Formulas for the volume of a regular triangular pyramid. Examples of problem solving
Theorem.
The volume of a pyramid is equal to one third of 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 pyramidOABCwith volume V, base areaS and height h. Draw an axis oh (OM2- height), consider the sectionA1 B1 C1pyramids with a plane perpendicular to the axisohand, therefore, parallel to the plane of the base. Denote byX abscissa point M1 intersection of this plane with the x-axis, and throughS(x)- cross-sectional area. Express S(x) through S, h and X. Note that triangles A1 AT1 With1 and ABC are similar. Indeed A1 AT1 II AB, so triangle OA 1 AT 1 similar to triangle OAB. With consequently, BUT1 AT1 : BUTB= OA 1: OA .
right triangles OA 1 AT 1 and OAB are also similar (they have a common acute angle with vertex O). Therefore, OA 1: OA = O 1 M1 : OM = x: h. Thus BUT 1 AT 1 : A B = x: h.Similarly, it is proved thatB1 C1:sun = X: h and A1 C1:AC = X: h.So the triangleA1 B1 C1 and ABCsimilar with coefficient of similarity X: h.Therefore, S(x) : S = (x: h)², or S(x) = S x²/ h².
Let us now apply the basic formula for calculating the volumes of bodies ata= 0, b=h we get
Thus, the volume of the original pyramid is 1/3Sh. The theorem has been proven.
Consequence:
Volume V of a truncated pyramid with height h and base areas S and S1 , are calculated by the formula
h - the height of the pyramid
S top - area of the upper base
S lower - area of the lower base
To find the volume of a pyramid, you need to know several formulas. Let's consider them.
How to find the volume of a pyramid - 1st way
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 \u003d 1/3 * 25 * 10 \u003d 1/3 * 250 \u003d 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 \u003d 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 we get an approximate answer - 1299 cm 3
How to find the volume of a pyramid - 3rd way
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 \u003d 10 * 25 / 4 √ 3 \u003d 250 / 4 √ 3. Usually, what happened in the denominator is not calculated and left in the same form. You can also multiply both the numerator and denominator by 4√3 to get 1000√3/48. Reducing we get 125√ 3/6 cm 3.
How to find the volume of a pyramid - 4th way
If a square lies at the base of the pyramid, then its volume can be found by the following formula: V = 1/3*h*a 2, where a are the sides of the square. For example: height - 5 cm, side of the square - 3 cm. V \u003d 1/3 * 5 * 9 \u003d 15 cm 3
How to find the volume of a pyramid - 5th way
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 an edge of the tetrahedron. For example: tetrahedron edge \u003d 7. V \u003d 7 * 7 * 7√2 / 12 \u003d 343 cm 3
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.
However, let's try to give it a geometric definition. Let us imagine several points (A1, A2,..., An) on the plane and one more (E) that does not belong to it. So, if point E (top) 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 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 with a polygon at the base, and triangles united by a common vertex as side faces.
Since the pyramid is a spatial figure, then it also has such a quantitative characteristic, as it is calculated from the well-known equal third of the product of the base of the pyramid and its height:
The volume of the pyramid, when deriving the formula, is initially calculated for a triangular one, taking as a basis a constant ratio relating this value to 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 above volume formula is obvious.
Standing apart among all the pyramids are the right ones, in which the base lies. 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 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 very simply.
For example, to calculate the volume of a quadrangular pyramid, if it is regular, the length of the side of a regular quadrangle (square) at the base is squared and, multiplying 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 the ball inscribed in the pyramid and the area of its total surface;
- as two thirds of the product of the distance between two arbitrarily taken crossing edges and the area of the parallelogram that forms the midpoints of the remaining four edges.
The volume of the pyramid is also 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 of pyramids, one cannot ignore the 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.
The first volume of the pyramid, although not quite in its modern form, but equal to 1/3 of the volume of the prism known to us, was found by Democritus. Archimedes called his counting method “without proof,” since Democritus approached the pyramid as if it were a figure made up of infinitely thin, similar plates.
Vector algebra also “addressed” the question of finding the volume of the pyramid, using the coordinates of its vertices for this. The pyramid built on the triplet of vectors a,b,c is equal to one sixth of the modulus of the mixed product of the given vectors.
Definition. Side face- this is a triangle in which one angle lies at the top of the pyramid, and the opposite side of 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 there are corners in a polygon.
Definition. pyramid height is a perpendicular dropped from the top to the base of the pyramid.
Definition. Apothem- this 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. Correct pyramid- This is a pyramid in which the base is a regular polygon, and the height descends to the center of the base.
Volume and surface area of the pyramid
Formula. pyramid volume through base area and height:
pyramid properties
If all side edges are equal, then a circle can be circumscribed around the base of the pyramid, 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.
The lateral ribs are equal when they form equal angles with the base plane, or if a circle can be described around the base of the pyramid.
If the side faces are inclined to the plane of the base at one angle, then a circle can be inscribed in the base of the pyramid, and the top of the pyramid is projected into its center.
If the side faces are inclined to the base plane at one 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 edges are equal.
