Definition
Pyramid is a polyhedron composed of a polygon \(A_1A_2...A_n\) and \(n\) triangles with a common vertex \(P\) (not lying in the plane of the polygon) and sides opposite it, coinciding with the sides of the polygon.
Designation: \(PA_1A_2...A_n\) .
Example: pentagonal pyramid \(PA_1A_2A_3A_4A_5\) .
Triangles \(PA_1A_2, \PA_2A_3\), etc. are called side faces pyramids, segments \(PA_1, PA_2\), etc. – lateral ribs, polygon \(A_1A_2A_3A_4A_5\) – basis, point \(P\) – top.
Height pyramids are a perpendicular descended from the top of the pyramid to the plane of the base.
A pyramid with a triangle at its base is called tetrahedron.
The pyramid is called correct, if its base is a regular polygon and one of the following conditions is met:
\((a)\) lateral ribs the pyramids are equal;
\((b)\) the height of the pyramid passes through the center of the circle circumscribed near the base;
\((c)\) the side ribs are inclined to the plane of the base at the same angle.
\((d)\) side faces inclined to the plane of the base at the same angle.
Regular tetrahedron is a triangular pyramid, all of whose faces are equal equilateral triangles.
Theorem
Conditions \((a), (b), (c), (d)\) are equivalent.
Proof
Let's find the height of the pyramid \(PH\) . Let \(\alpha\) be the plane of the base of the pyramid.
1) Let us prove that from \((a)\) it follows \((b)\) . Let \(PA_1=PA_2=PA_3=...=PA_n\) .
Because \(PH\perp \alpha\), then \(PH\) is perpendicular to any line lying in this plane, which means the triangles are right-angled. This means that these triangles are equal in common leg \(PH\) and hypotenuse \(PA_1=PA_2=PA_3=...=PA_n\) . So, \(A_1H=A_2H=...=A_nH\) . This means that the points \(A_1, A_2, ..., A_n\) are at the same distance from the point \(H\), therefore, they lie on the same circle with the radius \(A_1H\) . This circle, by definition, is circumscribed about the polygon \(A_1A_2...A_n\) .
2) Let us prove that \((b)\) implies \((c)\) .
\(PA_1H, PA_2H, PA_3H,..., PA_nH\) rectangular and equal on two legs. This means that their angles are also equal, therefore, \(\angle PA_1H=\angle PA_2H=...=\angle PA_nH\).
3) Let us prove that \((c)\) implies \((a)\) .
Similar to the first point, triangles \(PA_1H, PA_2H, PA_3H,..., PA_nH\) rectangular and along the leg and sharp corner. This means that their hypotenuses are also equal, that is, \(PA_1=PA_2=PA_3=...=PA_n\) .
4) Let us prove that \((b)\) implies \((d)\) .
Because in a regular polygon, the centers of the circumscribed and inscribed circles coincide (generally speaking, this point is called the center of a regular polygon), then \(H\) is the center of the inscribed circle. Let's draw perpendiculars from the point \(H\) to the sides of the base: \(HK_1, HK_2\), etc. These are the radii of the inscribed circle (by definition). Then, according to the TTP (\(PH\) is a perpendicular to the plane, \(HK_1, HK_2\), etc. are projections, perpendicular to the sides) oblique \(PK_1, PK_2\), etc. perpendicular to the sides \(A_1A_2, A_2A_3\), etc. respectively. So, by definition \(\angle PK_1H, \angle PK_2H\) equal to the angles between the side faces and the base. Because triangles \(PK_1H, PK_2H, ...\) are equal (as rectangular on two sides), then the angles \(\angle PK_1H, \angle PK_2H, ...\) are equal.
5) Let us prove that \((d)\) implies \((b)\) .
Similar to the fourth point, the triangles \(PK_1H, PK_2H, ...\) are equal (as rectangular along the leg and acute angle), which means the segments \(HK_1=HK_2=...=HK_n\) are equal. This means, by definition, \(H\) is the center of a circle inscribed in the base. But because For regular polygons, the centers of the inscribed and circumscribed circles coincide, then \(H\) is the center of the circumscribed circle. Chtd.
Consequence
The lateral faces of a regular pyramid are equal isosceles triangles.
Definition
The height of the lateral face of a regular pyramid drawn from its vertex is called apothem.
The apothems of all lateral faces of a regular pyramid are equal to each other and are also medians and bisectors.
