Volume of a pyramid with the base of a regular triangle. Volume of a triangular pyramid

One of the simplest three-dimensional figures is the triangular pyramid, since it consists of the smallest number of faces from which a figure can be formed in space. In this article we will look at formulas that can be used to find the volume of a triangular regular pyramid.

Triangular pyramid

According to the general definition, a pyramid is a polygon, all of whose vertices are connected to one point that is not located in the plane of this polygon. If the latter is a triangle, then the entire figure is called a triangular pyramid.

The pyramid in question consists of a base (triangle) and three side faces (triangles). The point at which the three side faces are connected is called the vertex of the figure. The perpendicular from this vertex dropped to the base is the height of the pyramid. If the point of intersection of the perpendicular with the base coincides with the point of intersection of the medians of the triangle at the base, then we speak of a regular pyramid. Otherwise it will be slanted.

As stated, the base of a triangular pyramid can be a general type of triangle. However, if it is equilateral, and the pyramid itself is straight, then they speak of a regular three-dimensional figure.

Any one has 4 faces, 6 edges and 4 vertices. If the lengths of all edges are equal, then such a figure is called a tetrahedron.

general type

Before writing down a regular triangular pyramid, we give an expression for this physical quantity for a general type pyramid. This expression looks like:

Here S o is the area of ​​the base, h is the height of the figure. This equality will be valid for any type of pyramid polygon base, as well as for a cone. If at the base there is a triangle with side length a and height h o lowered onto it, then the formula for volume will be written as follows:

Formulas for the volume of a regular triangular pyramid

Triangular has an equilateral triangle at the base. It is known that the height of this triangle is related to the length of its side by the equality:

Substituting this expression into the formula for the volume of a triangular pyramid written in the previous paragraph, we obtain:

V = 1/6*a*h o *h = √3/12*a 2 *h.

The volume of a regular pyramid with a triangular base is a function of the length of the side of the base and the height of the figure.

Since any regular polygon can be inscribed in a circle, the radius of which will uniquely determine the length of the side of the polygon, then this formula can be written in terms of the corresponding radius r:

This formula can be easily obtained from the previous one, if we take into account that the radius r of the circumscribed circle through the length of side a of the triangle is determined by the expression:

Problem of determining the volume of a tetrahedron

We will show how to use the above formulas when solving specific geometry problems.

It is known that a tetrahedron has an edge length of 7 cm. Find the volume of a regular triangular pyramid-tetrahedron.

Recall that a tetrahedron is a regular triangular pyramid in which all bases are equal to each other. To use the formula for the volume of a regular triangular pyramid, you need to calculate two quantities:

  • length of the side of the triangle;
  • height of the figure.

The first quantity is known from the problem statement:

To determine the height, consider the figure shown in the figure.

The marked triangle ABC is a right triangle, where angle ABC is 90 o. Side AC is the hypotenuse and its length is a. Using simple geometric reasoning, it can be shown that side BC has the length:

Note that the length BC is the radius of the circle circumscribed around the triangle.

h = AB = √(AC 2 - BC 2) = √(a 2 - a 2 /3) = a*√(2/3).

Now you can substitute h and a into the corresponding formula for volume:

V = √3/12*a 2 *a*√(2/3) = √2/12*a 3 .

Thus, we have obtained the formula for the volume of a tetrahedron. It can be seen that the volume depends only on the length of the edge. If we substitute the value from the problem conditions into the expression, then we get the answer:

V = √2/12*7 3 ≈ 40.42 cm 3.

If we compare this value with the volume of a cube having the same edge, we find that the volume of the tetrahedron is 8.5 times less. This indicates that the tetrahedron is a compact figure that occurs in some natural substances. For example, the methane molecule has a tetrahedral shape, and each carbon atom in diamond is connected to four other atoms to form a tetrahedron.

Homothetic pyramid problem

Let's solve one interesting geometric problem. Suppose that there is a triangular regular pyramid with a certain volume V 1. How many times should the size of this figure be reduced in order to obtain a homothetic pyramid with a volume three times smaller than the original?

Let's start solving the problem by writing the formula for the original regular pyramid:

V 1 = √3/12*a 1 2 *h 1 .

Let the volume of the figure required by the conditions of the problem be obtained by multiplying its parameters by the coefficient k. We have:

V 2 = √3/12*k 2 *a 1 2 *k*h 1 = k 3 *V 1 .

Since the ratio of the volumes of the figures is known from the condition, we obtain the value of the coefficient k:

k = ∛(V 2 /V 1) = ∛(1/3) ≈ 0.693.

Note that we would obtain a similar value for the coefficient k for a pyramid of any type, and not just for a regular triangular one.

