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 school curriculum. In this article we will try to consider different ways her location.
Parts of the pyramid
Each pyramid consists of the following elements:
- side faces, which have three angles and converge at the vertex;
- 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's remember 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 at regular pyramid, if the lengths of the diagonals of both bases are known, as well as the edge of the pyramid. Let the diagonal larger base is equal to d1, while the diagonal smaller base- 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.
One of the simplest volumetric figures is triangular pyramid, since it consists of smallest number 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 general definition a pyramid is a polygon, all of whose vertices are connected to one point 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 triangle general type. However, if it is equilateral, and the pyramid itself is straight, then they speak of a regular three-dimensional figure.
Any triangular pyramid 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 the 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
A regular triangular pyramid has 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.
Volume of a regular pyramid with 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 through 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'll show you how to use the above formulas to solve specific tasks geometry.
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 regular in which all bases are equal to each other. To use the triangular volume formula, 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 conditions:
To determine the height, consider the figure shown in the figure.
Marked triangle ABC is rectangular, 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 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, which is realized 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 curious 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 the pyramid arbitrary type, and not just for regular triangular.
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 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 triangular pyramid has 4 faces, 6 edges and 4 vertices. If the lengths of all edges are equal, then such a figure is called a tetrahedron.
Volume of a general triangular pyramid
Before writing down the formula for the volume of a regular triangular pyramid, we give an expression for this physical quantity for a general type pyramid. This expression looks like:
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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:
V = 1/6*a*h o *h.
Formulas for the volume of a regular triangular pyramid
A regular triangular pyramid 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:
V = √3/4*h*r 2 .
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:
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- length of the side of the triangle;
- height of the figure.
The first quantity is known from the problem conditions:
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
Goals and objectives of the lesson:
- derive formulas for the volume of a pyramid using the basic formula for the volume of bodies and the volume of a truncated pyramid.
- systematize theoretical knowledge on the topic of finding the volume of a pyramid.
- develop the skill of finding the volume of a pyramid whose vertex is projected into the center of a circle inscribed or circumscribed near the base.
- develop solution skills typical tasks on the application of formulas for the volume of a pyramid and a truncated pyramid.
Lesson progress
I.Explanationnew material.
The proof of the theorem is performed using a multimedia projector
Let's prove the theorem: the volume of the pyramid isone third, the product of the area of the base and the height.
Proof:
First we prove the theorem for a triangular pyramid, then for an arbitrary one.
1. Consider a triangular pyramid OABC with volume V, base area S and height h. Let's draw the axis oh (OM 2- height), consider the section A 1 B 1 C 1 pyramid with a plane perpendicular to the axis Oh and, therefore, parallel to the plane of the base. Let us denote by X abscissa point M 1
intersection of this plane with the x axis, and through S(x)- cross-sectional area. Let's express S(x) through S, h And X. Note that 
Indeed , hence, .
Right Triangles 
, are also similar (they have a common acute angle with top ABOUT).
Let's apply now basic formula to calculate the volumes of bodies at a = 0, b =h we get
2. Let us now prove the theorem for an arbitrary pyramid with height h and base area S. Such a pyramid can be divided into triangular pyramids with a total height h. Let us express the volume of each triangular pyramid using the formula we have proven and add these volumes. Bracketing common multiplier, we obtain in brackets the sum of the bases of triangular pyramids, i.e. area S of the bases of the original pyramid.
Thus, the volume of the original pyramid is . The theorem has been proven.
II. solve problems using ready-made drawings.
Task 1. (Fig. 3)
Given:ABCD- regular pyramid AB = 3; AD= . Find: A) Sbasic; b) JSC; V) DO G) V .
Task 2. (Fig. 4)
Given:ABCDF- regular pyramid .
Task 3. (Fig. 5)
Given:ABCDEKF- regular pyramid
Find: A) Sbasic ; b) V.
Task4. (fig.. 6)
Find: V.
Task verification is performed using a multimedia projector with detailed analysis step-by-step solution.
Task 1. (Fig. 3)
a) (the formula is used to calculate the area of a regular triangle)
AB = = 3, we have 
b) 
(formula for the radius of a circumscribed circle using the side of an equilateral triangle) 
.
Task 2. (Fig. 4)
1) Let us therefore consider
– isosceles, OS = FO = 2.
Task 3. (Fig. 5)
Task 4. (Fig. 6)
III. Checking the output of the formula for calculating the volume of a truncated pyramid (the student’s message at the blackboard is done using a multimedia projector)
Student answer:
We consider the volume of a truncated pyramid as the difference in volumes full pyramid and the one that is cut off from it by a plane, parallel to the base(Fig. 1).
Let's substitute this expression for X into the first formula,
Work in the form of a test, with verification through a multimedia projector.
