Pythagoras School studied spheres and regular polyhedrons, but before Plato School began to study them, people knew little about pyramids, prisms, cones and cylinders. Three-dimensional geometric ring
Eudoxus established their measuring method and proved that the volume of a cone is one third of that of a cylinder with equal bottom and equal height, and may be the first person to prove that the volume of a sphere is proportional to the cube of its radius.
Edit the basic theme of this paragraph.
Subject content
Including:
Various geometric solid figures (10 sheets)-coincidence of surfaces and lines-dihedral angle and solid angle-square, cuboid, parallelepiped-tetrahedron and other pyramids-prism-octahedron, dodecahedron, icosahedron-cone, cylinder-sphere-other quadric surfaces: ellipsoid of revolution, ellipsoid and paraboloid. The axiomatic solid geometry of hyperboloid includes If two points on a straight line are on a plane, then the straight line is on this plane. Axiom 2 passes through three points that are not on a straight line, and there is only one plane. Axiom 3 If two non-overlapping planes have a common point, then they have only one common line passing through that point. Axiom 4 Two straight lines parallel to the same straight line. List of surface areas and volumes of various three-dimensional figures Name symbol Area S Volume V
The side length of cube a is s = 6a 2 v = a 3.
Length of cuboid a
B width
C- height S=2(ab+ac+bc) V=abc.
Bottom-bottom area of prism
H- high S=S side +2S bottom V=Sh.
S-shaped bottom area of pyramid
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V=Sh/3
Prisms S 1 and S2- upper and lower bottom areas
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v = h[s 1+S2+√(s 1 S2)]/3
Prisma toid s 1- Upper bottom area
S2-bottom area
S0-middle cross-sectional area
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V=h(S 1+S2+4S0)/6
R- bottom radius of cylinder
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C-bottom circumference C=2πr
Bottom-bottom region
S-side-lateral area
S table-surface area s bottom = π r 2
S side =Ch
S table =Ch+2S bottom V=S bottom h = π r 2h.
Hollow cylinder R—— radius of external circle
R—— radius of inner circle
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V=πh(R^2-r^2)
R base radius of straight cone
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L-bus S = π r (r+L) V = π r 2h/3.
Cone r- upper bottom radius
R- bottom radius
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l-Bus S =π(R2+R2+RL+RL)V =πh(R2+RR+R2)/3
Sphere r radius
D- diameter s = 4πR2；; V=4/3πr^3=πd^3/6
Ball missing h- ball missing height
Sphere radius
A—— the radius of the bottom of the ball leakage a 2 = h (2r-h)
V=πh(3a^2+h^2)/6 =πh2(3r-h)/3
Tables r 1 and R2-the radius of the table top and table top.
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V=πh[3(r 12+r22)+h2]/6
Circle radius
D—— ring diameter
R—— the section radius of the ring.
D—— ring section diameter
V=2π^2Rr^2 =π^2Dd^2/4
Bucket D—— diameter of barrel belly
D—— diameter of barrel bottom
H- barrel height
V = π h (2d 2+D2)/ 12 (the bus is circular with the center of the barrel).
V = π h (2D 2+DD+3D 2/4)/ 15 (bus is parabolic)
Note: Beginners will find solid geometry difficult, but as long as you lay a good foundation, solid geometry will become easy. The key to learn solid geometry well is to establish a solid model, transform the solid into a plane, and use plane knowledge to solve problems. Solid geometry will definitely cause big problems in the college entrance examination, so it is very important to learn solid well.
Three vertical line theorems
If a straight line in a plane is perpendicular to the projection of a diagonal line passing through the plane in the plane, the straight line is also perpendicular to the diagonal line. Inverse theorem theorem of three perpendicular lines: If a straight line in a plane is perpendicular to a diagonal line passing through the plane, it is also perpendicular to the projection of the diagonal line in the plane. 1, the three vertical theorems describe the vertical relationship among PO (diagonal line), AO (projection) and a (straight line). 2. A and PO may intersect or not be in the plane. 3. The essence of the Three Perpendicular Theorem is the judgment theorem that a diagonal line in the plane is perpendicular to a straight line in the plane. The key to applying the three perpendicular theorem is to find the perpendicular of the plane (datum).
