Find the Surface Area pyramid (60)(24)(16)

 $\begin{array}{r} h = & 60 \\ l = & 16 \\ w = & 24 \end{array}$

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 $l = 16$ , the width $w = 24$ , and the height $h = 60$ into the formula for surface area of a pyramid.

$16 \cdot 24 + 24 \cdot \sqrt{{(\frac{16}{2})}^{2} + {(60)}^{2}} + 16 \cdot \sqrt{{(\frac{24}{2})}^{2} + {(60)}^{2}}$

Step 3

Simplify each term.

Multiply $16$ by $24$ .

$384 + 24 \cdot \sqrt{{(\frac{16}{2})}^{2} + {(60)}^{2}} + 16 \cdot \sqrt{{(\frac{24}{2})}^{2} + {(60)}^{2}}$

Divide $16$ by $2$ .

$384 + 24 \cdot \sqrt{8^{2} + {(60)}^{2}} + 16 \cdot \sqrt{{(\frac{24}{2})}^{2} + {(60)}^{2}}$

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

$384 + 24 \cdot \sqrt{64 + {(60)}^{2}} + 16 \cdot \sqrt{{(\frac{24}{2})}^{2} + {(60)}^{2}}$

Raise $60$ to the power of $2$ .

$384 + 24 \cdot \sqrt{64 + 3600} + 16 \cdot \sqrt{{(\frac{24}{2})}^{2} + {(60)}^{2}}$

Add $64$ and $3600$ .

$384 + 24 \cdot \sqrt{3664} + 16 \cdot \sqrt{{(\frac{24}{2})}^{2} + {(60)}^{2}}$

Rewrite $3664$ as $4^{2} \cdot 229$ .

Factor $16$ out of $3664$ .

$384 + 24 \cdot \sqrt{16 (229)} + 16 \cdot \sqrt{{(\frac{24}{2})}^{2} + {(60)}^{2}}$

Rewrite $16$ as $4^{2}$ .

$384 + 24 \cdot \sqrt{4^{2} \cdot 229} + 16 \cdot \sqrt{{(\frac{24}{2})}^{2} + {(60)}^{2}}$

Pull terms out from under the radical.

$384 + 24 \cdot (4 \sqrt{229}) + 16 \cdot \sqrt{{(\frac{24}{2})}^{2} + {(60)}^{2}}$

Multiply $4$ by $24$ .

$384 + 96 \sqrt{229} + 16 \cdot \sqrt{{(\frac{24}{2})}^{2} + {(60)}^{2}}$

Divide $24$ by $2$ .

$384 + 96 \sqrt{229} + 16 \cdot \sqrt{12^{2} + {(60)}^{2}}$

Raise $12$ to the power of $2$ .

$384 + 96 \sqrt{229} + 16 \cdot \sqrt{144 + {(60)}^{2}}$

Raise $60$ to the power of $2$ .

$384 + 96 \sqrt{229} + 16 \cdot \sqrt{144 + 3600}$

Add $144$ and $3600$ .

$384 + 96 \sqrt{229} + 16 \cdot \sqrt{3744}$

Rewrite $3744$ as $12^{2} \cdot 26$ .

Factor $144$ out of $3744$ .

$384 + 96 \sqrt{229} + 16 \cdot \sqrt{144 (26)}$

Rewrite $144$ as $12^{2}$ .

$384 + 96 \sqrt{229} + 16 \cdot \sqrt{12^{2} \cdot 26}$

Pull terms out from under the radical.

$384 + 96 \sqrt{229} + 16 \cdot (12 \sqrt{26})$

Multiply $12$ by $16$ .

$384 + 96 \sqrt{229} + 192 \sqrt{26}$

Step 4

Calculate the approximate solution to $4$ decimal places.

$2815.7554$

Find the Surface Area pyramid (60)(24)(16)

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