The approximate surface areas of corresponding prisms are listed below:
- A = 108 units²
- A = 229 units²
- A = 454 units²
How to match given surface areas with given figures
The surface area of the each figure ([tex]A[/tex]), in square units, is equal to the sum of the surface area of the two bases ([tex]A_{b}[/tex]), in square units, and the surface areas of the lateral sides ([tex]A_{l}[/tex]), in square units. Since bases are regular polygons, base surface area can be determined by this expression:
[tex]A_{b} = \frac{l^{2}\cdot n }{4\cdot \tan \frac{180}{n} }[/tex] (1)
Where:
- l - Side length, in units
- n - Number of sides
The area of one lateral side is expressed by this expression:
[tex]A_{l} =h\cdot l[/tex] (2)
Where h is the height of the rectangle, in units.
The total surface area is defined by the following formula:
[tex]A = 2\cdot A_{b} + n\cdot A_{l}[/tex] (3)
Now we proceed to calculate each surface area:
Case I ([tex]l = 2[/tex], [tex]n = 6[/tex], [tex]h = 9[/tex])
[tex]A = 2\cdot \left[\frac{2^{2}\cdot (6)}{4\cdot \tan \left(\frac{180}{6} \right)} \right]+6\cdot (2)\cdot (9)[/tex]
A = 108.136 units²
Case II ([tex]l = 4\sqrt{3}[/tex], [tex]n = 3[/tex], [tex]h = 9[/tex])
[tex]A = 2\cdot \left[\frac{(4\sqrt{3})^{2}\cdot (3)}{4\cdot \tan \left(\frac{180}{3} \right)} \right]+3\cdot (4\sqrt{3})\cdot (9)[/tex]
A = 228.631 units²
Case III ([tex]l = 6[/tex], [tex]n = 5[/tex], [tex]h = 11[/tex])
[tex]A = 2\cdot \left[\frac{6^{2}\cdot (5)}{4\cdot \tan \left(\frac{180}{5} \right)} \right]+5\cdot (6)\cdot (11)[/tex]
A = 453.874 units²
The approximate surface areas of corresponding prisms are listed below:
- A = 108 units²
- A = 229 units²
- A = 454 units²
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