A right circular cone has height 90 centimeters and slant height 106 centimeters. Find the circumference and area of the base in terms of 𝜋.
The base of a cone is a circle. This means we can calculate its area using the formula 𝜋 multiplied by the radius squared and its circumference by multiplying two by 𝜋 by the radius. In this question, we’re told that the slant height is 106 centimeters. The vertical height is 90 centimeters. At present, we don’t know the radius.
As these three sides form a right-angled triangle, we can use Pythagoras’s theorem to work out the radius. Pythagoras’s theorem states 𝑎 squared plus 𝑏 squared equals 𝑐 squared, where 𝑐 is the hypotenuse or longest side, in this case 106 centimeters.
Substituting in the values gives us an equation 𝑟 squared plus 90 squared is equal to 106 squared. 90 squared is equal to 8100. And 106 squared is equal to 11236. Subtracting 8100 from both sides of the equation gives us 𝑟 squared is equal to 3136. Square-rooting both sides of this new equation gives us a value for 𝑟, the radius of the base, of 56. This means that the radius is equal to 56 centimeters.
We can now use this value of 𝑟 to calculate the area of the base and the circumference of the base. The area is equal to 𝜋 multiplied by 56 squared. As 56 squared is equal to 3136, our area is 3136𝜋. The circumference can be calculated by multiplying two by 𝜋 by 56. Two times 56 is 112𝜋. So the circumference of the base is 112𝜋.
Therefore, a right circular cone with height 90 centimeters and slant height 106 centimeters has a circumference of 112𝜋 centimeters and a base area of 3136𝜋 centimeters squared.