1.B inclined prism side rib is equal to 7 cm, the perpendicular section is a right triangle with legs: 4 cm and 3 cm. Find the volume of the prism.
a) 10 cm 3, b) 42 cm 3, c) 60 cm 3, d) 30 cm 3.
2. In the right way hexagonal pyramid e side of its base is 2 cm. The volume of the pyramid is 6 cm 3. What is the height?
3. The volume of the pyramid is 56 cm 3, the base area is 14 cm 2. What is the height?
a) 14 cm, b) 12 cm, c) 16 cm.
4. In a regular triangular pyramid, the height is 5 cm, the sides of the base are 3 cm. What is the volume of the pyramid?
5. In the right way quadrangular pyramid the height is 9 cm. The side of the base is 4 cm. Find the volume of the pyramid.
a) 50 cm 3, b) 48 cm 3, c) 16 cm 3.
6. The volume of a regular quadrangular pyramid is 27 cm 3, height 9 cm. Find the side of the base.
a) 12 cm, b) 9 cm, c) 3 cm.
7. The volume of a truncated pyramid is 210 cm 3, the area of the lower base is 36 cm 2, the upper is 9 cm 2. Find the height of the pyramid.
a) 1cm, b) 15cm, c) 10cm.
8. Equal-sized prism and regular quadrangular pyramid have equal heights. What is the side of the base of the pyramid if the area of the base of the prism is S?
Answer table.
| Task | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| Answer | b | A | b | A | b | V | V | V |
Homework: 1. Solve problems No. 695v, No. 697, No. 690
2. Consider basic tasks
Task 1.
Prove that if the lateral edges of the pyramid are equal (or equal equal angles with the plane of the base), then the top of the pyramid is projected into the center of the circle described around the base.
Prove that if dihedral angles if the base of the pyramid is equal (or the heights of the side faces drawn from the top of the pyramid are equal), then the top of the pyramid is projected into the center of the circle inscribed in the base of the pyramid.
Here we will look at examples related to the concept of volume. To solve such tasks, you must know the formula for the volume of a pyramid:
S
h – height of the pyramid
The base can be any polygon. But in most problems Unified State Exam speech the condition, as a rule, refers to regular pyramids. Let me remind you of one of its properties:
The top of a regular pyramid is projected into the center of its base
Look at the projection of the regular triangular, quadrangular and hexagonal pyramids (TOP VIEW):
You can on the blog, where problems related to finding the volume of a pyramid were discussed.Let's consider the tasks:
27087. Find the volume of a regular triangular pyramid whose base sides are equal to 1 and whose height is equal to the root of three.
S– area of the base of the pyramid
h– height of the pyramid
Let's find the area of the base of the pyramid, this is regular triangle. Let's use the formula - the area of a triangle is equal to half the product of adjacent sides and the sine of the angle between them, which means:
Answer: 0.25
27088. Find the height of a regular triangular pyramid whose base sides are equal to 2 and whose volume is equal to the root out of three.
Concepts such as the height of a pyramid and the characteristics of its base are related by the volume formula:
S– area of the base of the pyramid
h– height of the pyramid
We know the volume itself, we can find the area of the base, since we know the sides of the triangle, which is the base. Knowing the indicated values, we can easily find the height.
To find the area of the base, we use the formula - the area of a triangle is equal to half the product of adjacent sides and the sine of the angle between them, which means:
Thus, by substituting these values into 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 and the side edge is 10. Find its volume.
The volume of the pyramid is calculated by the formula:
S– area of the base of the pyramid
h– height of the pyramid
We know the height. You need to find the area of the base. Let me remind you that the top of a regular pyramid is projected into 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 segment connecting the center of the square with point B is the leg, which equal to half diagonals of a square. We can calculate this leg using the Pythagorean theorem:
This means BD = 16. Let’s calculate the area of the square using the formula for the area of a quadrilateral:
Hence:
Thus, the volume of the pyramid is:
Answer: 256
27178. In a regular quadrangular pyramid, the height is 12 and the volume is 200. Find the side edge of this pyramid.
The height of the pyramid and its volume are known, which means we can find the area of the square, which is the base. Knowing the area of a square, we can find its diagonal. Next, considering a right triangle using the Pythagorean theorem, we calculate the side edge:
Let's find the area of the square (base of the pyramid):
Let's 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 the leg right triangle OB = 5.
Thus, we can calculate what the lateral 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 edges is perpendicular to the plane of the base and equal to 3.
As has been said many times, the volume of the pyramid is calculated by the formula:
S– area of the base of the pyramid
h– height of the pyramid
The side edge perpendicular to the base is equal to three, which means that the height of the pyramid is three. The base of the pyramid is a polygon whose area is equal to:
Thus:
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. Good luck to you!
Sincerely, Alexander Krutitskikh.
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