3. All side ribs are inclined at the same angles to the base.
4. Apothems of all side faces are equal.
5. The areas of all side faces are equal.
6. All faces have the same dihedral (flat) angles.
7. A sphere can be described around the pyramid. The center of the described sphere will be the intersection point of the perpendiculars that pass through the middle of the edges.
8. A sphere can be inscribed in a 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 circumscribed sphere, then the sum of the flat angles at the apex is equal to π or vice versa, one angle is equal to π / n, where n is the number of angles at the base of the pyramid.
The connection of the pyramid with the sphere
A sphere can be described around the pyramid when at the base of the pyramid lies a polyhedron around which a circle can be described (a necessary and sufficient condition). The center of the sphere will be the point of intersection of planes passing perpendicularly through the midpoints of the side edges of the pyramid.
A sphere can always be described around any triangular or regular pyramid.
A sphere can be inscribed in a pyramid if the bisector planes of the internal 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 a 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.
A cone is said to be circumscribed around a 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 side edges of the pyramid are equal to each other.
Connection of a pyramid with a cylinder
A pyramid is said to be 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 circumscribed around a pyramid if a circle can be circumscribed around the base of the pyramid.
A tetrahedron has four faces and four vertices and six edges, where any two edges have no common vertices but do not touch.
Each vertex consists of three faces and edges that form trihedral angle.
The segment connecting the vertex of the tetrahedron with the center of the opposite face is called median of the tetrahedron(GM).
Bimedian is called a segment connecting the midpoints of opposite edges that do not touch (KL).
All bimedians and medians of a tetrahedron intersect at one point (S). In this case, the bimedians are divided in half, and the medians in a ratio of 3: 1 starting from the top.
Definition. Acute Angled Pyramid is a pyramid in which the apothem is more than half the length of the side of the base.
Definition. obtuse pyramid is a pyramid in which the apothem is less than half the length of the side of the base.
Definition. regular tetrahedron A tetrahedron whose 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 a vertex) are equal.
Definition. Rectangular tetrahedron a tetrahedron is called which has a right angle between three edges at the vertex (the edges are perpendicular). Three faces form rectangular trihedral 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 on which the apothem falls.
Definition. Isohedral tetrahedron A tetrahedron is called in which the side faces are equal to each other, and the base is a regular triangle. The faces of such a tetrahedron are isosceles triangles.
Definition. Orthocentric tetrahedron a tetrahedron is called in which all the heights (perpendiculars) that are lowered from the top to the opposite face intersect at one point.
Definition. star pyramid A polyhedron whose base is a star is called.
One of the simplest volumetric figures is a 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 consider formulas with which you can 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 whole figure is called a triangular pyramid.
The considered pyramid consists of a base (triangle) and three side faces (triangles). The point where the three side faces are connected is called the vertex of the figure. The perpendicular dropped to the base from this vertex 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 they speak of a regular pyramid. Otherwise, it will be sloping.
As has been said, the base of a triangular pyramid can be a general triangle. However, if it is equilateral, and the pyramid itself is straight, then they talk about the correct three-dimensional figure.
Each has 4 faces, 6 edges and 4 vertices. If the lengths of all the 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 pyramid of a general type. 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 base of the pyramid polygon, as well as for the cone. If at the base there is a triangle having side length a and height h o lowered to 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 get:
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 whose radius uniquely determines the length of the side of the polygon, then this formula can be written in terms of the corresponding radius r:
This formula is easy to obtain from the previous one, given that the radius r of the circumscribed circle through the length of the side a of the triangle is determined by the expression:
The task of determining the volume of a tetrahedron
Let us show how to use the above formulas in solving specific geometry problems.
It is known that the 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:
- the length of the side of the triangle;
- figure height.
The first value is known from the condition of the problem:
To determine the height, consider the figure shown in the figure.
The marked triangle ABC is a right triangle where the angle ABC is 90o. The AC side is the hypotenuse, whose length is a. By simple geometric reasoning, it can be shown that the side BC has length:
Note that the length BC is the radius of the circumscribed circle around the triangle.
h \u003d AB \u003d √ (AC 2 - BC 2) \u003d √ (a 2 - a 2 / 3) \u003d 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 rib. If we substitute the value from the condition of the problem into the expression, then we get the answer:
V \u003d √2 / 12 * 7 3 ≈ 40.42 cm 3.
If we compare this value with the volume of a cube that has the same edge, we get that the volume of a 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 is tetrahedral, and each carbon atom in diamond is connected to four other atoms to form a tetrahedron.
Problem with homothetic pyramids
Let's solve one curious geometric problem. Assume that there is a triangular regular pyramid with some volume V 1 . By how many times should the size of this figure be reduced in order to obtain a pyramid homothetic to it with a volume three times smaller than the original one?
Let's start solving the problem by writing the formula for the original regular pyramid:
V 1 \u003d √3 / 12 * a 1 2 * h 1.
Let the volume of the figure required by the condition 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 figures is known from the condition, we obtain the value of the coefficient k:
k \u003d ∛ (V 2 / V 1) \u003d ∛ (1/3) ≈ 0.693.
Note that we would have obtained a similar value of the coefficient k for an arbitrary type of pyramid, and not just for a regular triangular one.
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