To find the volume of a pyramid, you need to know several formulas. Let's look at them.

How to find the volume of a pyramid - 1st method

The volume of a pyramid can be found using the height and area of ​​its base. V = 1/3*S*h. So, for example, if the height of the pyramid is 10 cm, and the area of ​​its base is 25 cm 2, then the volume will be equal to V = 1/3*25*10 = 1/3*250 = 83.3 cm 3

How to find the volume of a pyramid - 2nd method

If a regular polygon lies at the base of the pyramid, then its volume can be found using the following formula: V = na 2 h/12*tg(180/n), where a is the side of the polygon lying at the base, and n is the number of its sides. For example: The base is a regular hexagon, that is, n = 6. Since it is regular, all its sides are equal, that is, all a are equal. Let's say a = 10, and h - 15. We insert the numbers into the formula and get an approximate answer - 1299 cm 3


How to find the volume of a pyramid - 3rd method

If an equilateral triangle lies at the base of the pyramid, then its volume can be found using the following formula: V = ha 2 /4√3, where a is the side of the equilateral triangle. For example: the height of the pyramid is 10 cm, the side of the base is 5 cm. The volume will be equal to V = 10*25/4√ 3 = 250/4√ 3. Usually, what is in the denominator is not calculated and is left in the same form. You can also multiply both the numerator and denominator by 4√ 3. We get 1000√ 3/48. By reducing we get 125√ 3/6 cm 3.


How to find the volume of a pyramid - 4th method

If there is a square at the base of the pyramid, then its volume can be found using the following formula: V = 1/3*h*a 2, where a is the sides of the square. For example: height – 5 cm, square side – 3 cm. V = 1/3*5*9 = 15 cm 3


How to find the volume of a pyramid - 5th method

If the pyramid is a tetrahedron, that is, all its faces are equilateral triangles, you can find the volume of the pyramid using the following formula: V = a 3 √2/12, where a is the edge of the tetrahedron. For example: tetrahedron edge = 7. V = 7*7*7√2/12 = 343 cm 3

What is a pyramid?

How does she look?

You see: at the bottom of the pyramid (they say “ at the base") some polygon, and all the vertices of this polygon are connected to some point in space (this point is called " vertex»).

This whole structure still has side faces, side ribs And base ribs. Once again, let's draw a pyramid along with all these names:

Some pyramids may look very strange, but they are still pyramids.

Here, for example, is completely “oblique” pyramid.

And a little more about the names: if there is a triangle at the base of the pyramid, then the pyramid is called triangular, if it is a quadrangle, then quadrangular, and if it is a centagon, then... guess for yourself.

At the same time, the point where it fell height, called height base. Please note that in the “crooked” pyramids height may even end up outside the pyramid. Like this:

And there’s nothing wrong with that. It looks like an obtuse triangle.

Correct pyramid.

Lots of complicated words? Let's decipher: “At the base - correct” - this is understandable. Now let us remember that a regular polygon has a center - a point that is the center of and , and .

Well, the words “the top is projected into the center of the base” mean that the base of the height falls exactly into the center of the base. Look how smooth and cute it looks regular pyramid.

Hexagonal: at the base there is a regular hexagon, the vertex is projected into the center of the base.

Quadrangular: the base is a square, the top is projected to the point of intersection of the diagonals of this square.

Triangular: at the base there is a regular triangle, the vertex is projected to the point of intersection of the heights (they are also medians and bisectors) of this triangle.

Very important properties of a regular pyramid:

In the right pyramid

  • all side edges are equal.
  • all lateral faces are isosceles triangles and all these triangles are equal.

Volume of the pyramid

The main formula for the volume of a pyramid:

Where exactly did it come from? This is not so simple, and at first you just need to remember that a pyramid and a cone have volume in the formula, but a cylinder does not.

Now let's calculate the volume of the most popular pyramids.

Let the side of the base be equal and the side edge equal. We need to find and.

This is the area of ​​a regular triangle.

Let's remember how to look for this area. We use the area formula:

For us, “ ” is this, and “ ” is also this, eh.

Now let's find it.

According to the Pythagorean theorem for

What's the difference? This is the circumradius in because pyramidcorrect and, therefore, the center.

Since - the point of intersection of the medians too.

(Pythagorean theorem for)

Let's substitute it into the formula for.

And let’s substitute everything into the volume formula:

Attention: if you have a regular tetrahedron (i.e.), then the formula turns out like this:

Let the side of the base be equal and the side edge equal.

There is no need to look here; After all, the base is a square, and therefore.

We'll find it. According to the Pythagorean theorem for

Do we know? Almost. Look:

(we saw this by looking at it).

Substitute into the formula for:

And now we substitute and into the volume formula.

