Question

Match the value to the correct image. Not all values will be used.

Answer 1

The approximate surface areas of corresponding prisms are listed below:

1. A = 108 units²
2. A = 229 units²
3. A = 454 units²

How to match given surface areas with given figures

The surface area of the each figure ($A$), in square units, is equal to the sum of the surface area of the two bases ($A_{b}$), in square units, and the surface areas of the lateral sides ($A_{l}$), in square units. Since bases are regular polygons, base surface area can be determined by this expression:

$A_{b} = \frac{l^{2}\cdot n }{4\cdot \tan \frac{180}{n} }$  (1)

Where:

• l - Side length, in units
• n - Number of sides

The area of one lateral side is expressed by this expression:

$A_{l} =h\cdot l$   (2)

Where h is the height of the rectangle, in units.

The total surface area is defined by the following formula:

$A = 2\cdot A_{b} + n\cdot A_{l}$   (3)

Now we proceed to calculate each surface area:

Case I ($l = 2$, $n = 6$, $h = 9$)

$A = 2\cdot \left[\frac{2^{2}\cdot (6)}{4\cdot \tan \left(\frac{180}{6} \right)} \right]+6\cdot (2)\cdot (9)$

A = 108.136 units²

Case II ($l = 4\sqrt{3}$, $n = 3$, $h = 9$)

$A = 2\cdot \left[\frac{(4\sqrt{3})^{2}\cdot (3)}{4\cdot \tan \left(\frac{180}{3} \right)} \right]+3\cdot (4\sqrt{3})\cdot (9)$

A = 228.631 units²

Case III ($l = 6$, $n = 5$, $h = 11$)

$A = 2\cdot \left[\frac{6^{2}\cdot (5)}{4\cdot \tan \left(\frac{180}{5} \right)} \right]+5\cdot (6)\cdot (11)$

A = 453.874 units²

The approximate surface areas of corresponding prisms are listed below:

1. A = 108 units²
2. A = 229 units²
3. A = 454 units²

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