Question

a cone has a radius R equal to its height what is the volume of the frustum if the smaller cones radius is r​

Answer 1

Given : A cone has a radius R equal to its height. the radius of smaller cone is r.

To find : the volume of frustum.

solution : as radius of cone is equal to its height.

so, R = H

so, slant height , L = √(R² + H²) = √(R² + R²) = √2R

now, cutting a smaller cone from the top to form frustum.

see diagram,

here, ∆ABC ~ ∆ADE

⇒AB/AD = BC/DE

⇒H/h = R/r

⇒h = H(r/R)

but R = H so, h = r

now slant height of smaller cone, l = √(r² + h²) = √(r² + r²) = √2r

now volume of frustum = volume of bigger cone - volume of smaller cone

= πR²L/3 - πr²l/3

= πR²√2R/3 - πr²√2r/3

= √2π/3 [R³ - r³]

= √2π/3 (R - r)(R² + r² + rR)

Therefore the volume of frustum is √2π/3 (R - r)(R² + r² + rR)

Answer 2

Answer:

Given : A cone has a radius R equal to its height. the radius of smaller cone is r.

To find : the volume of frustum.

solution : as radius of cone is equal to its height.

so, R = H

so, slant height , L = √(R² + H²) = √(R² + R²) = √2R

now, cutting a smaller cone from the top to form frustum.

see diagram,

here, ∆ABC ~ ∆ADE

⇒AB/AD = BC/DE

⇒H/h = R/r

⇒h = H(r/R)

but R = H so, h = r

now slant height of smaller cone, l = √(r² + h²) = √(r² + r²) = √2r

now volume of frustum = volume of bigger cone - volume of smaller cone

= πR²L/3 - πr²l/3

= πR²√2R/3 - πr²√2r/3

= √2π/3 [R³ - r³]

= √2π/3 (R - r)(R² + r² + rR)

Therefore the volume of frustum is √2π/3 (R - r)(R² + r² + rR)

Step-by-step explanation: