When solving Problem C2 using the coordinate method, many students face the same problem. They can't calculate coordinates of points included in the formula dot product. The greatest difficulties arise pyramids. And if the base points are considered more or less normal, then the tops are a real hell.
Today we will work on a regular quadrangular pyramid. There is also a triangular pyramid (aka - tetrahedron). It's more complex design, so a separate lesson will be devoted to it.
First, let's remember the definition:
A regular pyramid is one that:
- The base is a regular polygon: triangle, square, etc.;
- An altitude drawn to the base passes through its center.
In particular, the base of a quadrangular pyramid is square. Just like Cheops, only a little smaller.
Below are calculations for a pyramid in which all edges are equal to 1. If this is not the case in your problem, the calculations do not change - just the numbers will be different.
Vertices of a quadrangular pyramid
So, let the correct one be given quadrangular pyramid SABCD, where S is the vertex, the base ABCD is a square. All edges are equal to 1. You need to enter a coordinate system and find the coordinates of all points. We have:
We introduce a coordinate system with origin at point A:
- The OX axis is directed parallel to the edge AB;
- OY axis is parallel to AD. Since ABCD is a square, AB ⊥ AD;
- Finally, we direct the OZ axis upward, perpendicular to the plane ABCD.
Now we calculate the coordinates. Additional construction: SH - height drawn to the base. For convenience, we will place the base of the pyramid in a separate drawing. Since points A, B, C and D lie in the OXY plane, their coordinate is z = 0. We have:
- A = (0; 0; 0) - coincides with the origin;
- B = (1; 0; 0) - step by 1 along the OX axis from the origin;
- C = (1; 1; 0) - step by 1 along the OX axis and by 1 along the OY axis;
- D = (0; 1; 0) - step only along the OY axis.
- H = (0.5; 0.5; 0) - the center of the square, the middle of the segment AC.
It remains to find the coordinates of point S. Note that the x and y coordinates of points S and H are the same, since they lie on a line parallel to the OZ axis. It remains to find the z coordinate for point S.
Consider triangles ASH and ABH:
- AS = AB = 1 by condition;
- Angle AHS = AHB = 90°, since SH is the height and AH ⊥ HB as the diagonals of the square;
- Side AH is common.
Therefore, right triangles ASH and ABH equal one leg and one hypotenuse each. This means SH = BH = 0.5 BD. But BD is the diagonal of a square with side 1. Therefore we have:
Total coordinates of point S:
In conclusion, we write down the coordinates of all the vertices of a regular rectangular pyramid:
What to do when the ribs are different
What if the side edges of the pyramid are not equal to the edges of the base? In this case, consider the triangle AHS:
Triangle AHS - rectangular, and the hypotenuse AS is also a side edge of the original pyramid SABCD. Leg AH is easily calculated: AH = 0.5 AC. We will find the remaining leg SH according to the Pythagorean theorem. This will be the z coordinate for point S.
Task. Given a regular quadrangular pyramid SABCD, at the base of which lies a square with side 1. Side edge BS = 3. Find the coordinates of point S.
We already know the x and y coordinates of this point: x = y = 0.5. This follows from two facts:
- The projection of point S onto the OXY plane is point H;
- At the same time, point H is the center of a square ABCD, all sides of which are equal to 1.
It remains to find the coordinate of point S. Consider triangle AHS. It is rectangular, with the hypotenuse AS = BS = 3, the leg AH being half the diagonal. For further calculations we need its length:
Pythagorean theorem for triangle AHS: AH 2 + SH 2 = AS 2. We have:
So, the coordinates of point S:
First level
Pyramid. Visual guide (2019)
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.
A lot of complex 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 - regular hexagon, the top is projected to 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 side edges are equal.
- All lateral faces are isosceles triangles and all these triangles are equal.
Here you can find basic information about pyramids and related formulas and concepts. All of them are studied with a mathematics tutor in preparation for the Unified State Exam.
Consider a plane, a polygon 
, lying in it and a point S, not lying in it. Let's connect S to all the vertices of the polygon. The resulting polyhedron is called a pyramid. The segments are called side ribs. 
The polygon is called the base, and point S is the top of the pyramid. Depending on the number n, the pyramid is called triangular (n=3), quadrangular (n=4), pentagonal (n=5) and so on. An alternative name for a triangular pyramid is tetrahedron. The height of a pyramid is the perpendicular descending from its top to the plane of the base. 
A pyramid is called regular if a regular polygon, and the base of the pyramid's altitude (the base of the perpendicular) is its center.
