crossection
2D plane(shape) with intersection of 3D shape(figure) - crosssection of solid
Horizontal crossection
Vertical crossection
*of sphere
**also exist different/other types of crossections(with random angles, complex planes)
polyhedrons
polygon - shape built by line segments
Prism
Based on an n-angle polygon, with two connected parallel bases
*lines of connection are shaping parallelograms
Formulas
S(all) = 2*S(base)+ S(side)
S(side) - n*shape
V = S(of the base or horizontal crossection)*l
*l - side edge
*S - surface area
*V - volume
Straight & regular prism
Diagonal in straight: d = a^2*b^2*l^2, l = h
*straigh with straigh angles to base
Cube
*all edges are equal
Pyramid
Straight and regular pyramid
*in straight - top vertex above the centroid of the base
Formulas
S(all) = S(sides)+S(base)
S(side) - S of triangle
V = ⅓*S(base)*h
*h - height
**for proof V:
Build cube with diagonals - will see 6 equal pyramids
Volume of cube = a*b*c; S(base) = a*b; c = H
In one pyramid h = H/2; H = 2*h; = > V = S*H/6; V = S*h*2/6 = S*h/3
Truncated pyramid
V = ⅓(S1+S2+sqrt(S1*S2))*h
*proof through coefficient of the big part and small part(they are common based)
x(volume) = V - v(1)
V = ⅓*S*H
v = ⅓*s*h
S/H = s/h
Sh=Hs
h = Hs/S - something like this, and to (1) with H-h
curvilinear based
Cylinder
Prism wit a circle base
S(all)=2*S(base)+S(side)
S(side) = 2*pi*r*l
S(base)=pi*r^2
*horizontal crossection - circle
*vertical crossection - parallelogram
**ca be oblique(regular)
Cone
S(all)=S(side)+S(base)
S(side) = pi*l*r
S(base)=S(circle)
V=⅓*pi*r^2*h
*when roll-up the side it is not a circle but a sector and possible to find area through formules for the sector
Truncated cone
S(side) = (R + r)sqrt((R-r)^2+h^2)*pi
V=⅓*pi*h(R^2+r^2+R*r)
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