Pre-Algebra Examples

 $\begin{array}{r} h = & 12 \\ l = & 15 \\ w = & 12 \end{array}$

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 $l = 15$ , the width $w = 12$ , and the height $h = 12$ into the formula for surface area of a pyramid.

$15 \cdot 12 + 12 \cdot \sqrt{{(\frac{15}{2})}^{2} + {(12)}^{2}} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3

Simplify each term.

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Step 3.1

Multiply $15$ by $12$ .

$180 + 12 \cdot \sqrt{{(\frac{15}{2})}^{2} + {(12)}^{2}} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.2

Apply the product rule to $\frac{15}{2}$ .

$180 + 12 \cdot \sqrt{\frac{15^{2}}{2^{2}} + {(12)}^{2}} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.3

Raise $15$ to the power of $2$ .

$180 + 12 \cdot \sqrt{\frac{225}{2^{2}} + {(12)}^{2}} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.4

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

$180 + 12 \cdot \sqrt{\frac{225}{4} + {(12)}^{2}} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.5

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

$180 + 12 \cdot \sqrt{\frac{225}{4} + 144} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.6

To write $144$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$180 + 12 \cdot \sqrt{\frac{225}{4} + 144 \cdot \frac{4}{4}} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.7

Combine $144$ and $\frac{4}{4}$ .

$180 + 12 \cdot \sqrt{\frac{225}{4} + \frac{144 \cdot 4}{4}} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.8

Combine the numerators over the common denominator.

$180 + 12 \cdot \sqrt{\frac{225 + 144 \cdot 4}{4}} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.9

Simplify the numerator.

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Step 3.9.1

Multiply $144$ by $4$ .

$180 + 12 \cdot \sqrt{\frac{225 + 576}{4}} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.9.2

Add $225$ and $576$ .

$180 + 12 \cdot \sqrt{\frac{801}{4}} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.10

Rewrite $\sqrt{\frac{801}{4}}$ as $\frac{\sqrt{801}}{\sqrt{4}}$ .

$180 + 12 \cdot \frac{\sqrt{801}}{\sqrt{4}} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.11

Simplify the numerator.

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Step 3.11.1

Rewrite $801$ as $3^{2} \cdot 89$ .

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Step 3.11.1.1

Factor $9$ out of $801$ .

$180 + 12 \cdot \frac{\sqrt{9 (89)}}{\sqrt{4}} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.11.1.2

Rewrite $9$ as $3^{2}$ .

$180 + 12 \cdot \frac{\sqrt{3^{2} \cdot 89}}{\sqrt{4}} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.11.2

Pull terms out from under the radical.

$180 + 12 \cdot \frac{3 \sqrt{89}}{\sqrt{4}} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.12

Simplify the denominator.

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Step 3.12.1

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

$180 + 12 \cdot \frac{3 \sqrt{89}}{\sqrt{2^{2}}} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.12.2

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

$180 + 12 \cdot \frac{3 \sqrt{89}}{2} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.13

Cancel the common factor of $2$ .

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Step 3.13.1

Factor $2$ out of $12$ .

$180 + 2 (6) \cdot \frac{3 \sqrt{89}}{2} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.13.2

Cancel the common factor.

$180 + 2 \cdot 6 \cdot \frac{3 \sqrt{89}}{2} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.13.3

Rewrite the expression.

$180 + 6 \cdot (3 \sqrt{89}) + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.14

Multiply $3$ by $6$ .

$180 + 18 \cdot \sqrt{89} + 15 \cdot \sqrt{{(\frac{12}{2})}^{2} + {(12)}^{2}}$

Step 3.15

Divide $12$ by $2$ .

$180 + 18 \sqrt{89} + 15 \cdot \sqrt{6^{2} + {(12)}^{2}}$

Step 3.16

Raise $6$ to the power of $2$ .

$180 + 18 \sqrt{89} + 15 \cdot \sqrt{36 + {(12)}^{2}}$

Step 3.17

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

$180 + 18 \sqrt{89} + 15 \cdot \sqrt{36 + 144}$

Step 3.18

Add $36$ and $144$ .

$180 + 18 \sqrt{89} + 15 \cdot \sqrt{180}$

Step 3.19

Rewrite $180$ as $6^{2} \cdot 5$ .

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Step 3.19.1

Factor $36$ out of $180$ .

$180 + 18 \sqrt{89} + 15 \cdot \sqrt{36 (5)}$

Step 3.19.2

Rewrite $36$ as $6^{2}$ .

$180 + 18 \sqrt{89} + 15 \cdot \sqrt{6^{2} \cdot 5}$

Step 3.20

Pull terms out from under the radical.

$180 + 18 \sqrt{89} + 15 \cdot (6 \sqrt{5})$

Step 3.21

Multiply $6$ by $15$ .

$180 + 18 \sqrt{89} + 90 \sqrt{5}$

Step 4

Calculate the approximate solution to $4$ decimal places.

$551.0578$