Important Notes
1. Height is correct triangular pyramid falls at the point of intersection of the altitudes (or bisectors, or medians) of the base (the base is a regular triangle).
2. The height of a regular quadrangular pyramid falls at the point of intersection of the diagonals of the base (the base is a square).
3. The height of a regular hexagonal pyramid falls at the point of intersection of the diagonals of the base (the base is a regular hexagon).
4. The height of the pyramid is perpendicular to any straight line lying at the base.
Definition
The pyramid is called rectangular, if one of its side edges is perpendicular to the plane of the base.
Important Notes
1. In a rectangular pyramid, the edge perpendicular to the base is the height of the pyramid. That is, \(SR\) is the height.
2. Because \(SR\) is perpendicular to any line from the base, then \(\triangle SRM, \triangle SRP\)– right triangles.
3. Triangles \(\triangle SRN, \triangle SRK\)- also rectangular.
That is, any triangle formed by this edge and the diagonal emerging from the vertex of this edge lying at the base will be rectangular.
\[(\Large(\text(Volume and surface area of the pyramid)))\]
Theorem
The volume of the pyramid is equal to one third of the product of the area of the base and the height of the pyramid: \
Consequences
Let \(a\) be the side of the base, \(h\) be the height of the pyramid.
1. The volume of a regular triangular pyramid is \(V_(\text(right triangle.pir.))=\dfrac(\sqrt3)(12)a^2h\),
2. The volume of a regular quadrangular pyramid is \(V_(\text(right.four.pir.))=\dfrac13a^2h\).
3. The volume of a regular hexagonal pyramid is \(V_(\text(right.six.pir.))=\dfrac(\sqrt3)(2)a^2h\).
4. The volume of a regular tetrahedron is \(V_(\text(right tetr.))=\dfrac(\sqrt3)(12)a^3\).
Theorem
The area of the lateral surface of a regular pyramid is equal to the half product of the perimeter of the base and the apothem.
\[(\Large(\text(Frustum)))\]
Definition
Consider an arbitrary pyramid \(PA_1A_2A_3...A_n\) . Let us draw a plane parallel to the base of the pyramid through a certain point lying on the side edge of the pyramid. This plane will split the pyramid into two polyhedra, one of which is a pyramid (\(PB_1B_2...B_n\)), and the other is called truncated pyramid(\(A_1A_2...A_nB_1B_2...B_n\) ).
The truncated pyramid has two bases - polygons \(A_1A_2...A_n\) and \(B_1B_2...B_n\) which are similar to each other.
The height of a truncated pyramid is a perpendicular drawn from some point of the upper base to the plane of the lower base.
Important Notes
1. All lateral faces of a truncated pyramid are trapezoids.
2. The segment connecting the centers of the bases of a regular truncated pyramid (that is, a pyramid obtained by cross-section of a regular pyramid) is the height.
Definition. Side edge- this is a triangle in which one angle lies at the top of the pyramid, and the opposite side coincides with the side of the base (polygon).
Definition. Side ribs- these are the common sides of the side faces. A pyramid has as many edges as the angles of a polygon.
Definition. Pyramid height- this is a perpendicular lowered from the top to the base of the pyramid.
Definition. Apothem- this is a perpendicular to 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 a pyramid by a plane passing through the top of the pyramid and the diagonal of the base.
Definition. Correct pyramid 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. Volume of the pyramid through base area and height:
Properties of the pyramid
If all the side edges are equal, then a circle can be drawn around the base of the pyramid, and the center of the base coincides with the center of the circle. Also, a perpendicular dropped from the top passes through the center of the base (circle).
If all the side edges are equal, then they are inclined to the plane of the base at the same angles.
The lateral ribs are equal when they form with the plane of the base equal angles 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 the same angle, then a circle can be inscribed into the base of the pyramid, and the top of the pyramid is projected into its center.
If the side faces are inclined to the plane of the base 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 edges are equal.
3. All side ribs are inclined at equal angles to the base.
4. The apothems 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. A sphere can be described around the pyramid. The center of the circumscribed sphere will be the intersection point of the perpendiculars that pass through the middle of the edges.
8. You can fit a sphere into a pyramid. The center of the inscribed sphere will be the point of intersection 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 plane angles at the vertex 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 between the pyramid and the sphere
A sphere can be described around a pyramid when at the base of the pyramid there is a polyhedron 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.
