Pre-Algebra Examples

 $\begin{array}{r} h = & 13 \\ l = & 20.3 \\ w = & 13 \end{array}$

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 $l = 20.3$ , the width $w = 13$ , and the height $h = 13$ into the formula for surface area of a pyramid.

$20.3 \cdot 13 + 13 \cdot \sqrt{{(\frac{20.3}{2})}^{2} + {(13)}^{2}} + 20.3 \cdot \sqrt{{(\frac{13}{2})}^{2} + {(13)}^{2}}$

Step 3

Simplify each term.

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Step 3.1

Multiply $20.3$ by $13$ .

$263.9 + 13 \cdot \sqrt{{(\frac{20.3}{2})}^{2} + {(13)}^{2}} + 20.3 \cdot \sqrt{{(\frac{13}{2})}^{2} + {(13)}^{2}}$

Step 3.2

Divide $20.3$ by $2$ .

$263.9 + 13 \cdot \sqrt{{10.15}^{2} + {(13)}^{2}} + 20.3 \cdot \sqrt{{(\frac{13}{2})}^{2} + {(13)}^{2}}$

Step 3.3

Raise $10.15$ to the power of $2$ .

$263.9 + 13 \cdot \sqrt{103.0225 + {(13)}^{2}} + 20.3 \cdot \sqrt{{(\frac{13}{2})}^{2} + {(13)}^{2}}$

Step 3.4

Raise $13$ to the power of $2$ .

$263.9 + 13 \cdot \sqrt{103.0225 + 169} + 20.3 \cdot \sqrt{{(\frac{13}{2})}^{2} + {(13)}^{2}}$

Step 3.5

Add $103.0225$ and $169$ .

$263.9 + 13 \cdot \sqrt{272.0225} + 20.3 \cdot \sqrt{{(\frac{13}{2})}^{2} + {(13)}^{2}}$

Step 3.6

Apply the product rule to $\frac{13}{2}$ .

$263.9 + 13 \sqrt{272.0225} + 20.3 \cdot \sqrt{\frac{13^{2}}{2^{2}} + {(13)}^{2}}$

Step 3.7

Raise $13$ to the power of $2$ .

$263.9 + 13 \sqrt{272.0225} + 20.3 \cdot \sqrt{\frac{169}{2^{2}} + {(13)}^{2}}$

Step 3.8

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

$263.9 + 13 \sqrt{272.0225} + 20.3 \cdot \sqrt{\frac{169}{4} + {(13)}^{2}}$

Step 3.9

Raise $13$ to the power of $2$ .

$263.9 + 13 \sqrt{272.0225} + 20.3 \cdot \sqrt{\frac{169}{4} + 169}$

Step 3.10

To write $169$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$263.9 + 13 \sqrt{272.0225} + 20.3 \cdot \sqrt{\frac{169}{4} + 169 \cdot \frac{4}{4}}$

Step 3.11

Combine $169$ and $\frac{4}{4}$ .

$263.9 + 13 \sqrt{272.0225} + 20.3 \cdot \sqrt{\frac{169}{4} + \frac{169 \cdot 4}{4}}$

Step 3.12

Combine the numerators over the common denominator.

$263.9 + 13 \sqrt{272.0225} + 20.3 \cdot \sqrt{\frac{169 + 169 \cdot 4}{4}}$

Step 3.13

Simplify the numerator.

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Step 3.13.1

Multiply $169$ by $4$ .

$263.9 + 13 \sqrt{272.0225} + 20.3 \cdot \sqrt{\frac{169 + 676}{4}}$

Step 3.13.2

Add $169$ and $676$ .

$263.9 + 13 \sqrt{272.0225} + 20.3 \cdot \sqrt{\frac{845}{4}}$

Step 3.14

Rewrite $\sqrt{\frac{845}{4}}$ as $\frac{\sqrt{845}}{\sqrt{4}}$ .

$263.9 + 13 \sqrt{272.0225} + 20.3 \cdot \frac{\sqrt{845}}{\sqrt{4}}$

Step 3.15

Simplify the numerator.

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Step 3.15.1

Rewrite $845$ as $13^{2} \cdot 5$ .

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Step 3.15.1.1

Factor $169$ out of $845$ .

$263.9 + 13 \sqrt{272.0225} + 20.3 \cdot \frac{\sqrt{169 (5)}}{\sqrt{4}}$

Step 3.15.1.2

Rewrite $169$ as $13^{2}$ .

$263.9 + 13 \sqrt{272.0225} + 20.3 \cdot \frac{\sqrt{13^{2} \cdot 5}}{\sqrt{4}}$

Step 3.15.2

Pull terms out from under the radical.

$263.9 + 13 \sqrt{272.0225} + 20.3 \cdot \frac{13 \sqrt{5}}{\sqrt{4}}$

Step 3.16

Simplify the denominator.

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Step 3.16.1

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

$263.9 + 13 \sqrt{272.0225} + 20.3 \cdot \frac{13 \sqrt{5}}{\sqrt{2^{2}}}$

Step 3.16.2

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

$263.9 + 13 \sqrt{272.0225} + 20.3 \cdot \frac{13 \sqrt{5}}{2}$

Step 3.17

Multiply $20.3 \frac{13 \sqrt{5}}{2}$ .

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Step 3.17.1

Combine $20.3$ and $\frac{13 \sqrt{5}}{2}$ .

$263.9 + 13 \sqrt{272.0225} + \frac{20.3 (13 \sqrt{5})}{2}$

Step 3.17.2

Multiply $13$ by $20.3$ .

$263.9 + 13 \sqrt{272.0225} + \frac{263.9 \sqrt{5}}{2}$

Step 3.17.3

Multiply $263.9$ by $\sqrt{5}$ .

$263.9 + 13 \sqrt{272.0225} + \frac{590.09833926}{2}$

Step 3.18

Divide $590.09833926$ by $2$ .

$263.9 + 13 \sqrt{272.0225} + 295.04916963$

Step 4

Simplify by adding terms.

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Step 4.1

Add $263.9$ and $13 \sqrt{272.0225}$ .

$478.31036005 + 295.04916963$

Step 4.2

Add $478.31036005$ and $295.04916963$ .

$773.35952968$

Step 5

Calculate the approximate solution to $4$ decimal places.

$773.3595$