How do you calculate the lateral and surface areas of a cone whose height is 12 in and whose slant height is 15 in.
Draw a cross section and you will have a right-angled triangle.
find the missing side which will be the radius of the base of the cone , r should be 9 in
You should have formulas for the surface area of a cone.
Could you show me the work for this one.
Thank you
r^2 + 12^2 = 15^2
r^2 = 225-144 = 81
r = √81 = 9
The surface area is made up of two parts,
the base, which is a circle, πr^2 and
the lateral surface (without the base) which has an area of πrs, where s is the slant height of 15 in
So total area is
πr^2 + πrs
= π(9^2) + π(9)(15)
= .....
To calculate the lateral and surface areas of a cone, you'll need the height and the slant height. The lateral area refers to the curved surface area of the cone, while the surface area includes the lateral area as well as the base.
1. To calculate the lateral area (A_lateral), you'll need to find the slant height (l) and the circumference of the base (C). The formula for the lateral area is A_lateral = π × l × r, where r is the radius of the base.
2. To find the radius (r) of the base, you'll need to use the Pythagorean theorem. The height (h), the slant height (l), and the radius (r) form a right triangle. The Pythagorean theorem states that r^2 = l^2 - h^2.
3. Once you have the radius (r), you can calculate the circumference of the base (C) using the formula C = 2πr.
4. Now that you have the slant height (l) and the circumference of the base (C), you can calculate the lateral area (A_lateral) using the formula A_lateral = π × l × r.
5. To calculate the surface area (A_surface), add the area of the base (A_base) to the lateral area. The formula for the surface area is A_surface = A_lateral + A_base.
Let's calculate the lateral and surface areas using the given values:
Given:
Height (h) = 12 in
Slant height (l) = 15 in
Step 1: Calculate the radius (r)
Using the Pythagorean theorem:
r^2 = l^2 - h^2 = 15^2 - 12^2 = 225 - 144 = 81
Taking the square root of both sides: r = √81 = 9
Step 2: Calculate the circumference of the base (C)
Using the formula C = 2πr:
C = 2π × 9 = 18π
Step 3: Calculate the lateral area (A_lateral)
Using the formula A_lateral = π × l × r:
A_lateral = π × 15 × 9 = 135π
Step 4: Calculate the surface area (A_surface)
To calculate the surface area, we need to find the area of the base (A_base). The area of a circle is given by the formula A_base = πr^2. Since the radius is 9, the area of the base is A_base = π × 9^2 = 81π.
Now, we can calculate the surface area using the formula A_surface = A_lateral + A_base:
A_surface = 135π + 81π = 216π
Therefore, the lateral area of the cone is 135π square inches and the surface area of the cone is 216π square inches.