A sphere can always be described around any triangular or regular pyramid.
A sphere can be inscribed into 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.
Connection of a pyramid with a cone
A cone is said to be 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 to each other.
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 the lateral edges of the pyramid are equal to each other.
Relationship between a pyramid and a cylinder
A pyramid is called 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.
A tetrahedron has four faces and four vertices and six edges, where any two edges do not have common vertices but do not touch.
Each vertex consists of three faces and edges that form triangular angle.
The segment connecting the vertex of a tetrahedron with the center of the opposite face is called median of the tetrahedron(GM).
Bimedian 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 are divided in a ratio of 3:1 starting from the top.
Definition. Acute angled pyramid- a pyramid in which the apothem is more than half the length of the side of the base.
Definition. Obtuse pyramid- a pyramid in which the apothem is less than half the length of the side of the base.
Definition. Regular tetrahedron- a tetrahedron in which all four faces are equilateral triangles. It is one of the five regular polygons. In a regular tetrahedron, all dihedral angles (between faces) and trihedral angles (at the vertex) are equal.
Definition. Rectangular tetrahedron is a tetrahedron with a right angle between 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 on which the apothem falls.
Definition. Isohedral tetrahedron is called a tetrahedron whose side faces are equal to each other, and the base is a regular triangle. Such a tetrahedron has faces that are isosceles triangles.
Definition. Orthocentric tetrahedron is 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.
A three-dimensional figure that often appears in geometric problems is the pyramid. The simplest of all the figures in this class is triangular. In this article we will analyze in detail the basic formulas and properties of the correct
Geometric ideas about the figure
Before moving on to considering the properties of a regular triangular pyramid, let’s take a closer look at what kind of figure we are talking about.
Let's assume that there is an arbitrary triangle in three-dimensional space. Let us select any point in this space that does not lie in the plane of the triangle and connect it with the three vertices of the triangle. We got a triangular pyramid.
It consists of 4 sides, all of which are triangles. The points where three faces meet are called vertices. The figure also has four of them. The lines of intersection of two faces are edges. The pyramid in question has 6 edges. The figure below shows an example of this figure.
Since the figure is formed by four sides, it is also called a tetrahedron.
Correct pyramid
Was discussed above arbitrary figure with a triangular base. Now suppose that we draw a perpendicular segment from the top of the pyramid to its base. This segment is called height. Obviously, you can draw 4 different heights for the figure. If the height intersects the triangular base at the geometric center, then such a pyramid is called straight.
A straight pyramid, the base of which is an equilateral triangle, is called regular. For her, all three triangles forming lateral surface figures are isosceles and equal to each other. A special case of a regular pyramid is the situation when all four sides are equilateral identical triangles.
Let's consider the properties of a regular triangular pyramid and give the corresponding formulas for calculating its parameters.
Base side, height, lateral edge and apothem
Any two of the listed parameters uniquely determine the remaining two characteristics. Let us present formulas that relate these quantities.
Let us assume that the side of the base of a regular triangular pyramid is a. The length of its lateral edge is b. What will be the height of a regular triangular pyramid and its apothem?
For height h we get the expression:
This formula follows from the Pythagorean theorem for which the side edge, the height and 2/3 of the height of the base are.
The apothem of a pyramid is the height for any side triangle. The length of the apothem a b is equal to:
a b = √(b 2 - a 2 /4)
From these formulas it is clear that whatever the side of the base of a triangular regular pyramid and the length of its side edge, the apothem will always be greater than the height of the pyramid.
The two formulas presented contain all four linear characteristics of the figure in question. Therefore, given the known two of them, you can find the rest by solving the system of written equalities.
Figure volume
For absolutely any pyramid (including an inclined one), the value of the volume of space limited by it can be determined by knowing the height of the figure and the area of its base. The corresponding formula is:
Applying this expression to the figure in question, we obtain the following formula:
Where the height of a regular triangular pyramid is h and its base side is a.
It is not difficult to obtain a formula for the volume of a tetrahedron in which all sides are equal to each other and represent equilateral triangles. In this case, the volume of the figure is determined by the formula:
That is, it is determined uniquely by the length of side a.
Surface area
Let us continue to consider the properties of a regular triangular pyramid. The total area of all the faces of a figure is called its surface area. The latter can be conveniently studied by considering the corresponding development. The figure below shows what the development of a regular triangular pyramid looks like.
