How to find the surface area of a pentagonal prism?
Understand the Problem
The question is asking how to calculate the surface area of a pentagonal prism, which involves finding the area of the two pentagonal bases and the five rectangular lateral faces connecting these bases. The approach typically involves using the formula for the area of a pentagon, alongside the dimensions of the prism (height and side lengths).
Answer
The total surface area of the pentagonal prism is given by: $$ A_{total} = \frac{1}{2} \sqrt{5(5 + 2\sqrt{5})} s^2 + 5sh $$
Answer for screen readers
The total surface area of the pentagonal prism is given by:
$$ A_{total} = \frac{1}{2} \sqrt{5(5 + 2\sqrt{5})} s^2 + 5sh $$
Steps to Solve
- Calculate the Area of the Pentagon Base
To find the area of the pentagonal base, use the formula for the area of a regular pentagon:
$$ A_{pentagon} = \frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} s^2 $$
where $s$ is the length of one side of the pentagon.
- Determine the Height of the Prism
Identify the height ($h$) of the pentagonal prism. This is a dimension that is usually provided in the problem.
- Calculate the Area of the Rectangular Faces
Each lateral rectangular face has an area equal to the length of one side of the pentagon multiplied by the height of the prism. The area of one rectangular face is given by:
$$ A_{rectangle} = s \times h $$
Since there are five rectangular faces, the total area of all lateral faces will be:
$$ A_{lateral} = 5 \times (s \times h) $$
- Combine the Areas for Total Surface Area
Now, you can find the total surface area of the prism by adding the area of the two pentagonal bases and the area of the five rectangular faces:
$$ A_{total} = 2 \times A_{pentagon} + A_{lateral} $$
Substituting the formulas from the previous steps gives:
$$ A_{total} = 2 \times \left(\frac{1}{4} \sqrt{5(5 + 2\sqrt{5})} s^2\right) + 5 \times (s \times h) $$
- Simplify and Solve for Total Surface Area
Finally, simplify the equation above to find the total surface area of the pentagonal prism.
The total surface area of the pentagonal prism is given by:
$$ A_{total} = \frac{1}{2} \sqrt{5(5 + 2\sqrt{5})} s^2 + 5sh $$
More Information
Finding the surface area of a prism involves knowing both the area of the bases and the lateral face area. The formula for the area of a pentagon includes a constant derived from geometry that relates the side length to the area directly.
Tips
- Forgetting to include both bases in the surface area calculation.
- Miscalculating the area of the rectangular lateral faces.
- Confusing the height of the prism with the side length of the base.
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