Question

圆台上下底面半径r,R,侧面积等于两底面积之和,圆台母线长＝

Answer 1

设圆台母线长为X,展开的扇形大圆弧半径为R1,小圆弧半径为R2.

则：X=R1-R2

PI*R^2+PI*r^2=2*PI*R*R1/2-2*PI*r*R2/2

=PI*R*R1-PI*r*R2

R^2+r^2=R*R1-r*R2

R2=(R*R1-R^2-r^2)/r

2*PI*R/R1=2*PI*r/R2

R/R1=r/R2

R1/R=R2/r=(R*R1-R^2-r^2)/r^2

R*R1/r^2-R1/R=(R^2+r^2)/r^2

R1*(R^2-r^2)/(R*r^2)=(R^2+r^2)/r^2

R1=R*(R^2+r^2)/(R^2-r^2)

R2=R*R1/r-(R^2+r^2)/r

=R*(R*(R^2+r^2)/(R^2-r^2))/r-(R^2+r^2)/r

=((R^2+r^2)/r)*(R^2/(R^2-r^2)-1)

=((R^2+r^2)/r)*(r^2/(R^2-r^2))

=r*(R^2+r^2)/(R^2-r^2)

X=R1-R2

=R*(R^2+r^2)/(R^2-r^2)-r*(R^2+r^2)/(R^2-r^2)

=((R^2+r^2)/(R^2-r^2))*(R-r)

=(R^2+r^2)/(R+r)

圆台母线长为(R^2+r^2)/(R+r).