A regular pyramid at the base is a square. Pyramid and its elements
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.
Side ribs are equal when they form with the plane of the base equal angles or if a circle can be circumscribed 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 edges 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.
Here are collected basic information about the pyramids and related formulas and concepts. All of them are studied with a tutor in mathematics in preparation for the exam.
Consider a plane, a polygon 
lying in it and a point S not lying in it. Connect S to all vertices of the polygon. The resulting polyhedron is called a pyramid. The segments are called lateral edges. 
The polygon is called the base, and the point S is called 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. Alternative name triangular pyramid - tetrahedron. The height of a pyramid is the perpendicular drawn from its apex to the base plane. 
A pyramid is called correct if a regular polygon, and the base of the height of the pyramid (the base of the perpendicular) is its center.
Tutor's comment:
Do not confuse the concept 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 of the edges are equal. This is his definition. It is easy to prove that the equality implies that the center P of the polygon with a height base, 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. 
Mathematics tutor about his terminology: work with pyramids is 80% 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 name the first of them apothemic, and second costal. Unfortunately, you will not find this terminology in any of the textbooks, and the teacher has to introduce it unilaterally.
Pyramid volume formula:
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 total surface area of the pyramid.
3) 
, where MN is the distance of any two crossing edges, and is the area of the parallelogram formed by the midpoints of the four remaining edges.
Pyramid Height Base Property:
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 towards 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 commentary: note that all items are united by one common property: one way or another, side faces participate everywhere (apothems are their elements). Therefore, the tutor can offer a less precise, but more convenient formulation for memorization: the 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 suffices to show that all apothemical triangles are equal.
The point P coincides with the center of the circumscribed circle near the base of the pyramid, if one of the three conditions is true:
1) All side edges are equal
2) All side ribs are equally inclined towards the base
3) All side ribs are equally inclined to the height
Pyramid. Truncated pyramid
Pyramid is called a polyhedron, one of whose faces is a polygon ( base ), and all other faces are triangles with a common vertex ( side faces ) (Fig. 15). The pyramid is called correct , if its base is a regular polygon and the top of the pyramid is projected into the center of the base (Fig. 16). A triangular pyramid in which all edges are equal is called tetrahedron .
Side rib pyramid is called the side of the side face that does not belong to the base Height pyramid is the distance from its top to the plane of the base. All side edges of a regular pyramid are equal to each other, all side faces are equal isosceles triangles. The height of the side face of a regular pyramid drawn from the vertex is called apothema . diagonal section A section of a pyramid is called a plane passing through two side edges that do not belong to the same face.
Side surface area pyramid is called the sum of the areas of all side faces. Full surface area is the sum of the areas of all the side faces and the base.
Theorems
1. If in a pyramid all lateral edges are equally inclined to the plane of the base, then the top of the pyramid is projected into the center of the circumscribed circle near the base.
2. If in a pyramid all lateral edges have equal lengths, then the top of the pyramid is projected into the center of the circumscribed circle near the base.
3. If in the pyramid all faces are equally inclined to the plane of the base, then the top of the pyramid is projected into the center of the circle inscribed in the base.
To calculate the volume of an arbitrary pyramid, the formula is correct:
where V- volume;
S main- base area;
H is the height of the pyramid.
For a regular pyramid, the following formulas are true:
where p- the perimeter of the base;
h a- apothem;
H- height;
S full
S side
S main- base area;
V is the volume of a regular pyramid.
truncated pyramid called the part of the pyramid enclosed between the base and the cutting plane parallel to the base of the pyramid (Fig. 17). Correct truncated pyramid called the part of a regular pyramid, enclosed between the base and a cutting plane parallel to the base of the pyramid.
Foundations truncated pyramid - similar polygons. Side faces - trapezoid. Height truncated pyramid is called the distance between its bases. Diagonal A truncated pyramid is a segment connecting its vertices that do not lie on the same face. diagonal section A section of a truncated pyramid is called a plane passing through two side edges that do not belong to the same face.
For a truncated pyramid, the formulas are valid:
(4)
where S 1 , S 2 - areas of the upper and lower bases;
S full is the total surface area;
S side is the lateral surface area;
H- height;
V is the volume of the truncated pyramid.
For a regular truncated pyramid, the following formula is true:
where p 1 , p 2 - base perimeters;
h a- the apothem of a regular truncated pyramid.
Example 1 In a regular triangular pyramid, the dihedral angle at the base is 60º. Find the tangent of the angle of inclination of the side edge to the plane of the base.
