\ Určete\ objem\ a\ povrch\ komolého\ kužele, 
\jehož\ podstavy\ jsou\ kruh\ opsaný\ a\ kruh\ vepsaný 
\protilehlým\ stěnám\ krychle\ s\ hranou\ délky\ a.

Zadání

(\frac{a}{2})^{2}+(\frac{a}{2})^{2}=r_1
\r_1=\frac{\sqrt{2}}{2}a

r_2=\frac{a}{2}

Objem\ komolého\ jehlanu\ získáme\ z\ vzorce
\S=πr_1^2+πr_2^2+πs(r_1+r_2)
\stranu\ s\ získáme\ z\ troúhelníku\ APQ
\\left| AP \right|=r_1-r_2\ \left|PQ  \right|=a
\s^2=a^2+(r_1-r_2)^2\ 
\s=\sqrt{a^2+(\frac{\sqrt{2}a}{2}-\frac{a}{2})^2}  
\s=a\frac{\sqrt{7-2\sqrt{2}}}{2}
\po\ dosazení\ do\ vzorce
\S=π(\frac{\sqrt{2}a}{2})^2+π(\frac{a}{2})^2+πa\frac{\sqrt{7-2\sqrt{2}}}{2}(\frac{\sqrt{2}a}{2}+\frac{a}{2})
\S=\frac{1}{4}πa^2\left [3+(\sqrt{2}+1)(\frac{\sqrt{7-2\sqrt{2}}}{2})  \right ] 

a_1 = 5 (slider from 0 to 20, step 1)
A = (2, -2, 0)
B = Point(Circle[A, a_1, xOyPlane])
f = Segment(A, B)
C = Point(Circle[B, Distance[A, B], Segment[A, B]])
b, D, E, F, G, H, stenaABCD, stenaADHE, stenaABFE, stenaBCGF, stenaCDHG, stenaEFGH, hranaAD, hranaAB, hranaBC, hranaCD, hranaAE, hranaDH, hranaEH, hranaEF, hranaBF, hranaFG, hranaCG, hranaGH = Cube(A, B, C)
I = Midpoint(hranaEF)
J = Midpoint(hranaGH)
S_2 = Midpoint(I, J)
L = Midpoint(f)
M = Midpoint(hranaCD)
S_1 = Midpoint(M, L)
g = Segment(S_1, A)
r = Segment(J, S_2)
i = Line(S_1, S_2)
c = Circle(S_1, A, xOyPlane)
j = Line(S_1, M)
K, N = Intersect(j, c)
k = Line(K, J)
O = Intersect(i, k)
l = Segment(S_1, O)
m = Segment(S_2, O)
d, e, n = Cone(c, l)
p = Circle(i, J)
o, q, r_4 = Cone(p, m)
vysledek: d - o
s_1 = true
P_1 = Intersect(hranaEF, hranaEH)
t = Plane(F, G, E)
c_1 = IntersectPath(t, o)
r_1 = Segment(S_1, A)
h = Plane(A, S_1, E)
f_1 = Line(E, S_2)
Q = Intersect(f_1, c_1, 2)
R = Point(f_1)
r_2 = Segment(R, Q)
S = Point(r_1)
T = Point(r_1)
s = Segment(Q, T)
Q' = Mirror(Q, l)
U = Point(r_1)
g_1 = Segment(U, C)
V = Point(r_2)
h_1 = Segment(V, Q')
i_1 = Segment(C, Q')
W = Point(r_1)
text3: "Objem\ komolého\ kužele\ je\  " + (LaTeX[vysledek]) + ""
α = Angle(S_1, M, C, xOyPlane)
text5: "v=a
\\Objem\ vyniklého\ komolého\ kuželu\ určíme\ z\ vzorce
\\ V=\frac{1}{3}πv(r_1^2+r_1r_2+r_2^2)=\frac{1}{3}πa(\frac{a^2}{2}+\frac{a^2\sqrt{2}}{4}+\frac{a^2}{4})
\\V=\frac{πa^3(3+\sqrt{2})}{12}
\\" + (LaTeX[text3]) + ""
j_1 = OrthogonalLine(Q, r_1, h)
P = Intersect(j_1, r_1)
a = Segment(Q, P)
k_1: x = (((1 / 4 * π) * a_1^(2)) * (3 + ((2^(1 / 2) + 1) * (7 - (2 * 2^(1 / 2)))^(1 / 2) / 2)))
u: (((1 / 4 * π) * a_1^(2)) * (3 + ((2^(1 / 2) + 1) * (7 - (2 * 2^(1 / 2)))^(1 / 2) / 2)))
text7: "Obsah\ komolého\ kuželu\ je\ " + (LaTeX[u]) + " "