Let's assume that we know the height h and the side of the base a of the figure. Then the area of its base will be equal to:
Every schoolchild can obtain this expression if he remembers how to find the area of a triangle and also takes into account that the height equilateral triangle is also a bisector and a median.
The lateral surface area formed by three identical isosceles triangles is:
S b = 3/2*√(a 2 /12+h 2)*a
This equality follows from the expression of the apothem of the pyramid in terms of the height and length of the base.
The total surface area of the figure is:
S = S o + S b = √3/4*a 2 + 3/2*√(a 2 /12+h 2)*a
Note that for a tetrahedron in which all four sides are identical equilateral triangles, the area S will be equal to:
Properties of a regular truncated triangular pyramid
If the top of the considered triangular pyramid is cut off with a plane parallel to the base, then the remaining lower part will be called a truncated pyramid.
In the case of a triangular base, the result of the described sectioning method is a new triangle, which is also equilateral, but has a shorter side length than the side of the base. A truncated triangular pyramid is shown below.
We see that this figure is already limited to two triangular bases and three isosceles trapezoids.
Let us assume that the height of the resulting figure is equal to h, the lengths of the sides of the lower and upper bases are a 1 and a 2, respectively, and the apothem (height of the trapezoid) is equal to a b. Then the surface area of the truncated pyramid can be calculated using the formula:
S = 3/2*(a 1 +a 2)*a b + √3/4*(a 1 2 + a 2 2)
Here the first term is the area of the lateral surface, the second term is the area of the triangular bases.
The volume of the figure is calculated as follows:
V = √3/12*h*(a 1 2 + a 2 2 + a 1 *a 2)
To unambiguously determine the characteristics of a truncated pyramid, you need to know its three parameters, which is demonstrated by the given formulas.
This video tutorial will help users get an idea of the Pyramid theme. Correct pyramid. In this lesson we will get acquainted with the concept of a pyramid and give it a definition. Let's look at what it is regular pyramid and what properties it has. Then we prove the theorem about the lateral surface of a regular pyramid.
In this lesson we will get acquainted with the concept of a pyramid and give it a definition.
Consider a polygon A 1 A 2...A n, which lies in the α plane, and the point P, which does not lie in the α plane (Fig. 1). Let's connect the dots P with peaks A 1, A 2, A 3, … A n. We get n triangles: A 1 A 2 R, A 2 A 3 R and so on.
Definition. Polyhedron RA 1 A 2 ...A n, made up of n-square A 1 A 2...A n And n triangles RA 1 A 2, RA 2 A 3 …RA n A n-1 is called n-coal pyramid. Rice. 1.
Rice. 1
Consider a quadrangular pyramid PABCD(Fig. 2).
R- the top of the pyramid.
ABCD- the base of the pyramid.
RA- side rib.
AB- base rib.
From point R let's drop the perpendicular RN to the base plane ABCD. The drawn perpendicular is the height of the pyramid.
Rice. 2
Full surface The pyramid consists of a lateral surface, that is, the area of all lateral faces, and the area of the base:
S full = S side + S main
A pyramid is called correct if:
- its base is a regular polygon;
- the segment connecting the top of the pyramid with the center of the base is its height.
Explanation using the example of a regular quadrangular pyramid
Consider a regular quadrangular pyramid PABCD(Fig. 3).
R- the top of the pyramid. Base of the pyramid ABCD- a regular quadrilateral, that is, a square. Dot ABOUT, the point of intersection of the diagonals, is the center of the square. Means, RO is the height of the pyramid.
Rice. 3
Explanation: in the correct n In a triangle, the center of the inscribed circle and the center of the circumcircle coincide. This center is called the center of the polygon. Sometimes they say that the vertex is projected into the center.
The height of the lateral face of a regular pyramid drawn from its vertex is called apothem and is designated h a.
1. all lateral edges of a regular pyramid are equal;
2. The side faces are equal isosceles triangles.
We will give a proof of these properties using the example of a regular quadrangular pyramid.
Given: PABCD- regular quadrangular pyramid,
ABCD- square,
RO- height of the pyramid.
Prove:
1. RA = PB = RS = PD
2.∆ABP = ∆BCP =∆CDP =∆DAP See Fig. 4.
Rice. 4
Proof.
RO- height of the pyramid. That is, straight RO perpendicular to the plane ABC, and therefore direct JSC, VO, SO And DO lying in it. So triangles ROA, ROV, ROS, ROD- rectangular.
