how do you calculate the surface area of a cone with a slant height of 13cm and a height of 12cm.?

To calculate the curved surface area of the cone, we need to know the base radius R.

The slant height (L) of a cone is 13 cm
The vertical height (H) is 12 cm
Use Pythagoras theorem to find the radius of the base, R
R² = L² - H²
R = sqrt(13²-12²)
=5

The curved surface area of a cone is the product πRL = π*5*13 = 65π

To this, we should add the surface area of the base, equal to πR².

To calculate the surface area of a cone with a given slant height and height, you can use the following formula:

Surface Area = πr(r + h),

where π is a constant approximately equal to 3.14159, r is the radius of the base of the cone, and h is the height of the cone.

To find the radius, we need to use the slant height. The slant height, height, and radius of a right circular cone form a right triangle, with the slant height as the hypotenuse, the height as one of the legs, and the radius as the other leg.

Using the Pythagorean Theorem, we can find the radius:

r = √(slant height^2 - height^2).

In this case, the slant height is given as 13 cm, and the height is given as 12 cm. Plugging these values into the formula, we get:

r = √(13^2 - 12^2) = √(169 - 144) = √25 = 5 cm.

Now that we have the radius, we can calculate the surface area using the formula:

Surface Area = πr(r + h) = π(5)(5 + 12) = π(5)(17) = 85π cm^2.

Therefore, the surface area of the cone is 85π square centimeters.