Please help me to solve this equation for d: rl=sqrt(rc^2-(rc-d)^2)?

Question

This is a formula for calculating the radius of a sphere at a particular level. rl = radius at level, rc = radius of curvature, d = distance to level from center of sphere. see http://www.monolithic.com/construction/formulas.pdf

Answer 1

Please help me to solve this equation for d: rl=sqrt(rc^2-(rc-d)^2)?

This is a formula for calculating the radius of a sphere at a particular level. rl = radius at level, rc = radius of curvature, d = distance to level from center of sphere.



rl = √(rc² -(rc-d)²)



So rl² = rc² -(rc-d)²



Thus (rc-d)² = rc² - rl²



ie rc-d = √(rc² - rl²)



So rc = √(rc² - rl²) + d



Therfore d = rc - √(rc² - rl²)

Answer 2

Step by step inverses:

rl^2=rc^2-(rc-d)^2

rl^2-rc^2=-(rc-d)^2

rc^2-rl^2=(rc-d)^2

sqrt(rc^2-rl^2)=rc-d....need to think about + or - here

rc-sqrt(rc^2-rl^2)=-d

sqrt(rc^2-rl^2)-rc=d

Answer 3

d