**Math Labs with Activity – Total Surface Area of a Right-Circular Cylinder Formula**

**OBJECTIVE**

To demonstrate a method to derive a formula for finding the total surface area of a right-circular cylinder

**Materials Required**

- A right-circular cylinder made of chart paper whose total surface area is to be determined
- A geometry box
- A pair of scissors

**Theory**

When a right-circular cylinder of height h and base radius r is unfolded, we get two circular sheets each of radius r (the top and bottom faces) and one rectangular sheet of length 2πr and breadth h (the curved surface).

**Procedure**

**Step 1:** Take a right-circular cylinder of height h and base radius r as shown in Figure 36.1.

**Step 2:** Remove its two circular faces on the top and bottom. Cut the curved surface vertically and open it to get a rectangular sheet (see Figure 36.2).

**Step 3:** Measure the length and breadth of the rectangle formed.

**Observations and Calculations**

We observe that the length of the rectangular sheet = 2πr and the breadth of the rectangular sheet = h.

the area of the rectangular sheet = 2πrh.

So, the curved surface area of the cylinder = 2πrh.

Also, the area of each circular sheet = πr².

the total area of the top and bottom surfaces of the cylinder = 2 x πr²=2πr².

Hence, the total surface area of the cylinder = 2πrh + 2πr²=2πr(h+r).

**Result**

The total surface area of the cylinder is given by S = 2πr(h+r).

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