Decision. Let's make a drawing (Fig. 18).
 |
The pyramid is regular, which means that the base is an equilateral triangle and all the side faces are equal isosceles triangles. The dihedral angle at the base is the angle of inclination of the side face of the pyramid to the plane of the base. The linear angle will be the angle a between two perpendiculars: i.e. The top of the pyramid is projected at the center of the triangle (the center of the circumscribed circle and the inscribed circle in the triangle ABC). The angle of inclination of the side rib (for example SB) is the angle between the edge itself and its projection onto the base plane. For rib SB this angle will be the angle SBD. To find the tangent you need to know the legs SO and OB. Let the length of the segment BD is 3 a. dot O line segment BD is divided into parts: and From we find SO: 
From we find: 
Answer:
Example 2 Find the volume of a regular truncated quadrangular pyramid if the diagonals of its bases are cm and cm and the height is 4 cm.
Decision. To find the volume of a truncated pyramid, we use formula (4). To find the areas of the bases, you need to find the sides of the base squares, knowing their diagonals. The sides of the bases are 2 cm and 8 cm, respectively. This means the areas of the bases and Substituting all the data into the formula, we calculate the volume of the truncated pyramid:
Answer: 112 cm3.
Example 3 Find the area of the lateral face of a regular triangular truncated pyramid whose sides of the bases are 10 cm and 4 cm, and the height of the pyramid is 2 cm.
Decision. Let's make a drawing (Fig. 19).
The side face of this pyramid is an isosceles trapezoid. To calculate the area of a trapezoid, you need to know the bases and the height. The bases are given by condition, only the height remains unknown. Find it from where BUT 1 E perpendicular from a point BUT 1 on the plane of the lower base, A 1 D- perpendicular from BUT 1 on AC. BUT 1 E\u003d 2 cm, since this is the height of the pyramid. For finding DE we will make an additional drawing, in which we will depict a top view (Fig. 20). Dot O- projection of the centers of the upper and lower bases. since (see Fig. 20) and On the other hand OK is the radius of the inscribed circle and 
OM is the radius of the inscribed circle:
MK=DE.
According to the Pythagorean theorem from
Side face area: 
Answer:
Example 4 At the base of the pyramid lies an isosceles trapezoid, the bases of which a and b (a> b). Each side face forms an angle equal to the plane of the base of the pyramid j. Find the total surface area of the pyramid.
Decision. Let's make a drawing (Fig. 21). Total surface area of the pyramid SABCD is equal to the sum of the areas and the area of the trapezoid ABCD.
We use the statement that if all the faces of the pyramid are equally inclined to the plane of the base, then the vertex is projected into the center of the circle inscribed in the base. Dot O- vertex projection S at the base of the pyramid. Triangle SOD is the orthogonal projection of the triangle CSD to the base plane. According to the theorem on the area of the orthogonal projection of a flat figure, we get:
Similarly, it means 
Thus, the problem was reduced to finding the area of the trapezoid ABCD. Draw a trapezoid ABCD separately (Fig. 22). Dot O is the center of a circle inscribed in a trapezoid.
Since a circle can be inscribed in a trapezoid, then or By the Pythagorean theorem we have
A three-dimensional figure that often appears in geometric problems is a pyramid. The simplest of all the figures of this class is triangular. In this article, we will analyze in detail the basic formulas and properties of the correct
Geometric representations of the figure
Before proceeding to consider the properties of a regular triangular pyramid, let's take a closer look at what figure we are talking about.
Let's assume that there is an arbitrary triangle in three-dimensional space. We choose any point in this space that does not lie in the plane of the triangle, and connect it to 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 intersection lines of two faces are edges. The pyramid under consideration has 6 ribs. 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
Above, an arbitrary figure with a triangular base was considered. Now suppose we draw a perpendicular line from the top of the pyramid to its base. This segment is called the height. It is obvious that it is possible to spend 4 different heights for the figure. If the height intersects the triangular base in the geometric center, then such a pyramid is called a straight pyramid.
A straight pyramid whose base is an equilateral triangle is called a regular pyramid. For her, all three triangles forming side 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.
Consider the properties of a regular triangular pyramid and give the appropriate formulas for calculating its parameters.
Base side, height, lateral edge and apothem
Any two of the listed parameters uniquely determine the other two characteristics. We give formulas that connect the named quantities.
Suppose that the side of the base of a regular triangular pyramid is a. The length of its side edge is equal to b. What will be the height of a regular triangular pyramid and its apothem?
