Question

Find the total surface area and lateral surface area of the cylinder, if c. Diameter = 56 mm, Height = 32 cm​

Answer 1

Total Surface Area = 612.2 cm³

Lateral Surface Area = 563.0 cm³

Explanation:

The formula for the surface area of a cylinder is:

`\boxed{\begin{array}{l}\underline{\textsf{Surface Area of a Cylinder}}\\\\SA=2\pi rh+2\pi r^2\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$SA$ is the surface area.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius of the circular base.}\\\phantom{ww}\bullet\;\textsf{$h$ is the height.}\end{array}}`

Given that the diameter of a circle is twice its radius, and the diameter of the cylinder is 56 mm, then r = 28 mm.

Given that the height of the cylinder is given in centimeters, we need to convert the radius into centimeters by dividing by 10. Therefore, r = 2.8 cm.

To find the surface area of the cylinder, substitute r = 2.8 and h = 32 into the surface area formula:

`SA=2\pi(2.8)(32)+2\pi(2.8)^2`

`SA=179.2\pi+15.68\pi`

`SA=194.88\pi`

`SA=612.233576...`

`SA=612.2\; \sf cm^2\; (nearest\;tenth)`

Therefore, the surface area of the cylinder is 612.2 cm², rounded to the nearest tenth.

The lateral surface area of a cylinder specifically refers to the area of the curved surface or "side" of the cylinder, excluding the areas of the circular bases:

`\boxed{\begin{array}{l}\underline{\textsf{Lateral Surface Area of a Cylinder}}\\\\LSA=2\pi rh\\\\\textsf{where:}\\\phantom{ww}\bullet\;\textsf{$LSA$ is the lateral surface area.}\\\phantom{ww}\bullet\;\textsf{$r$ is the radius of the circular base.}\\\phantom{ww}\bullet\;\textsf{$h$ is the height.}\end{array}}`

To find the lateral surface area of the cylinder, substitute r = 2.8 and h = 32 into the LSA formula:

`LSA=2\pi(2.8)(32)`

`LSA=179.2\pi`

`LSA=562.97340352...`

`LSA=563.0\; \sf cm^2\; (nearest\;tenth)`

Therefore, the lateral surface area of the cylinder is 563.0 cm², rounded to the nearest tenth.

`\hrulefill`

Additional Notes

If you want the areas in terms of millimeters squared, simply multiply the areas given in centimeters squared by 100:

`SA=194.88\pi \cdot 100=19488\pi \approx 61223.4\; \sf mm^2`

`LSA=179.2\pi \cdot 100=17920\pi \approx 56297.3\; \sf mm^2`

Answer 2

Total surface area of the cylinder =
`\sf 61223.36 \, \textsf{mm}^2`

Lateral surface area of the cylinder =
`\sf 56297.34 \, \textsf{mm}^2`

Explanation:

To find the total surface area (TSA) and lateral surface area (LSA) of a cylinder, we can use the following formulas:

Total Surface Area (TSA):

`\sf TSA = 2\pi r^2 + 2\pi rh`

Lateral Surface Area (LSA):

`\sf LSA = 2\pi rh`

Where:

• 
  `\sf r` is the radius of the base of the cylinder.
• 
  `\sf h` is the height of the cylinder.

Given that the diameter (
`\sf D`) is 56 mm, the radius (
`\sf r`) is half of the diameter:
`\sf r = (D)/(2) = (56)/(2) = 28` mm.

Given that the height (
`\sf h`) is 32 cm, convert it to millimeters (1 cm = 10 mm):

`\sf h = 32 * 10 = 320` mm.

Now, substitute these values into the formulas:

1. Total Surface Area (TSA):

`\sf TSA = 2\pi (28)^2 + 2\pi (28)(320)`

2. Lateral Surface Area (LSA):

`\sf LSA = 2\pi (28)(320)`

Now, calculate these values:

`\sf TSA = 2\pi (784) + 2\pi (8960) \\\\= 1568\pi + 17920\pi \\\\= 19488\pi \\\\ \approx 61223.357633157 \\\\ \approx 61223.36 \, \textsf{mm}^2\textsf{ ( in 2 d.p.)}`

`\sf LSA = 2\pi (28)(320)\\\\ = 17920\pi \\\\= 56297.340352329\\\\ \approx 56297.34\, \textsf{mm}^2 \textsf{in 2 d.p.)}`

Therefore, the total surface area of the cylinder is
`\sf 61223.36 \, \textsf{mm}^2`,

and

the lateral surface area is
`\sf 56297.34 \, \textsf{mm}^2`.