First, find that the plane (datum plane) is perpendicular to the plane. Second, find the projective line. At this time, A and B become a straight line and a diagonal line on the plane. Thirdly, it is proved that the projective straight line is perpendicular to the straight line A, so that A and B are perpendicular. Note: 1. The key to applying the theorem is to find the reference frame of "datum plane" and prove that the theorem of three perpendicular lines is known by vectors. PA is the vertical line of plane A, oblique line, o A is the projection of PA in A, B belongs to A, and B is perpendicular to OA. Prove that b is perpendicular to PA. Prove: Because PO is perpendicular to A, PO is perpendicular to B, and o A is perpendicular to B, vector PA times b= (vector PO+ vector OA) times b= (vector PO times B) plus (vector OA times B). 2) it is known that PO and PA are perpendicular to plane a, diagonal, OA is the projection of PA in a, b belongs to a, and b is perpendicular to PA. Proof: B is perpendicular to OA. It is proved that because PO is perpendicular to A, PO is perpendicular to B, and PA is perpendicular to B, vector OA= (vector PA- vector PO) times vector OA by b = = (vector PA- vector PO) times b =. 2. It is known that three planes OAB, OBC and OAC intersect at point O, and angle AOB= angle BOC= angle COA=60 degrees. Find the angle between the intersection line OA and the plane OBC. Vector OA= (vector OB+ vector AB), O is the center, because AB=BC=CA, the angle formed by OA on the plane OBC is 30 degrees.
Edit the dihedral corner of this paragraph.
definition
A straight line in a plane divides the plane into two parts, each part is called a half plane, and a figure composed of two half planes from a straight line is called a dihedral angle. This straight line is called the edge of dihedral angle, and each half plane is called the face of dihedral angle.
Plane angle of dihedral angle
Take any point on the edge of dihedral angle as the endpoint, and make two rays perpendicular to the edge in two planes. The angle formed by these two rays is called the plane angle of dihedral angle. A dihedral angle whose plane angle is a right angle is called a straight dihedral angle. Definition of two planes perpendicular: Two planes intersect, and if the dihedral angle they form is a straight dihedral angle, they are said to be perpendicular to each other.
The size range of dihedral angle
0 R+R ② The circumference of two circles d = r+r.
③ the intersection of two circles r-r < d < r+r (r > r).
④ inscribed circle D = R-R (R > R) ⑤ two circles contain D < R-R (R > R).
Theorem 136 The intersection of two circles bisects the common chord of two circles vertically.
Theorem 137 divides a circle into n (n ≥ 3);
(1) The polygon obtained by connecting the points in turn is the inscribed regular N polygon of this circle.
(2) The tangent of a circle passing through each point, and the polygon whose vertex is the intersection of adjacent tangents is the circumscribed regular N polygon of the circle.
Theorem 138 Any regular polygon has a circumscribed circle and an inscribed circle, which are concentric circles.
139 every inner angle of a regular n-polygon is equal to (n-2) ×180/n.
140 Theorem Radius and apothem Divides a regular N-polygon into 2n congruent right triangles.
14 1 the area of the regular n polygon Sn = PNRN/2 P represents the perimeter of the regular n polygon.
142 The area of a regular triangle √ 3a/4a indicates the side length.
143 if there are k positive n corners around a vertex, then the sum of these angles should be
360, so k× (n-2) 180/n = 360 is changed to (n-2)(k-2)=4.
The formula for calculating the arc length of 144 is L = NR/ 180.
145 sector area formula: s sector =n r 2/360 = LR/2.
146 inner common tangent length = d-(R-r) outer common tangent length = d-(R+r)