Volume of irregular pyramid. The volume of the triangular pyramid. Formulas and an example of solving the problem

Definition Side face   - this is a triangle in which one corner lies at the top of the pyramid, and the side opposite to it coincides with the side of the base (polygon).

Definition Side ribs   are the common sides of the side faces. A pyramid has as many edges as angles of a polygon.

Definition Pyramid height   - This is a perpendicular, lowered from the top to the base of the pyramid.

Definition Apothem   is the 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 section of the pyramid by a plane passing through the top of the pyramid and the diagonal of the base.

Definition Regular pyramid   - This is a pyramid in which the base is a regular polygon, and the height drops to the center of the base.

The volume and surface area of \u200b\u200bthe pyramid

Formula. Pyramid volume   through the base area and height:

Pyramid properties

If all the side edges are equal, then around the base of the pyramid you can describe a circle, and the center of the base coincides with the center of the circle. Also, the perpendicular dropped from the top passes through the center of the base (circle).

If all side ribs are equal, then they are inclined to the base plane at the same angles.

Side ribs are equal when they form equal angles with the plane of the base, or if a circle can be described around the base of the pyramid.

If the side faces are inclined to the base plane at the same angle, then a circle can be entered into the base of the pyramid, and the top of the pyramid is projected to its center.

If the side faces are inclined to the base plane at the same angle, then the apothems of the side faces are equal.

Properties of a regular pyramid

1. The top of the pyramid is equidistant from all corners of the base.

2. All side ribs are equal.

3. All side ribs are inclined at equal angles to the base.

4. The apofems of all lateral faces are equal.

5. The areas of all side faces are equal.

6. All faces have the same dihedral (flat) angles.

7. Around the pyramid can describe the sphere. The center of the described sphere will be the intersection point of the perpendiculars that pass through the middle of the ribs.

8. A sphere can be entered into the pyramid. The center of the inscribed sphere will be the intersection point of the bisectors, emanating from the angle 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 the flat angles at the vertex is π or vice versa, one angle is π / n, where n is the number of angles at the base of the pyramid.

The connection of the pyramid with the sphere

Around a pyramid, one can describe a sphere when the polyhedron lies around the base of the pyramid, around which a circle can be described (a necessary and sufficient condition). The center of the sphere will be the intersection point of planes passing perpendicularly through the midpoints of the side edges of the pyramid.

Around any triangular or regular pyramid you can always describe a sphere.

A sphere can be entered into the pyramid if the bisector planes of the inner dihedral angles of the pyramid intersect at one point (a necessary and sufficient condition). This point will be the center of the sphere.

The connection of the pyramid with the cone

A 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.

A cone can be inscribed in a pyramid if the apothems of the pyramid are equal to each other.

A cone is called circumscribed around the pyramid if their vertices coincide and the base of the cone is circumscribed around the base of the pyramid.

A cone can be described around a pyramid if, all the side edges of the pyramid are equal to each other.

The connection of the pyramid with the cylinder

A pyramid is inscribed in a cylinder if the top of the pyramid lies on one base of the cylinder, and the base of the pyramid is inscribed in another base of the cylinder.

A cylinder can be described around a pyramid if a circle can be described around the base of the pyramid.

The tetrahedron has four faces and four vertices and six edges, where any two edges do not have common vertices but do not touch each other.

Each vertex consists of three faces and edges that form trihedral angle.

The segment connecting the vertex of the tetrahedron to the center of the opposite face is called the tetrahedron   (GM).

Bimediana   A segment is called connecting the midpoints of opposing edges that are not touching (KL).

All bimedians and medians of the tetrahedron intersect at one point (S). In this case, the bimedians are divided in half, and the medians in a 3: 1 ratio starting from the top.

Definition Pointed pyramid   - This is a pyramid in which the apothem is more than half the length of the side of the base.

Definition Obtuse pyramid   - This is a pyramid in which the apothem is less than half the length of the side of the base.