Consider a square ABCD. From the properties of a square it follows that AO = VO = CO = DO.
Then the right triangles ROA, ROV, ROS, ROD leg RO- general and legs JSC, VO, SO And DO are equal, which means that these triangles are equal on two sides. From the equality of triangles follows the equality of segments, RA = PB = RS = PD. Point 1 has been proven.
Segments AB And Sun are equal because they are sides of the same square, RA = PB = RS. So triangles AVR And VSR - isosceles and equal on three sides.
In a similar way we find that triangles ABP, VCP, CDP, DAP are isosceles and equal, as required to be proved in paragraph 2.
The area of the lateral surface of a regular pyramid is equal to half the product of the perimeter of the base and the apothem:
To prove this, let’s choose a regular triangular pyramid.
Given: RAVS- regular triangular pyramid.
AB = BC = AC.
RO- height.
Prove: 
. See Fig. 5.
Rice. 5
Proof.
RAVS- regular triangular pyramid. That is AB= AC = BC. Let ABOUT- center of the triangle ABC, Then RO is the height of the pyramid. At the base of the pyramid lies an equilateral triangle ABC. notice, that 
.
Triangles RAV, RVS, RSA- equal isosceles triangles (by property). A triangular pyramid has three side faces: RAV, RVS, RSA. This means that the area of the lateral surface of the pyramid is:
S side = 3S RAW
The theorem has been proven.
The radius of a circle inscribed at the base of a regular quadrangular pyramid is 3 m, the height of the pyramid is 4 m. Find the area of the lateral surface of the pyramid.
Given: regular quadrangular pyramid ABCD,
ABCD- square,
r= 3 m,
RO- height of the pyramid,
RO= 4 m.
Find: S side. See Fig. 6.
Rice. 6
Solution.
According to the proven theorem, .
Let's first find the side of the base AB. We know that the radius of a circle inscribed at the base of a regular quadrangular pyramid is 3 m.
Then, m.
Find the perimeter of the square ABCD with a side of 6 m:
Consider a triangle BCD. Let M- middle of the side DC. Because ABOUT- middle BD, That 
(m).
Triangle DPC- isosceles. M- middle DC. That is, RM- median, and therefore height in the triangle DPC. Then RM- apothem of the pyramid.
RO- height of the pyramid. Then, straight RO perpendicular to the plane ABC, and therefore direct OM, lying in it. Let's find the apothem RM from a right triangle ROM.
Now we can find the lateral surface of the pyramid:
Answer: 60 m2.
The radius of the circle circumscribed around the base of a regular triangular pyramid is equal to m. The area of the lateral surface is 18 m 2. Find the length of the apothem.
Given: ABCP- regular triangular pyramid,
AB = BC = SA,
R= m,
S side = 18 m2.
Find: . See Fig. 7.
Rice. 7
Solution.
In a right triangle ABC The radius of the circumscribed circle is given. Let's find a side AB this triangle using the law of sines.
Knowing the side of a regular triangle (m), we find its perimeter.
By the theorem on the lateral surface area of a regular pyramid, where h a- apothem of the pyramid. Then:
Answer: 4 m.
So, we looked at what a pyramid is, what a regular pyramid is, and we proved the theorem about the lateral surface of a regular pyramid. In the next lesson we will get acquainted with the truncated pyramid.
Bibliography
- Geometry. Grades 10-11: textbook for students of general education institutions (basic and specialized levels) / I. M. Smirnova, V. A. Smirnov. - 5th ed., rev. and additional - M.: Mnemosyne, 2008. - 288 p.: ill.
- Geometry. 10-11 grade: Textbook for general education educational institutions/ Sharygin I.F. - M.: Bustard, 1999. - 208 p.: ill.
- Geometry. Grade 10: Textbook for general education institutions with in-depth and specialized study of mathematics /E. V. Potoskuev, L. I. Zvalich. - 6th ed., stereotype. - M.: Bustard, 008. - 233 p.: ill.
- Internet portal "Yaklass" ()
- Internet portal “Festival of pedagogical ideas “First of September” ()
- Internet portal “Slideshare.net” ()
Homework
- Can a regular polygon be the base of an irregular pyramid?
- Prove that disjoint edges of a regular pyramid are perpendicular.
