A cone has a slant height of 15cm. The true height is 12 and the radius "r" is unknown. True height 12 and the radius intersect at a perpendicular angle inside the cone. What is the surface area of the entire cone?
if r^2 = 81, r=9
To find the surface area of the entire cone, we need to calculate the curved surface area and then add it to the area of the base.
First, let's find the slant height of the cone, which is given as 15 cm.
The slant height, true height, and radius form a right triangle inside the cone. Using the Pythagorean theorem, we can find the radius.
The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In this case, the slant height is the hypotenuse, the radius is one side, and the true height is the other side. Let's use the Pythagorean theorem to solve for the radius "r".
Applying the Pythagorean theorem:
r^2 + 12^2 = 15^2
Simplifying the equation:
r^2 + 144 = 225
Subtracting 144 from both sides:
r^2 = 81
Taking the square root of both sides:
r = √81 = 9
Now that we have the value of the radius, we can calculate the curved surface area and the base area.
The curved surface area (CSA) of the cone can be calculated using the formula:
CSA = π * r * l,
where r is the radius and l is the slant height.
Substituting the values:
CSA = π * 9 * 15 = 135π cm^2
The base area (BA) of the cone can be calculated using the formula:
BA = π * r^2,
where r is the radius.
Substituting the value:
BA = π * 9^2 = 81π cm^2
Finally, we can find the surface area of the entire cone by adding the curved surface area to the base area:
Surface Area = CSA + BA
= 135π + 81π
= 216π cm^2
Therefore, the surface area of the entire cone is 216π cm^2.
r^2 + 12^2 = 15^2
r^2 = 225 - 144 = 81
r = 4
area of bottom = pi (16)
area of sides = (1/2)(2 pi r) (L) = pi * 4 * 15
total area = pi (16 + 60)