# Total Surface Area of Cylinder ## Total Surface Area of Cylinder

The total surface area (TSA) of a cylinder is the sum of its curved surface area and the area of its two circular bases. The curvedsurface area (CSA) of a cylinder is given by the product of its circumference of circular base (2πr) and the height (h) of the cylinder i.e. CSA = 2πrh. And the area of circular base is πr2.

Hence, the total surface area of cylinder = Curved surface area + 2 × area of circular base
= 2πrh + 2πr2
= 2πr(h + r)

Total surface area (TSA) of cylinder = 2πr(r + h)

Since circumference (c) = 2πr, we can also write the formula of total surface area of cylinder as,
Total surface area (TSA) of cylinder = c(r + h)

### Workout Examples

Example 1: Find the total surface area of the given cylinder.

Solution: Here,
Diameter of the cylinder (d) = 14cm
Radius of the cylinder (r) = 14cm/2 = 7cm
Height of the cylinder (h) = 20cm

We know,
Total surface area (TSA) = 2πr(r + h)
= 2 × 22/7 × 7 (7 + 20)
= 2 × 22 × 27
= 1188cm2

Thus, the total surface area of cylinder is 1188cm2.

Example 2: The total surface area of a cylinder is 231 cm2. Its CSA (curved surface area) is 2/3 of total surface area. Find the radius of its base and height.

Solution: Here,
Total surface area of the cylinder = 231cm2
Curved surface area = 2/3 of the total surface area of cylinder
i.e.     2πrh = 2/3 × 231
or,     2 × 22/7 × rh = 154
or,     rh = 154 × 7/44
or,     rh = 24.5 ………………… (i)

Again, Curved surface area = 2/3 of total surface area
i.e.    2πrh = 2/3 × 2πr(r + h)
or,     h = 2/3 × (r + h)
or,     3h = 2r + 2h
or,     h = 2r ………………….. (i)

From (i) and (ii),
r × 2r = 24.5
or,      r2 = 24.5/2
or,      r2 = 12.25
or,      r = 3.5cm

From (ii),
h = 2r = 2 × 3.5 = 7cm

Thus, the radius is 3.5cm and height is 7cm.

Example 3: 50 circular plates, each of radius 7cm and thickness 0.5cm are placed one above the other to form a right circular cylinder. Find its total surface area.

Solution: Here,
Number of circular plates (N) = 50
Thickness of each plate = 0.5cm
Total height of cylinder (h) = 50 × 0.5cm = 25cm
Radius of each plate (r) = 7cm

Now,
The total surface area (TSA) of cylinder = 2πr(r + h)
= 2 × 22/7 × 7 × (7 + 25)
= 2 × 22 × 32
= 1408cm2

Thus, the total surface area of cylinder is 1408cm2.

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