V=S(A1A2A3)*SO/3
S(A1A2A3)=A1B*A2A3/2
A1B^2=A1A3^2-(A3A2/2)^2=10^2-6^2=64
A1B=8
S(A1A2A3)=12*8/2=48
SO=OB*tg60
ОВ=r-радиус вписанной окружности в ΔA1A2A3
S(A1A2A3)=p*r; p=(A1A2+A2A3+A3A1)/2=(10+10+12)/2=16
48=16*r; r=48/16=3
SO=r*tg<B=3√3
V=(48*3√3)/3=48√3
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Verified answer
V=S(A1A2A3)*SO/3
S(A1A2A3)=A1B*A2A3/2
A1B^2=A1A3^2-(A3A2/2)^2=10^2-6^2=64
A1B=8
S(A1A2A3)=12*8/2=48
SO=OB*tg60
ОВ=r-радиус вписанной окружности в ΔA1A2A3
S(A1A2A3)=p*r; p=(A1A2+A2A3+A3A1)/2=(10+10+12)/2=16
48=16*r; r=48/16=3
SO=r*tg<B=3√3
V=(48*3√3)/3=48√3