Definition Correct tetrahedron   - a tetrahedron in which all four faces are equilateral triangles. It is one of five regular polygons. In a regular tetrahedron, all dihedral angles (between faces) and trihedral angles (at the apex) are equal.

Definition Rectangular tetrahedron   a tetrahedron is called in which the right angle between the three edges at the apex (the edges are perpendicular). Three faces form rectangular triangular angle   and the faces are right triangles, and the base is an arbitrary triangle. The apothem of any face is equal to half the side of the base upon which the apothem falls.

Definition Isometric tetrahedron   called a tetrahedron whose side faces are equal to each other, and the base is a regular triangle. In such a tetrahedron, the faces are isosceles triangles.

Definition Orthocentric tetrahedron   called a tetrahedron in which all the heights (perpendiculars) that are lowered from the top to the opposite face intersect at one point.

Definition Star pyramid   called a polyhedron whose base is a star.

Here we will analyze examples related to the concept of volume. To solve such tasks, you must know the formula for the volume of the pyramid:

S

h - pyramid height

The base can be any polygon. But in most tasks on the exam, the condition, as a rule, is about the correct pyramids. Let me remind you of one of its properties:

The top of the regular pyramid is projected to the center of its base.

Look at the projection of the regular triangular, quadrangular and hexagonal pyramids (TOP VIEW):

You can on the blog, which dealt with the tasks associated with finding the volume of the pyramid.Consider the tasks:

27087. Find the volume of the regular triangular pyramid, the sides of the base of which are equal to 1, and the height is equal to the root of three.

S   - area of \u200b\u200bthe base of the pyramid

h   - pyramid height

Find the area of \u200b\u200bthe base of the pyramid, this is a regular triangle. We use the formula - the area of \u200b\u200bthe triangle is equal to half the product of the neighboring sides by the sine of the angle between them, which means:

Answer: 0.25

27088. Find the height of the regular triangular pyramid, the sides of the base of which are equal to 2, and the volume is equal to the root of three.

Such concepts as the height of the pyramid and the characteristics of its base are connected by the volume formula:

S   - area of \u200b\u200bthe base of the pyramid

h   - pyramid height

The volume itself is known to us, we can find the base area, since the sides of the triangle, which is the base, are known. Knowing the indicated values, we can easily find the height.

To find the base area, we use the formula - the area of \u200b\u200bthe triangle is equal to half the product of the neighboring sides by the sine of the angle between them, which means:

Thus, substituting these values \u200b\u200bin the volume formula, we can calculate the height of the pyramid:

The height is three.

Answer: 3

27109. In a regular quadrangular pyramid, the height is 6, the side edge is 10. Find its volume.

The volume of the pyramid is calculated by the formula:

S   - area of \u200b\u200bthe base of the pyramid

h   - pyramid height

We know the height. It is necessary to find the area of \u200b\u200bthe base. Let me remind you that the top of the regular pyramid is projected to the center of its base. The base of a regular quadrangular pyramid is a square. We can find its diagonal. Consider a right triangle (highlighted in blue):

The line connecting the center of the square to point B is a leg that is equal to half the diagonal of the square. This leg can be calculated by the Pythagorean theorem:

So BD \u003d 16. We calculate the area of \u200b\u200bthe square using the quadrangle area formula:

Hence:

Thus, the volume of the pyramid is equal to:

Answer: 256

27178. In a regular quadrangular pyramid, the height is 12, the volume is 200. Find the side edge of this pyramid.

The height of the pyramid and its volume are known, so we can find the area of \u200b\u200bthe square, which is the base. Knowing the area of \u200b\u200bthe square, we can find its diagonal. Further, considering a right triangle by Pythagorean theorem, we calculate the side edge:

Find the area of \u200b\u200bthe square (base of the pyramid):

We calculate the diagonal of the square. Since its area is 50, the side will be equal to the root of fifty and according to the Pythagorean theorem:

Point O divides the diagonal BD in half, which means that the leg of the right triangle OB \u003d 5.

