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 [take pie to be 22/7]
the slant height s is found using the Pythagorean Theorem:
s^2 = h^2 +r^2
If the angle from the base to the top is θ, then of course the base radius is
r = h tanθ
the area of the base is πr^2
and that's pi, not pie!
Slant height is 24cm and the area of it base is 38cm^2
Am here to find the vertical height of a cone
To calculate the slant height of the cone, we can use the formula:
Slant height = Height ÷ sin(Semi-vertical angle)
Given:
Height (h) = 21 cm
Semi-vertical angle (θ) = 30 degrees
Before we proceed further, we need to convert the angle from degrees to radians, as the trigonometric functions in most programming languages work with radians.
Since 180 degrees = π radians,
30 degrees = (30/180) * π radians = (1/6) * π radians
So, we have:
θ (in radians) = (1/6) * 3.1416 ≈ 0.5236 radians
Now, we can substitute the values into the formula:
Slant height = 21 cm ÷ sin(0.5236 radians)
We can use the trigonometric function sine (sin) to calculate the slant height. However, most programming languages and calculators take angles in radians by default. Therefore, we need to convert the angle from degrees to radians to use it in the formula.
Using the approximate value of π = 3.1416, we can calculate the slant height:
Slant height = 21 cm ÷ sin(0.5236)
= 21 cm ÷ sin(30 degrees)
≈ 21 cm ÷ 0.5
= 42 cm
So, the slant height of the cone is approximately 42 cm.
To calculate the area of the cone's base, we need to use the formula:
Area of base = π * radius^2
We are given the semi-vertical angle, which is the angle formed between the height and the slant height, and we can calculate the radius using trigonometry:
radius = slant height * sin(semi-vertical angle)
Using the approximate value of π = 22/7, we can calculate the area of the base:
radius = slant height * sin(30 degrees)
= 42 cm * 0.5
= 21 cm
Area of base = (22/7) * (21 cm)^2
= (22/7) * (21 cm * 21 cm)
= (22/7) * (441 cm^2)
≈ 1,386 cm^2
So, the area of the base of the cone is approximately 1,386 square centimeters.