Cylinder surface area

Surface Area of a Cylinder

Definition of Surface Area in a Cylinder

• The surface area of a cylinder consists of:

  • Two circular bases (top and bottom)

  • A rectangular side that wraps around the circular bases

Surface Area Formula

• The formula for the surface area (SA) of a cylinder is given by: $SA = 2 ext{π}r^2 + 2 ext{π}rh$ where:

  • $r$ = radius of the base of the cylinder

  • $h$ = height of the cylinder

  • $ext{π}$ (Pi) is approximately 3.14 (or the more precise value of $rac{22}{7}$, depending on the context of the problem)

Example Calculation

• To find the surface area of a cylinder with:

  • Radius $r = 5$ cm

  • Height $h = 10$ cm

Step-by-Step Calculation

1. Calculate the area of the two circles:

   • Use the formula for the area of a circle, which is $ext{Area} = ext{π}r^2$:

   • Area of one circle = $ext{π}(5)^2 = 3.14 imes 25 = 78.5$ cm²

   • Area of two circles = $2 imes 78.5 = 157$ cm²

2. Calculate the area of the rectangular side:

   • The area of the rectangle is given by the formula: $ext{Area} = ext{circumference} imes h$, where circumference of the base is $2 ext{π}r$:

   • Circumference = $2 imes 3.14 imes 5 = 31.4$ cm

   • Area of the rectangular side = $31.4 imes 10 = 314$ cm²

3. Add both areas together to find total surface area:

   • Total surface area = Area of two circles + Area of rectangle

   • Total surface area = $157 + 314 = 471$ cm²

Conclusion

• Therefore, the surface area of a cylinder with a radius of 5 cm and height of 10 cm is:
  $471 ext{ cm}^2$

Key Takeaways

• The surface area of a cylinder can be found by combining the areas of its two circular bases and the rectangular side.

• Understanding the formula is crucial for applying it to various problems and scenarios involving cylinders.