Thus, we can calculate what the side edge of the pyramid is equal to:

Answer: 13

245353. Find the volume of the pyramid shown in the figure. Its base is a polygon, the adjacent sides of which are perpendicular, and one of the side ribs is perpendicular to the plane of the base and is equal to 3.

As has already been repeatedly said - the volume of the pyramid is calculated by the formula:

S   - area of \u200b\u200bthe base of the pyramid

h   - pyramid height

The lateral edge perpendicular to the base is three, which means that the height of the pyramid is three. The base of the pyramid is a polygon whose area is:

In this way:

Answer: 27

27086. The base of the pyramid is a rectangle with sides 3 and 4. Its volume is 16. Find the height of this pyramid.

That's all. Success to you!

Sincerely, Alexander Krutitsky.

P.S: I would be grateful if you talk about the site on social networks.

A pyramid is a polyhedron based on a polygon. All faces in turn form triangles that converge at one vertex. Pyramids are triangular, quadrangular and so on. In order to determine which pyramid is in front of you, it is enough to calculate the number of angles at its base. The definition of "pyramid height" is very common in geometry problems in the school curriculum. In the article, we will try to consider different ways of finding it.

Parts of the pyramid

Each pyramid consists of the following elements:

• lateral faces that have three angles and converge at the apex;
• apothem is the height that descends from its top;
• the top of the pyramid is a point that connects the side ribs, but does not lie in the plane of the base;
• the base is a polygon on which the vertex does not lie;
• the height of the pyramid is a segment that intersects the top of the pyramid and forms a right 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 is the volume, S is the base area, h is the height of the pyramid) we find that h \u003d (3 * V) / S. To fix the material, let's immediately solve the problem. The triangular base is 50 cm 2, while its volume is 125 cm 3. The height of the triangular pyramid, which we need to find, is unknown. Everything is simple here: we insert the data into our formula. We get 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 edges are known

As we recall, the height of the pyramid forms a right angle with its base. And this means that the height, edge and half of the diagonal together form Many, of course, remember the Pythagorean theorem. Knowing two dimensions, it will not be difficult to find a third value. Recall the well-known theorem a² \u003d b² + c², where a is the hypotenuse, and in our case the edge of the pyramid; b - the first leg or half of the diagonal and c - respectively, the second leg, or the height of the pyramid. From this formula, c² \u003d a² - b².

Now the problem: in the correct pyramid, the diagonal is 20 cm, when the length of the rib is 30 cm. It is necessary to find the height. We decide: c² \u003d 30² - 20² \u003d 900-400 \u003d 500. Hence, c \u003d √ 500 \u003d about 22.4.

How to find the height of a truncated pyramid

It is a polygon that has a section parallel to its base. The height of a truncated pyramid is a segment that connects its two bases. The height can be found at the regular pyramid, if the lengths of the diagonals of both bases, as well as the edge of the pyramid, are known. Suppose that the diagonal of the larger base is d1, while the diagonal of the smaller base is d2, and the edge has a length of - l. To find the height, you can lower the heights at its base from the two upper opposite points of the diagram. We see that we have obtained two rectangular triangles, it remains to find the lengths of their legs. To do this, subtract the smaller from the larger diagonal and divide by 2. So we find one leg: a \u003d (d1-d2) / 2. Then, according to the Pythagorean theorem, we can only find the second leg, which is the height of the pyramid.

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

The word "pyramid" is involuntarily associated with the majestic giants in Egypt, faithfully keeping the peace of the pharaohs. Maybe that’s why the pyramid is unmistakably recognized by everyone, even children.

Nevertheless, we will try to give it a geometric definition. We represent several points on the plane (A1, A2, ..., An) and one more (E) that does not belong to it. So, if we connect the point E (vertex) with the vertices of the polygon formed by the points A1, A2, ..., An (base), we get a polyhedron, which is called a pyramid. Obviously, the vertices of the polygon at the base of the pyramid can be any number, and depending on their number, the pyramid can be called triangular and quadrangular, pentagonal, etc.

If you look closely at the pyramid, it will become clear why it is also defined differently - as a geometric figure having a polygon at the base, and triangles united by a common vertex as side faces.

