Question

If the diameter of the base of a closed right circular cylinder be equal to its height, h, then its whole surface area is :

• A. \(2\pi h^2\)
• B. \(\frac{3}{2}\pi h^2\)
• C. \(\frac{4}{3}\pi h^2\)
• D. \(\pi h^2\)

Hint:

1. Given: diameter \(d = h\). 2. Radius \(r = d/2 = h/2\). 3. Total Surface Area of cylinder (TSA) = \(2\pi r (\text{bases}) + 2\pi rh (\text{curved surface})\). 4. Substitute \(r = h/2\): \(TSA = 2\pi (h/2)^2 + 2\pi (h/2)h\) \(TSA = 2\pi (h^2/4) + \pi h^2\) \(TSA = \pi h^2/2 + \pi h^2\) \(TSA = \pi h^2/2 + 2\pi h^2/2 = 3\pi h^2/2\).

Answer 1

The Correct Option is B

Concept: The whole surface area (total surface area) of a closed right circular cylinder includes the area of its two circular bases and the area of its curved surface. Formula for Total Surface Area (TSA) of a cylinder: \(TSA = 2\pi r^2 + 2\pi rh\), where \(r\) is the radius of the base and \(h\) is the height. Step 1: Relate diameter, radius, and height Let \(d\) be the diameter of the base and \(r\) be the radius. We know that diameter \(d = 2r\). The problem states that the diameter of the base is equal to its height, \(h\). So, \(d = h\). Since \(d = 2r\), we have \(2r = h\). This implies that the radius \(r = \frac{h}{2}\). Step 2: Substitute \(r = \frac{h}{2}\) into the TSA formula The TSA formula is \(TSA = 2\pi r^2 + 2\pi rh\). Substitute \(r = \frac{h}{2}\): \[ TSA = 2\pi \left(\frac{h}{2}\right)^2 + 2\pi \left(\frac{h}{2}\right)h \] Step 3: Simplify the expression \[ TSA = 2\pi \left(\frac{h^2}{4}\right) + 2\pi \left(\frac{h^2}{2}\right) \] Simplify the fractions: \[ TSA = \frac{2\pi h^2}{4} + \frac{2\pi h^2}{2} \] \[ TSA = \frac{\pi h^2}{2} + \pi h^2 \] To add these terms, find a common denominator (which is 2): \(\pi h^2 = \frac{2\pi h^2}{2}\) So, \[ TSA = \frac{\pi h^2}{2} + \frac{2\pi h^2}{2} \] \[ TSA = \frac{\pi h^2 + 2\pi h^2}{2} \] \[ TSA = \frac{3\pi h^2}{2} \] This can also be written as \( \frac{3}{2}\pi h^2 \).