Find the Surface Area pyramid (8)(5)(3)

 $\begin{array}{r} h = & 8 \\ l = & 3 \\ w = & 5 \end{array}$

Step 1

The surface area of a pyramid is equal to the sum of the areas of each side of the pyramid. The base of the pyramid has area $l w$ , and $s_{l}$ and $s_{w}$ represent the slant height on the length and slant height on the width.

$(l e n g t h) \cdot (w i d t h) + (w i d t h) \cdot s_{l} + (l e n g t h) \cdot s_{w}$

Step 2

Substitute the values of the length $l = 3$ , the width $w = 5$ , and the height $h = 8$ into the formula for surface area of a pyramid.

$3 \cdot 5 + 5 \cdot \sqrt{{(\frac{3}{2})}^{2} + {(8)}^{2}} + 3 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Step 3

Simplify each term.

Multiply $3$ by $5$ .

$15 + 5 \cdot \sqrt{{(\frac{3}{2})}^{2} + {(8)}^{2}} + 3 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Apply the product rule to $\frac{3}{2}$ .

$15 + 5 \cdot \sqrt{\frac{3^{2}}{2^{2}} + {(8)}^{2}} + 3 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Raise $3$ to the power of $2$ .

$15 + 5 \cdot \sqrt{\frac{9}{2^{2}} + {(8)}^{2}} + 3 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Raise $2$ to the power of $2$ .

$15 + 5 \cdot \sqrt{\frac{9}{4} + {(8)}^{2}} + 3 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Raise $8$ to the power of $2$ .

$15 + 5 \cdot \sqrt{\frac{9}{4} + 64} + 3 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

To write $64$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$15 + 5 \cdot \sqrt{\frac{9}{4} + 64 \cdot \frac{4}{4}} + 3 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Combine $64$ and $\frac{4}{4}$ .

$15 + 5 \cdot \sqrt{\frac{9}{4} + \frac{64 \cdot 4}{4}} + 3 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Combine the numerators over the common denominator.

$15 + 5 \cdot \sqrt{\frac{9 + 64 \cdot 4}{4}} + 3 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Simplify the numerator.

Multiply $64$ by $4$ .

$15 + 5 \cdot \sqrt{\frac{9 + 256}{4}} + 3 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Add $9$ and $256$ .

$15 + 5 \cdot \sqrt{\frac{265}{4}} + 3 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Rewrite $\sqrt{\frac{265}{4}}$ as $\frac{\sqrt{265}}{\sqrt{4}}$ .

$15 + 5 \cdot \frac{\sqrt{265}}{\sqrt{4}} + 3 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$15 + 5 \cdot \frac{\sqrt{265}}{\sqrt{2^{2}}} + 3 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Pull terms out from under the radical, assuming positive real numbers.

$15 + 5 \cdot \frac{\sqrt{265}}{2} + 3 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

$15 + \frac{5 \sqrt{265}}{2} + 3 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(8)}^{2}}$

Apply the product rule to $\frac{5}{2}$ .

$15 + \frac{5 \sqrt{265}}{2} + 3 \cdot \sqrt{\frac{5^{2}}{2^{2}} + {(8)}^{2}}$

Raise $5$ to the power of $2$ .

$15 + \frac{5 \sqrt{265}}{2} + 3 \cdot \sqrt{\frac{25}{2^{2}} + {(8)}^{2}}$

Raise $2$ to the power of $2$ .

$15 + \frac{5 \sqrt{265}}{2} + 3 \cdot \sqrt{\frac{25}{4} + {(8)}^{2}}$

Raise $8$ to the power of $2$ .

$15 + \frac{5 \sqrt{265}}{2} + 3 \cdot \sqrt{\frac{25}{4} + 64}$

To write $64$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$15 + \frac{5 \sqrt{265}}{2} + 3 \cdot \sqrt{\frac{25}{4} + 64 \cdot \frac{4}{4}}$

Combine $64$ and $\frac{4}{4}$ .

$15 + \frac{5 \sqrt{265}}{2} + 3 \cdot \sqrt{\frac{25}{4} + \frac{64 \cdot 4}{4}}$

Combine the numerators over the common denominator.

$15 + \frac{5 \sqrt{265}}{2} + 3 \cdot \sqrt{\frac{25 + 64 \cdot 4}{4}}$

Simplify the numerator.

Multiply $64$ by $4$ .

$15 + \frac{5 \sqrt{265}}{2} + 3 \cdot \sqrt{\frac{25 + 256}{4}}$

Add $25$ and $256$ .

$15 + \frac{5 \sqrt{265}}{2} + 3 \cdot \sqrt{\frac{281}{4}}$

Rewrite $\sqrt{\frac{281}{4}}$ as $\frac{\sqrt{281}}{\sqrt{4}}$ .

$15 + \frac{5 \sqrt{265}}{2} + 3 \cdot \frac{\sqrt{281}}{\sqrt{4}}$

Simplify the denominator.

Rewrite $4$ as $2^{2}$ .

$15 + \frac{5 \sqrt{265}}{2} + 3 \cdot \frac{\sqrt{281}}{\sqrt{2^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$15 + \frac{5 \sqrt{265}}{2} + 3 \cdot \frac{\sqrt{281}}{2}$

$15 + \frac{5 \sqrt{265}}{2} + \frac{3 \sqrt{281}}{2}$

Step 4

Calculate the approximate solution to $4$ decimal places.

$80.8416$

Find the Surface Area pyramid (8)(5)(3)

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