Surface Area of a Frustrum:

To find the surface area of a frustum, we need to understand what a frustum is and how it is different from a regular cone.

A frustum is a geometric shape that is formed when the top of a cone is cut off parallel to the base. It has two circular bases, a smaller top base, and a larger bottom base, with a curved lateral surface connecting them.

To calculate the surface area of a frustum, we need to find the area of the two bases and the lateral surface.

Step 1: Find the Area of the Top and Bottom Bases:
The area of a circle is given by the formula A = πr^2, where A is the area and r is the radius.

Let's assume the radius of the top base is r1 and the radius of the bottom base is r2.

- The area of the top base (A1) = πr1^2
- The area of the bottom base (A2) = πr2^2

Step 2: Find the Lateral Surface Area:
The lateral surface area is the curved surface connecting the two bases.

The lateral surface area of a frustum can be found using the formula A = π(R + r)l, where R is the radius of the bottom base, r is the radius of the top base, and l is the slant height of the frustum.

Step 3: Calculate the Surface Area:
The total surface area (A) of the frustum can be calculated by summing up the areas of the two bases and the lateral surface.

A = A1 + A2 + lateral surface area

Example:
Let's say we have a frustum with a top base radius of 4 cm (r1), a bottom base radius of 6 cm (r2), and a slant height of 10 cm (l).

Using the formulas mentioned above, we can calculate the surface area as follows:

- Area of the top base (A1) = π(4^2) = 16π cm^2
- Area of the bottom base (A2) = π(6^2) = 36π cm^2
- Lateral surface area = π(6 + 4)(10) = 100π cm^2

Total surface area (A) = 16π + 36π + 100π = 152π cm^2

Therefore, the surface area of the given frustum is 152π cm^2.

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