For the height h we get the expression:
This formula follows from the Pythagorean theorem for which are the side edge, the height and 2/3 of the height of the base.
The apothem of a pyramid is the height for any lateral triangle. The length of apotema a b is:
a b \u003d √ (b 2 - a 2 / 4)
From these formulas it can be seen that whatever the side of the base of a triangular regular pyramid and the length of its lateral edge, the apotema will always be more height pyramids.
The two formulas presented contain all four linear characteristics the figure in question. Therefore, from the known two of them, you can find the rest by solving the system from the written equalities.
figure volume
For absolutely any pyramid (including an inclined one), the value of the volume of space bounded by it can be determined by knowing the height of the figure and the area of its base. The corresponding formula looks like:
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 uniquely determined by the length of side a.
Surface area
We continue to consider the properties of a triangular regular pyramid. total area of all the faces of a figure is called its surface area. It is convenient to study the latter by considering the corresponding development. The figure below shows what a regular triangular pyramid looks like.
Suppose 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 student can get this expression if he remembers how to find the area of a triangle, and also takes into account that the height of an equilateral triangle is also a bisector and a median.
The area of the lateral surface 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 apotema 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 the same 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 by a plane parallel to the base, then the remaining Bottom part will be called a truncated pyramid.
In the case of a triangular base, as a result of the section method described, a new triangle is obtained, which is also equilateral, but has a smaller side length than the base side. Truncated triangular pyramid shown below.
We see that this figure is already limited by two triangular bases and three isosceles trapezoids.
Suppose that the height of the resulting figure is 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 by 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, it is necessary to know its three parameters, which is demonstrated by the above formulas.
Introduction
When we began to study stereometric figures, we touched on the topic "Pyramid". We liked this theme because the pyramid is very often used in architecture. And since our future profession architect, inspired by this figure, we think that she will be able to push us to great projects.
The strength of architectural structures, their most important quality. Associating strength, firstly, with the materials from which they are created, and, secondly, with the features constructive solutions, it turns out that the strength of the structure is directly related to the geometric shape that is basic for it.
In other words, we are talking about that geometric figure, which can be considered as a model of the corresponding architectural form. It turns out that the geometric shape also determines the strength of the architectural structure.
The Egyptian pyramids have long been considered the most durable architectural structure. As you know, they have the shape of regular quadrangular pyramids.
It is this geometric shape that provides the greatest stability due to the large base area. On the other hand, the shape of the pyramid ensures that the mass decreases as the height above the ground increases. It is these two properties that make the pyramid stable, and therefore strong in the conditions of gravity.
Objective of the project: learn something new about the pyramids, deepen knowledge and find practical applications.
To achieve this goal, it was necessary to solve the following tasks:
Learn historical information about the pyramid
Consider the pyramid as a geometric figure
Find application in life and architecture
Find the similarities and differences between the pyramids located in different parts Sveta
Theoretical part
The beginning of the geometry of the pyramid was laid in ancient Egypt and Babylon, but it was actively developed in Ancient Greece. The first to establish what the volume of the pyramid is equal to was Democritus, and Eudoxus of Cnidus proved it. The ancient Greek mathematician Euclid systematized knowledge about the pyramid in the XII volume of his "Beginnings", and also brought out the first definition of the pyramid: a bodily figure bounded by planes that converge from one plane at one point.
The tombs of the Egyptian pharaohs. The largest of them - the pyramids of Cheops, Khafre and Mikerin in El Giza in ancient times were considered one of the Seven Wonders of the World. The erection of the pyramid, in which the Greeks and Romans already saw a monument to the unprecedented pride of kings and cruelty, which doomed the entire people of Egypt to senseless construction, was the most important cult act and was supposed to express, apparently, the mystical identity of the country and its ruler. The population of the country worked on the construction of the tomb in the part of the year free from agricultural work. A number of texts testify to the attention and care that the kings themselves (albeit of a later time) paid to the construction of their tomb and its builders. It is also known about the special cult honors that turned out to be the pyramid itself.
Basic concepts
Pyramid A polyhedron is called, the base of which is a polygon, and the remaining faces are triangles having a common vertex.
Apothem- the height of the side face of a regular pyramid, drawn from its top;
Side faces- triangles converging at the top;
Side ribs- common sides of the side faces;
top of the pyramid- a point connecting the side edges and not lying in the plane of the base;
Height- a segment of a perpendicular drawn through the top of the pyramid to the plane of its base (the ends of this segment are the top of the pyramid and the base of the perpendicular);
Diagonal section of a pyramid- section of the pyramid passing through the top and the diagonal of the base;
Base- a polygon that does not belong to the top of the pyramid.
