perpendicular to the plane ABC, which means that direct Oh.lying in it. Find apophem RM. From a rectangular triangle Rum.

Now we can find the side surface of the pyramid:

Answer: 60 m 2.

The radius of the circle described near the base of the right triangular pyramid is m. The side surface area is 18 m 2. Find the length of the apophem.

Dano: AVSP - Proper triangular pyramids,

Av \u003d sun \u003d sa

R. \u003d m,

S side \u003d 18 m 2.

To find:. See fig. 7.

Fig. 7.

Decision.

In the right triangle ABC Dan radius of the described circle. Find side AU This triangle using the sinus theorem.

Knowing the side of the right triangle (m), we will find its perimeter.

By the theorem on the side surface area of \u200b\u200bthe correct pyramid, where h a.- Apperam pyramids. Then:

Answer: 4 m.

So, we considered that such a pyramid, which is the correct pyramid, proved the theorem about the side surface of the right pyramid. In the next lesson, we will get acquainted with a truncated pyramid.

Bibliography

1. Geometry. 10-11 Class: Textbook for students of general educational institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th ed., Act. and add. - M.: Mnemozina, 2008. - 288 p.: Il.
2. Geometry. 10-11 Class: Textbook for general educational institutions / Sharygin I. F. - M.: Drop, 1999. - 208 p.: Il.
3. Geometry. Grade 10: Textbook for general education institutions with in-depth and profile study of mathematics / e. V. Potoskuev, L. I. Zvalich. - 6th ed., Stereotype. - M.: Drop, 008. - 233 p.: Il.

1. Internet portal "Yaclass" ()
2. Internet portal "Festival of pedagogical ideas" First September "()
3. Internet portal "Slideshare.net" ()

Homework

1. Can the correct polygon be the basis of the wrong pyramid?
2. Prove that the wrong edges of the right pyramid are perpendicular.
3. Find the magnitude of the dihedral angle with the base side of the right quadrangular pyramid if the apophem of the pyramid is equal to the side of its base.
4. Ravox - Proper triangular pyramid. Build the linear angle of the dwarfrani angle at the base of the pyramid.

Pyramid - This is a polyhedron, in which one face is the base of the pyramid - an arbitrary polygon, and the remaining side faces are triangles with a total vertex called the peak of the pyramid. Perpendicular omeged from the top of the pyramid on its base is called pyramid height. The pyramid is called triangular, quadrangular, etc., if the base of the pyramid is a triangle, a quadrangle, etc. The triangular pyramid is a tetrahedron - tetrahedron. Quadrangular - five-meter, etc.

Pyramid, Truncated pyramid

Right pyramid

If the base of the pyramid is the correct polygon, and the height falls into the center of the base, then the pyramid is correct. In the right pyramid, all side ribs are equal, all side faces are equal to equal triangles. The height of the triangle side face of the right pyramid is called - apothem right pyramid.

Truncated pyramid

The cross section parallel to the base of the pyramid divides the pyramid into two parts. Part of the pyramid between its foundation and this section is truncated pyramid . This section for a truncated pyramid is one of its foundations. The distance between the bases of the truncated pyramid is called the height of a truncated pyramid. The truncated pyramid is called correctly, if the pyramid from which it was obtained was correct. All side faces of the correct truncated pyramid are equal equal to the trapezoid. The height of the trapezoid side of the right truncated pyramid is called - appeham of the right truncated pyramid.

Here are the basic information about the pyramids and its associated formulas and concepts. All of them are studied with a tutor in mathematics when preparing for the exam.

Consider a plane, polygon lying in it and point S, not lying in it. Connect S with all the vertices of the polygon. The polyhedron obtained is called a pyramid. Segments are called side ribs. The polygon is called the base, and the point S is the peak of the pyramid. Depending on the number of N, the pyramid is called triangular (n \u003d 3), quadagonal (n \u003d 4), pthairan (n \u003d 5) and so on. Alternative title Triangular Pyramid - tetrahedron. The height of the pyramid is called perpendicular, lowered from its vertex to the base plane.

The pyramid is called correct if The correct polygon, and the base of the height of the pyramid (the base of the perpendicular) is its center.

Tutor comment:
Do not confuse the concept of "Right Pyramid" and "Right Tetrahedron". In the right pyramid, the side ribs are not necessarily equal to the ribs of the base, and in the right tetrahedra all 6 ribs are equal. This is his definition. It is easy to prove that from equality it follows the coincidence of the center P of the polygon With the base of height, therefore the correct tetrahedron is the right pyramid.

What is apophem?
Apophistician pyramid is called the height of her side face. If the pyramid is correct, then all its apophems are equal. The opposite is incorrect.

Tutor in mathematics about its terminology: Working with pyramids by 80% is built through two types of triangles:
1) containing apophem SK and height SP
2) containing side edge SA and its projection PA

To simplify links to these triangles to the mathematics tutor more convenient to call the first one of them. apophhemny, and second rib. Unfortunately, you will not meet this terminology in any of the textbooks, and the teacher has to introduce it unilaterally.

Pyramid volume formula:
1) , where - the base area of \u200b\u200bthe pyramid, and is a pyramid
2), where - the radius of the inscribed ball, and the area of \u200b\u200bthe full surface of the pyramid.
3) where Mn is the distance by any two cross-country ribs, and the area of \u200b\u200bthe parallelogram formed by the middle of the four remaining ribs.

Property base of the height of the pyramid:

Point P (see Figure) coincides with the center of the inscribed circle into the base of the pyramid if one of the following conditions is satisfied:
1) All apophems are equal
2) All side faces are equally tilted to the base
3) All apophims are equally inclined to the height of the pyramid.
4) The height of the pyramid is equally inclined to all side faces.

Comment Tutor in mathematics: Please note that all items combines one common property: somehow, side edges are involved everywhere (apophems are their elements). Therefore, the tutor can offer less accurate, but more user-friendly wording: point P coincides with the center of the inscribed circumference. The base of the pyramid, if there is any equal information about its side faces. To prove, it is enough to show that all apophemical triangles are equal.

The point P coincides with the center described near the base of the pyramid circumference, if the one of their three conditions is true:
1) all side ribs are equal
2) all side ribs are equally tilted to the base
3) all side ribs are equally tilted to height