Since the pyramid is a spatial figure, it also has such a quantitative characteristic as is calculated from the well-known equal third of the product of the base of the pyramid to its height:

When deriving the formula, the volume of the pyramid is initially calculated for a triangular one, taking as a basis a constant relation connecting this value with the volume of a triangular prism having the same base and height, which, as it turns out, is three times this volume.

And since any pyramid is divided into triangular ones, and its volume does not depend on the constructions performed in the proof, the validity of the given volume formula is obvious.

Apart from all the pyramids are the regular ones, in which the base lies. As for, it should "end" in the center of the base.

In the case of an irregular polygon in the base, to calculate the base area you will need:

• break it into triangles and squares;

• calculate the area of \u200b\u200beach of them;

• add up the data.

In the case of a regular polygon at the base of the pyramid, its area is calculated using ready-made formulas, so the volume of a regular pyramid is calculated very simply.

For example, to calculate the volume of a quadrangular pyramid, if it is correct, erect the length of the side of the regular quadrangle (square) at the base into a square and, multiplying by the height of the pyramid, divide the resulting product by three.

The volume of the pyramid can be calculated using other parameters:

• as a third of the product of the radius of a ball inscribed in a pyramid by the area of \u200b\u200bits full surface;

• as two-thirds of the product of the distance between two arbitrarily taken intersecting edges and the area of \u200b\u200bthe parallelogram, which form the middle of the remaining four edges.

The volume of the pyramid is calculated simply and in the case when its height coincides with one of the side edges, that is, in the case of a rectangular pyramid.

Speaking of the pyramids, one cannot ignore the truncated pyramids obtained by the section of the pyramid parallel to the base of the plane. Their volume is almost equal to the difference in the volumes of the whole pyramid and the cut off top.

The first volume of the pyramid, though not entirely in its modern form, however, equal to 1/3 of the volume of the prism known to us, was found by Democritus. Archimedes called his method of counting “without proof”, since Democritus approached the pyramid as a figure composed of infinitely thin, similar plates.

Vector algebra also “turned” to the question of finding the volume of the pyramid, using the coordinates of its vertices for this. A pyramid built on a triple of vectors a, b, c is equal to one sixth of the module of the mixed product of given vectors.

Quadrangular pyramid   called a polyhedron, the base of which is a square, and all side faces are the same isosceles triangles.

This polyhedron has many different properties:

• Its lateral ribs and the dihedral angles adjacent to them are equal to each other;
• The areas of the side faces are the same;
• At the base of a regular quadrangular pyramid lies a square;
• The height dropped from the top of the pyramid intersects with the intersection point of the base diagonals.

All of these properties make it easy to find. However, quite often in addition to it, it is required to calculate the volume of the polyhedron. For this, the formula for the volume of the quadrangular pyramid is used:

That is, the volume of the pyramid is equal to one third of the product of the height of the pyramid by the area of \u200b\u200bthe base. Since it is equal to the product of its equal sides, we immediately enter the square area formula into the expression of volume.
  Consider an example of calculating the volume of a quadrangular pyramid.

Let a quadrangular pyramid be given, at the base of which lies a square with a side of a \u003d 6 cm. The side face of the pyramid is b \u003d 8 cm. Find the volume of the pyramid.

  To find the volume of a given polyhedron, we need the length of its height. Therefore, we will find it by applying the Pythagorean theorem. To begin, calculate the length of the diagonal. In the blue triangle, it will be hypotenuse. It is also worth remembering that the diagonals of a square are equal to each other and are divided in half at the intersection point:


  Now from the red triangle we find the height h we need. It will be equal to:

  Substitute the necessary values \u200b\u200band find the height of the pyramid:

Now, knowing the height, we can substitute all the values \u200b\u200bin the formula for the volume of the pyramid and calculate the necessary value:

In this way, knowing a few simple formulas, we were able to calculate the volume of a regular quadrangular pyramid. Do not forget that this value is measured in cubic units.