A cone has a circular base, a perpendicular height of 21cm, and a semi-vertical angle of 30degree. calculate the slant height of the cone. find the area of it's base
so the radius is r = 21 cot30° = 21/√3
that makes the
slant height s = 21 sec30° = 42/√3
area = πr^2 + πrs
To find the slant height of the cone, we can use the formula:
Slant Height = Height / sin(Semi-vertical Angle)
First, let's find the sine of the semi-vertical angle of 30 degrees:
sin(30 degrees) = 0.5
Now, we can substitute the values into the formula:
Slant Height = 21 cm / 0.5 = 42 cm
So, the slant height of the cone is 42 cm.
To find the area of the base of the cone, we use the formula for the area of a circle:
Area of the base = π * r^2
But to calculate the radius, we need either the diameter or the circumference of the circle. Since the diameter is not given, we will calculate it using the slant height.
Using the formula for the slant height of a cone, we can apply the Pythagorean theorem:
Slant Height^2 = Radius^2 + Height^2
Substituting the known values:
42^2 = Radius^2 + 21^2
1764 = Radius^2 + 441
Rearranging the equation:
Radius^2 = 1764 - 441
Radius^2 = 1323
Now, we can find the radius:
Radius = √1323 ≈ 36.38 cm
Finally, substituting the radius into the formula for the area of the base:
Area of the base = π * (36.38 cm)^2
Area of the base ≈ 4153.7 cm^2
Therefore, the area of the base of the cone is approximately 4153.7 cm^2.