Math - Geometry Lesson Plans : Surface Area of Cylinders

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Mensuration - Formulae for Surface Area of Cylinders

A cylinder is a solid which has a uniform, circular cross-section.

Curved surface area of a cylinder = 2 π rh
where r is the radius and h is the height.
In the figure alongside of the cylinder, r = PS = QR.
In the figure alongside of the cylinder, h = PQ = RS.
In the figure alongside of the cylinder, curved surface area = 2 π × PS × PQ.

Example
Find the curved surface area (in cm^{2}) of a cylinder of diameter 14 cm and height 20 cm. Solution.
Let the radius of the cylinder be r and the height be h.
Curved surface area of a cylinder = 2 π rh
= 2 × 22 / 7 × 7 × 20 = 880 cm^{2}.

Total surface area of a cylinder = 2 π rh + 2 π r^{2}
In the figure alongside of the cylinder, total surface area = 2 π × PS × PQ + 2 π × PS^{2}.

Example
Find the total surface area (in cm^{2}) of a cylinder of radius 3.5 cm and height 10 cm. Solution.
Let the radius of the cylinder be r and the height be h.
Total surface area of a cylinder = 2 π rh + 2 π r^{2}
= 2 π r (h + r) = 2 × 22 / 7 × 3.5 (10 + 3.5) = 297 cm^{2}.

Curved surface area of a hollow cylinder = 2 π Rh + 2 π rh
where R is the external radius and r is the internal radius.

Example
Find the curved surface area (in cm^{2}) of a hollow cylinder of external radius 7 cm given that its thickness is 1 cm and height is 21 cm. Solution.
Let the external radius, the internal radius and the height of the hollow cylinder be R, r and h respectively. r = 7 − 1 = 6 cm.
Curved surface area of a hollow cylinder = 2 π Rh + 2 π rh
= 2 π h (R + r) = 2 × 22 / 7 × 21 (7 + 6) = 1716 cm^{2}.

Total surface area of a hollow cylinder = 2 π Rh + 2 π rh + 2 (π R^{2} − π r^{2})

Example
A hollow cylinder (open at both ends) has an external diameter of 28 cm, length of 14 cm and thickness of 2 cm. Find its total surface area (in cm^{2} upto two decimal places). Solution.
Let the external radius, the internal radius and the height of the hollow cylinder be R, r and h respectively. r = 14 − 2 = 12 cm.
Total surface area of a hollow cylinder = 2 π Rh + 2 π rh + 2 (π R^{2} − π r^{2})
= 2 × 22 / 7 × 14 × 14 + 2 × 22 / 7 × 12 × 14 + 2 (22 / 7 × 14^{2} − 22 / 7 × 12^{2}) = 2614.86 cm^{2}.