The main properties of the correct pyramid
Side edges, side faces and apothems are equal, respectively.
The dihedral angles at the base are equal.
The dihedral angles at the side edges are equal.
Each height point is equidistant from all base vertices.
Each height point is equidistant from all side faces.
Basic pyramid formulas
The area of the lateral and full surface of the pyramid.
The area of the lateral surface of the pyramid (full and truncated) is the sum of the areas of all its lateral faces, the total surface area is the sum of the areas of all its faces.
Theorem: 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 of the pyramid.
p- perimeter of the base;
h- apothem.
The area of the lateral and full surfaces of a truncated pyramid.
p1, p 2 - base perimeters;
h- apothem.
R- total surface area of a regular truncated pyramid;
S side- area of the lateral surface of a regular truncated pyramid;
S1 + S2- base area
Pyramid Volume
Form The volume scale is used for pyramids of any kind.
H is the height of the pyramid.
Angles of the pyramid
The angles that are formed by the side face and the base of the pyramid are called dihedral angles at the base of the pyramid.
A dihedral angle is formed by two perpendiculars.
To determine this angle, you often need to use the three perpendiculars theorem.
The angles that are formed by a side edge and its projection onto the plane of the base are called angles between the lateral edge and the plane of the base.
The angle formed by two side faces is called dihedral angle at the lateral edge of the pyramid.
The angle, which is formed by two side edges of one face of the pyramid, is called corner at the top of the pyramid.
Sections of the pyramid
The surface of a pyramid is the surface of a polyhedron. Each of its faces is a plane, so the section of the pyramid given by the secant plane is a broken line consisting of separate straight lines.
Diagonal section
The section of a pyramid by a plane passing through two lateral edges that do not lie on the same face is called diagonal section pyramids.
Parallel sections
Theorem:
If the pyramid is crossed by a plane parallel to the base, then the side edges and heights of the pyramid are divided by this plane into proportional parts;
The section of this plane is a polygon similar to the base;
The areas of the section and the base are related to each other as the squares of their distances from the top.
Types of pyramid
Correct pyramid- a pyramid, the base of which is a regular polygon, and the top of the pyramid is projected into the center of the base.
At the correct pyramid:
1. side ribs are equal
2. side faces are equal
3. apothems are equal
4. dihedral angles at the base are equal
5. dihedral angles at side edges are equal
6. each height point is equidistant from all base vertices
7. each height point is equidistant from all side faces
Truncated pyramid- the part of the pyramid enclosed between its base and a cutting plane parallel to the base.
The base and corresponding section of a truncated pyramid are called bases of a truncated pyramid.
A perpendicular drawn from any point of one base to the plane of another is called the height of the truncated pyramid.
Tasks
No. 1. In the right quadrangular pyramid point O is the center of the base, SO=8 cm, BD=30 cm. Find the side edge SA.
Problem solving
No. 1. AT right pyramid all faces and edges are equal.
Let's consider OSB: OSB-rectangular rectangle, because.
SB 2 \u003d SO 2 + OB 2
SB2=64+225=289
Pyramid in architecture
Pyramid - a monumental structure in the form of an ordinary regular geometric pyramid, in which sides converge at one point. According to the functional purpose, the pyramids in ancient times were a place of burial or worship. The base of a pyramid can be triangular, quadrangular, or polygonal with an arbitrary number of vertices, but the most common version is the quadrangular base.
A considerable number of pyramids are known, built different cultures ancient world mostly as temples or monuments. The largest pyramids are the Egyptian pyramids.
All over the Earth you can see architectural structures in the form of pyramids. Pyramid buildings are reminiscent of ancient times and look very beautiful.
Egyptian pyramids greatest architectural monuments ancient egypt, among which one of the "Seven Wonders of the World" is the pyramid of Cheops. From the foot to the top, it reaches 137.3 m, and before it lost the top, its height was 146.7 m.
The building of the radio station in the capital of Slovakia, resembling an inverted pyramid, was built in 1983. In addition to offices and service premises, there is a fairly spacious concert hall inside the volume, which has one of the largest organs in Slovakia.
The Louvre, which "is as silent and majestic as a pyramid" has undergone many changes over the centuries before becoming the greatest museum in the world. It was born as a fortress, erected by Philip Augustus in 1190, which soon turned into a royal residence. In 1793 the palace became a museum. Collections are enriched through bequests or purchases.