- Find the value of the dihedral angle at the side of the base of a regular quadrangular pyramid if the apothem of the pyramid is equal to the side of its base.
- RAVS- regular triangular pyramid. Construct the linear angle of the dihedral angle at the base of the pyramid.
A triangular pyramid is a pyramid that has a triangle at its base. The height of this pyramid is the perpendicular that is lowered from the top of the pyramid to its base.
Finding the height of a pyramid
How to find the height of a pyramid? Very simple! To find the height of any triangular pyramid, you can use the volume formula: V = (1/3)Sh, where S is the area of the base, V is the volume of the pyramid, h is its height. From this formula, derive the height formula: to find the height of a triangular pyramid, you need to multiply the volume of the pyramid by 3, and then divide the resulting value by the area of the base, it will be: h = (3V)/S. Since the base of a triangular pyramid is a triangle, you can use the formula to calculate the area of a triangle. If we know: the area of the triangle S and its side z, then according to the area formula S=(1/2)γh: h = (2S)/γ, where h is the height of the pyramid, γ is the edge of the triangle; the angle between the sides of the triangle and the two sides themselves, then using the following formula: S = (1/2)γφsinQ, where γ, φ are the sides of the triangle, we find the area of the triangle. The value of the sine of angle Q needs to be looked at in the table of sines, which is available on the Internet. Next, we substitute the area value into the height formula: h = (2S)/γ. If the task requires calculating the height of a triangular pyramid, then the volume of the pyramid is already known.
Regular triangular pyramid
Find the height of a regular triangular pyramid, that is, a pyramid in which all faces are equilateral triangles, knowing the edge size γ. In this case, the edges of the pyramid are the sides of equilateral triangles. The height of a regular triangular pyramid will be: h = γ√(2/3), where γ is the edge of the equilateral triangle, h is the height of the pyramid. If the area of the base (S) is unknown, and only the length of the edge (γ) and the volume (V) of the polyhedron are given, then the necessary variable in the formula from the previous step must be replaced by its equivalent, which is expressed in terms of the length of the edge. The area of a triangle (regular) is equal to 1/4 of the product of the side length of this triangle squared by the square root of 3. We substitute this formula instead of the area of the base in the previous formula, and we obtain the following formula: h = 3V4/(γ 2 √3) = 12V/(γ 2 √3). The volume of a tetrahedron can be expressed through the length of its edge, then from the formula for calculating the height of a figure, you can remove all variables and leave only the side of the triangular face of the figure. The volume of such a pyramid can be calculated by dividing by 12 from the product the cubed length of its face by the square root of 2.
Substituting this expression into the previous formula, we obtain the following formula for calculation: h = 12(γ 3 √2/12)/(γ 2 √3) = (γ 3 √2)/(γ 2 √3) = γ√(2 /3) = (1/3)γ√6. Also correct triangular prism can be inscribed in a sphere, and knowing only the radius of the sphere (R) one can find the height of the tetrahedron itself. The length of the edge of the tetrahedron is: γ = 4R/√6. We replace the variable γ with this expression in the previous formula and get the formula: h = (1/3)√6(4R)/√6 = (4R)/3. The same formula can be obtained by knowing the radius (R) of a circle inscribed in a tetrahedron. In this case, the length of the edge of the triangle will be equal to 12 ratios between square root of 6 and radius. We substitute this expression into the previous formula and we have: h = (1/3)γ√6 = (1/3)√6(12R)/√6 = 4R.
How to find the height of a regular quadrangular pyramid
To answer the question of how to find the length of the height of a pyramid, you need to know what a regular pyramid is. Quadrangular pyramid is a pyramid with a quadrangle at its base. If in the conditions of the problem we have: the volume (V) and the area of the base (S) of the pyramid, then the formula for calculating the height of the polyhedron (h) will be as follows - divide the volume multiplied by 3 by the area S: h = (3V)/S. Given a square base of a pyramid with a given volume (V) and side length γ, replace the area (S) in the previous formula with the square of the side length: S = γ 2 ; H = 3V/γ2. The height of a regular pyramid h = SO passes exactly through the center of the circle that is circumscribed near the base. Since the base of this pyramid is a square, point O is the intersection point of diagonals AD and BC. We have: OC = (1/2)BC = (1/2)AB√6. Next, we are in right triangle We find SOC (using the Pythagorean theorem): SO = √(SC 2 -OC 2). Now you know how to find the height of a regular pyramid.