A cone has a circular base, a perpendicular height equal to 21cm, and a semi-vertical angle of 30degrees. Calculate the slant height of the cone. Find the area of its base.(take pie to be 22/7)

Pi r^2

solve it complete

To calculate the slant height of the cone, we can use the formula:

Slant height = √(radius² + height²)

Given information:
- Height (h) = 21 cm
- Semi-vertical angle (θ) = 30 degrees

To find the radius, we can use trigonometric functions. We know that the angle formed by the radius and the slant height is 90 degrees (since it is a right triangle). The sine function relates the opposite side (the radius) and the hypotenuse (the slant height).

sin(θ) = opposite/hypotenuse
sin(30°) = radius/slant height

Using the given semi-vertical angle (θ = 30 degrees) and rearranging the formula, we can find the radius:

radius = sin(30°) * slant height

Using the formula for the slant height, we can substitute the known values:

slant height = √(radius² + height²)

Now, let's calculate the slant height and the radius step by step:

1. Find the radius:
radius = sin(30°) * slant height

2. Calculate the slant height using the formula above:
slant height = √(radius² + height²)

Next, let's find the area of the base of the cone. Since the base of the cone is circular, the area can be found using the formula:

Area = π * radius²

Substituting the known values, we can calculate the area of the base.

Please note that "π" in the formula represents the mathematical constant, and for this calculation, we will use an approximate value of 22/7.

To recap:
1. Calculate the slant height using the formula slant height = √(radius² + height²).
2. Calculate the radius using the formula radius = sin(30°) * slant height.
3. Calculate the area of the base using the formula Area = π * radius², where π = 22/7.

if the slant height is s,

21/s = cos 30°

similarly, the radius r of the base is
r/21 = tan 30°

and, naturally, the area of the base
a = pi r^2