MindMap Gallery solid geometry
In mathematics, solid geometry is the traditional name for the geometry of 3-dimensional Euclidean space - because in fact this is roughly the space we live in. Generally taken as a follow-up course in plane geometry. Stereometry deals with the measurement of volumes of different shapes: cylinders, cones, frustum, spheres, prisms, wedges, bottle caps, etc. The Pythagoreans had dealt with spheres and regular polyhedra, but pyramids, prisms, cones, and cylinders were little known before the Platonists. Eudoxus established their measurement method, proving that a cone is one-third the volume of a cylinder with equal bases and equal heights, and was probably the first to prove that the volume of a sphere is proportional to the cube of its radius.
Edited at 2023-03-19 10:43:29El cáncer de pulmón es un tumor maligno que se origina en la mucosa bronquial o las glándulas de los pulmones. Es uno de los tumores malignos con mayor morbilidad y mortalidad y mayor amenaza para la salud y la vida humana.
La diabetes es una enfermedad crónica con hiperglucemia como signo principal. Es causada principalmente por una disminución en la secreción de insulina causada por una disfunción de las células de los islotes pancreáticos, o porque el cuerpo es insensible a la acción de la insulina (es decir, resistencia a la insulina), o ambas cosas. la glucosa en la sangre es ineficaz para ser utilizada y almacenada.
El sistema digestivo es uno de los nueve sistemas principales del cuerpo humano y es el principal responsable de la ingesta, digestión, absorción y excreción de los alimentos. Consta de dos partes principales: el tracto digestivo y las glándulas digestivas.
El cáncer de pulmón es un tumor maligno que se origina en la mucosa bronquial o las glándulas de los pulmones. Es uno de los tumores malignos con mayor morbilidad y mortalidad y mayor amenaza para la salud y la vida humana.
La diabetes es una enfermedad crónica con hiperglucemia como signo principal. Es causada principalmente por una disminución en la secreción de insulina causada por una disfunción de las células de los islotes pancreáticos, o porque el cuerpo es insensible a la acción de la insulina (es decir, resistencia a la insulina), o ambas cosas. la glucosa en la sangre es ineficaz para ser utilizada y almacenada.
El sistema digestivo es uno de los nueve sistemas principales del cuerpo humano y es el principal responsable de la ingesta, digestión, absorción y excreción de los alimentos. Consta de dos partes principales: el tracto digestivo y las glándulas digestivas.
solid geometry
basic formula
cylinder
cone
Taiwan body
sphere
sector area
The formula for the side area of a circular cone
basic concept
prism
oblique prism
right prism
regular prism
pyramid
Right pyramid
regular tetrahedron
prism
fundamental issue
distance
distance between two points
point-line distance
line distance
distance between parallel lines
point-to-surface distance
Line-area distance
Face to face distance
angle
line angle
Directly establish a coordinate system to find the angle between the two straight line direction vectors (note: the angle between the two straight lines should be less than 90°)
line angle
coordinate method
geometric method
face angle
dihedral angle
geometric method
Select a point on one of the half-planes (point A) and draw a perpendicular to the other half-plane to get a vertical foot (point B). Then draw a perpendicular to the intersection of the two half-planes through the vertical foot to get another vertical foot. (Point C), connect AC, and the obtained ∠ACB is the plane angle of the dihedral angle
coordinate method
How to find area and volume
method
area
Equal base conversion
Contour conversion
volume
Transform vertices
Vertex translation (contour transformation)
Equal base conversion
formula
(triangle) area
x, y represent the coordinates of two vectors with a common starting point (applicable to finding the area in two-dimensional space)
x, y, z represent the vector coordinates of two common starting points (suitable for area calculation in three-dimensional space)
volume
cross-section problem
Determine the cross section
translation method
extension method
coordinate method
The calculation amount is very large, but it can ensure that every intersection point is found
Use the coplanarity theorem to express the point coordinates on a certain plane, then find the intersection coordinates of the plane and the edge of the geometry (usually a cuboid), and finally connect the points to obtain the cross section (you can also use the collinearity theorem to find a certain The intersection point of the edge and the plane)
Find the perimeter, area, and volume of the section
Outside ball and inside ball
Solve the problem by using the equal distance from the points on the ball to the center of the ball (center perpendicular)
External ball (the distance from the center of the ball to each vertex is equal)
Use the theorem that the distance from a point on the mid-perpendicular to both ends of the line segment is equal to extend it Draw a straight line m perpendicular to the base through the circumcenter of the base ABC (i.e., the center of the circumscribed circle). Then the points on the straight line m are at the same distance from each vertex on the bottom surface. If you require the radius of the circumscribed sphere of the triangular pyramid P-ABC, you only need to add As the perpendicular line of PA (or PB, PC) (actually a plane composed of perpendicular lines, because there are countless perpendicular lines for a line segment in space, there are also countless perpendicular lines), and the perpendicular line is the same as The intersection point of the straight line m is the center of the circumscribed sphere of the triangular pyramid P-ABC.
The principle of finding the center position of the outside ball
The distance from the center of the outer ball to each vertex is equal
The distance from a point on the mid-perpendicular to both ends of the line segment is equal
Inscribed ball (the distance from the center of the ball to all faces is equal)
Since the distances from the center of the inscribed sphere of the triangular pyramid P-ABC to each surface of the triangular pyramid are equal (all equal to the inner radius r), we can use the equal volume method to find r, that is, r=3V/S (V represents the triangular pyramid The volume of , S represents the sum of the areas of all faces of the triangular pyramid),
A sphere that is tangent to all edges of a triangular pyramid (the distance from the center of the sphere to each edge is equal)
The principle of finding the center of the ball
The distances from the points on the angle bisector to both sides of the angle are equal → draw a straight line m perpendicular to the base ABC through the center of the base triangle ABC, then the points on the straight line m are equidistant from all sides of the base triangle ABC
theorem
Lines and planes are parallel
property theorem
decision theorem
Face to face parallel
property theorem
decision theorem
vertical line
property theorem
decision theorem
Face to face vertical
property theorem
decision theorem
Positional relationship judgment
Straight lines and straight lines
Observe whether the plane of one of the straight lines a and the other straight line b has a unique intersection point. If the intersection point is not on a, then the two straight lines a and b are in different planes.
three points collinear
collinearity theorem
four points coplanar
coplanarity theorem
optimal value problem
Maximum value problems of area, volume and perimeter
Solve by constructing the functional relationship between area (or volume, perimeter) and the unique variable, using monotonicity, basic inequality or derivative methods
The problem of the maximum value of the perimeter can also be solved by side expansion and using the shortest distance between two points.