Let the side of the base be equal and the side edge.

How to find? Look, a hexagon consists of exactly six identical regular triangles. We have already looked for the area of ​​a regular triangle when calculating the volume of a regular triangular pyramid; here we use the formula we found.

Now let's find (it).

According to the Pythagorean theorem for

But what does it matter? It's simple because (and everyone else too) is correct.

Let's substitute:

\displaystyle V=\frac(\sqrt(3))(2)((a)^(2))\sqrt(((b)^(2))-((a)^(2)))

PYRAMID. BRIEFLY ABOUT THE MAIN THINGS

A pyramid is a polyhedron that consists of any flat polygon (), a point not lying in the plane of the base (top of the pyramid) and all segments connecting the top of the pyramid with points of the base (side edges).

A perpendicular dropped from the top of the pyramid to the plane of the base.

Correct pyramid- a pyramid in which a regular polygon lies at the base, and the top of the pyramid is projected into the center of the base.

Property of a regular pyramid:

  • In a regular pyramid, all lateral edges are equal.
  • All lateral faces are isosceles triangles and all these triangles are equal.

Pyramid volume:

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A pyramid is a polyhedron with a polygon at its base. 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 count the number of angles at its base. The definition of “height of a pyramid” is very often found in geometry problems in the school curriculum. In this article we will try to look at different ways to find it.

Parts of the pyramid

Each pyramid consists of the following elements:

  • side faces, which have three corners and converge at the apex;
  • the apothem represents the height that descends from its apex;
  • 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 a 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 a pyramid if its volume is known

Through the formula V = (S*h)/3 (in the formula V is the volume, S is the area of ​​the base, h is the height of the pyramid) we find that h = (3*V)/S. To consolidate the material, let's immediately solve the problem. The triangular base is 50 cm 2 , whereas its volume is 125 cm 3 . The height of the triangular pyramid is unknown, which is what we need to find. Everything is simple here: we insert the data into our formula. We get h = (3*125)/50 = 7.5 cm.

How to find the height of a pyramid if the length of the diagonal and its edges are known

As we remember, the height of the pyramid forms a right angle with its base. 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 the third quantity. Let us recall the well-known theorem a² = 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 = a² - b².

Now the problem: in a regular pyramid the diagonal is 20 cm, when the length of the edge is 30 cm. You need to find the height. We solve: c² = 30² - 20² = 900-400 = 500. Hence c = √ 500 = about 22.4.

How to find the height of a truncated pyramid

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

Now let's look at this whole thing in practice. We have a task ahead of us. A 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. You need to find the height. First, we find one leg: a = (10-6)/2 = 2 cm. One leg is equal to 2 cm, and the hypotenuse is 4 cm. It turns out that the second leg or height will be equal to 16-4 = 12, that is, h = √12 = about 3.5 cm.

The word “pyramid” is involuntarily associated with the majestic giants in Egypt, faithfully guarding the peace of the pharaohs. Maybe that’s why everyone, even children, recognizes the pyramid unmistakably.

Nevertheless, let's try to give it a geometric definition. Let us imagine several points on the plane (A1, A2,..., An) and one more (E) that does not belong to it. So, if point E (vertex) is connected to the vertices of the polygon formed by points A1, A2,..., An (base), you get a polyhedron, which is called a pyramid. Obviously, the polygon at the base of the pyramid can have any number of vertices, and depending on their number, the pyramid can be called triangular, quadrangular, pentagonal, etc.

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

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

When deriving the formula, the volume of a pyramid is initially calculated for a triangular one, taking as a basis a constant ratio 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 during the proof, the validity of the given volume formula is obvious.

Standing apart among all the pyramids are the correct ones, which have at their base As for, it should “end” in the center of the base.

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

  • break it into triangles and squares;
  • calculate the area of ​​each of them;
  • add up the received 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 quite simply.

For example, to calculate the volume of a quadrangular pyramid, if it is regular, the length of the side of a regular quadrilateral (square) at the base is squared and, multiplied by the height of the pyramid, the resulting product is divided 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 and its total surface area;
  • as two-thirds of the product of the distance between two arbitrarily chosen crossing edges and the area of ​​the parallelogram that forms the midpoints of the remaining four edges.

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

Speaking about pyramids, we cannot ignore truncated pyramids, obtained by cutting the pyramid with a plane parallel to the base. Their volume is almost equal to the difference between the volumes of the whole pyramid and the cut off top.

Democritus was the first to find the volume of the pyramid, although not exactly in its modern form, but equal to 1/3 of the volume of the prism known to us. Archimedes called his method of calculation “without proof,” since Democritus approached the pyramid as a figure composed of infinitely thin, similar plates.

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