Tutor's comment:
Do not confuse the concepts of “regular pyramid” and “regular tetrahedron”. In a regular pyramid, the side edges are not necessarily equal to the edges of the base, but in a regular tetrahedron, all 6 edges are equal. This is his definition. It is easy to prove that the equality implies that the center P of the polygon coincides with a base height, so a regular tetrahedron is a regular pyramid.
What is an apothem?
The apothem of a pyramid is the height of its side face. If the pyramid is regular, then all its apothems are equal. The reverse is not true. 
A mathematics tutor about his terminology: 80% of work with pyramids is built through two types of triangles:
1) Containing apothem SK and height SP
2) Containing the lateral edge SA and its projection PA 
To simplify references to these triangles, it is more convenient for a math tutor to call the first of them apothemal, and second costal. Unfortunately, you will not find this terminology in any of the textbooks, and the teacher has to introduce it unilaterally.
Formula for the volume of a pyramid:
1) 
, where is the area of the base of the pyramid, and is the height of the pyramid
2) , where is the radius of the inscribed sphere, and is the area full surface pyramids.
3) 
, where MN is the distance between any two crossing edges, and is the area of the parallelogram formed by the midpoints of the four remaining edges.
Property of the base of the height of a pyramid:
Point P (see figure) coincides with the center of the inscribed circle at the base of the pyramid if one of the following conditions is met:
1) All apothems are equal
2) All side faces are equally inclined to the base
3) All apothems are equally inclined to the height of the pyramid
4) The height of the pyramid is equally inclined to all side faces
Math tutor's comment: Please note that all points are united by one common property: one way or another, lateral faces are involved everywhere (apothems are their elements). Therefore, the tutor can offer a less precise, but more convenient for learning, formulation: point P coincides with the center of the inscribed circle, the base of the pyramid, if there is any equal information about its lateral faces. To prove it, it is enough to show that all apothem triangles are equal.
Point P coincides with the center of a circle circumscribed near the base of the pyramid if one of three conditions is true:
1) All side edges are equal
2) All side ribs are equally inclined to the base
3) All side ribs are equally inclined to the height
- apothem- the height of the side face of a regular pyramid, which is drawn from its vertex (in addition, the apothem is the length of the perpendicular, which is lowered from the middle of the regular polygon to one of its sides);
- side faces (ASB, BSC, CSD, DSA) - triangles that meet at the vertex;
- lateral ribs ( AS , B.S. , C.S. , D.S. ) — common sides of the side faces;
- top of the pyramid (t. S) - a point that connects the side ribs and which does not lie in the plane of the base;
- height ( SO ) - a perpendicular segment drawn through the top of the pyramid to the plane of its base (the ends of such a segment will be the top of the pyramid and the base of the perpendicular);
- diagonal section of the pyramid- a section of the pyramid that passes through the top and the diagonal of the base;
- base (ABCD) - a polygon that does not belong to the vertex of the pyramid.
Properties of the pyramid.
1. When all the side edges have the same size, then:
- it is easy to describe a circle near the base of the pyramid, and the top of the pyramid will be projected into the center of this circle;
- the side ribs form equal angles with the plane of the base;
- Moreover, the opposite is also true, i.e. when the lateral ribs form with the plane of the base equal angles, or when a circle can be described near the base of the pyramid and the top of the pyramid will be projected into the center of this circle, which means that all the side edges of the pyramid are the same size.
2. When the side faces have an angle of inclination to the plane of the base of the same value, then:
- it is easy to describe a circle near the base of the pyramid, and the top of the pyramid will be projected into the center of this circle;
- the heights of the side faces are of equal length;
- the area of the side surface is equal to ½ the product of the perimeter of the base and the height of the side face.
3. A sphere can be described around a pyramid if at the base of the pyramid there is a polygon around which a circle can be described (a necessary and sufficient condition). The center of the sphere will be the point of intersection of the planes that pass through the middles of the edges of the pyramid perpendicular to them. From this theorem we conclude that a sphere can be described both around any triangular and around any regular pyramid.
4. A sphere can be inscribed into a pyramid if the bisector planes of the internal dihedral angles of the pyramid intersect at the 1st point (a necessary and sufficient condition). This point will become the center of the sphere.
The simplest pyramid.
Based on the number of angles, the base of the pyramid is divided into triangular, quadrangular, and so on.
There will be a pyramid triangular, quadrangular, and so on, when the base of the pyramid is a triangle, a quadrangle, and so on. Triangular pyramid there is a tetrahedron - a tetrahedron. Quadrangular - pentagonal and so on.
