Acone without a base is made from a half-circle of radius 10 cm. determine the volume of the cone. explain your reasoning.

The volume of the cone is approximately 226.7 cm³

The reasoning for obtaining the volume of the cone is as follows;

The known parameters are;

The shape from which the cone is made = A half circle

The radius of the half circle from which the cone is made, R = 10 cm

The required parameter;

The volume of the cone

Method;

We note that the circumference of circular base of the cone = Half the circumference of a circle

The slant height of the cone, l = The radius of the circle, R

The volume of a cone, V = (1/3)·π·r²·h

Where;

r = The radius of the base of the cone

h = The height of the cone = √(R² - r²)

Therefore, we find r as follows;

The circumference of the cone base = 2·π·r

Half the circumference of the circle from which the one is made = π·R

Therefore;

2·π·r = π·R

r = R/2

∴ r = 10/2 = 5

r = 5 cm

h = √(R² - r²)

∴ h = √(10² - 5²) = 5·√3

The height of the cone, h = 5·√3

∴ V = (1/3) × π × 5² × (5·√3) = (125/3)·π·√3 ≈ 226.7

The volume of the cone, V ≈ 226.7 cm³

Learn more about volumes of cones here;

link

  125π√3/3 cm³ ≈ 226.72 cm³

Step-by-step explanation:

The length of the circular edge of the half-circle is ...

  (1/2)C = (1/2)(2πr) = πr = 10π . . . . cm

This is the circumference of the circular edge of the cone, so the radius of the cone is found from ...

  C = 2πr

  10π = 2πr . . . . fill in the numbers; next, solve for r

  r = 5 . . . . cm

The slant height of the cone is the original radius, 10 cm, so the height of the cone from base to apex is found from the Pythagorean theorem.

  (10 cm)² = h² + r²

  h = √((10 cm)² -(5 cm)²) = 5√3 cm

And the cone's volume is ...

  V = 1/3·πr²h = (1/3)π(5 cm)²(5√3 cm)

  V = 125π√3/3 cm³ ≈ 226.72 cm³