Find in terms of π the curved surface area of a cone with circular base diameter 10 cm and height 12cm
The curved surface area of a cone can be found using the formula:
CSA = πrℓ
where CSA is the curved surface area, r is the radius of the base, and ℓ is the slant height of the cone.
Given that the diameter of the base is 10 cm, the radius (r) can be found by dividing the diameter by 2:
r = 10 cm / 2 = 5 cm.
To find the slant height (ℓ), we can use the Pythagorean theorem. The slant height, height, and radius form a right triangle, with the height as one of the legs and the slant height as the hypotenuse. The radius is the other leg.
Using the Pythagorean theorem:
ℓ^2 = h^2 + r^2
ℓ^2 = 12 cm^2 + (5 cm)^2
ℓ^2 = 144 cm^2 + 25 cm^2
ℓ^2 = 169 cm^2
Taking the square root of both sides to solve for ℓ:
ℓ = √169 cm
ℓ = 13 cm.
Now, we can substitute the values for r and ℓ into the formula for CSA:
CSA = π(5 cm)(13 cm)
CSA = 65π cm^2.
Therefore, the curved surface area of the cone is 65π cm^2.
To find the curved surface area of a cone, we need to calculate the lateral or curved surface area. The formula to calculate the lateral surface area of a cone is given as:
Lateral Surface Area = π * r * l
where r is the radius of the circular base, and l is the slant height of the cone.
Given that the diameter of the circular base is 10 cm, we can find the radius by halving the diameter:
radius (r) = diameter / 2 = 10 cm / 2 = 5 cm
Now, to find the slant height (l), you can use the Pythagorean theorem, where the height (h) is the perpendicular height and the base radius (r) is the base of the right triangle:
l = √(r^2 + h^2)
Given that the height (h) is 12 cm and the radius (r) is 5 cm, we can calculate the slant height (l):
l = √(5^2 + 12^2) = √(25 + 144) = √169 = 13 cm
Now, substituting the values of the radius (r = 5 cm) and slant height (l = 13 cm) into the formula of the lateral surface area:
Lateral Surface Area = π * r * l = π * 5 cm * 13 cm = 65π cm^2
Therefore, the curved surface area of the cone is 65π cm².