Basic Math Examples

 $\begin{array}{r} h = & 5 \\ l = & 3 \\ w = & 4 \end{array}$

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 $w = 4$ , and the height $h = 5$ into the formula for surface area of a pyramid.

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

Step 3

Simplify each term.

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Step 3.1

Multiply $3$ by $4$ .

$12 + 4 \cdot \sqrt{{(\frac{3}{2})}^{2} + {(5)}^{2}} + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(5)}^{2}}$

Step 3.2

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

$12 + 4 \cdot \sqrt{\frac{3^{2}}{2^{2}} + {(5)}^{2}} + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(5)}^{2}}$

Step 3.3

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

$12 + 4 \cdot \sqrt{\frac{9}{2^{2}} + {(5)}^{2}} + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(5)}^{2}}$

Step 3.4

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

$12 + 4 \cdot \sqrt{\frac{9}{4} + {(5)}^{2}} + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(5)}^{2}}$

Step 3.5

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

$12 + 4 \cdot \sqrt{\frac{9}{4} + 25} + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(5)}^{2}}$

Step 3.6

To write $25$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$12 + 4 \cdot \sqrt{\frac{9}{4} + 25 \cdot \frac{4}{4}} + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(5)}^{2}}$

Step 3.7

Combine $25$ and $\frac{4}{4}$ .

$12 + 4 \cdot \sqrt{\frac{9}{4} + \frac{25 \cdot 4}{4}} + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(5)}^{2}}$

Step 3.8

Combine the numerators over the common denominator.

$12 + 4 \cdot \sqrt{\frac{9 + 25 \cdot 4}{4}} + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(5)}^{2}}$

Step 3.9

Simplify the numerator.

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Step 3.9.1

Multiply $25$ by $4$ .

$12 + 4 \cdot \sqrt{\frac{9 + 100}{4}} + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(5)}^{2}}$

Step 3.9.2

Add $9$ and $100$ .

$12 + 4 \cdot \sqrt{\frac{109}{4}} + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(5)}^{2}}$

Step 3.10

Rewrite $\sqrt{\frac{109}{4}}$ as $\frac{\sqrt{109}}{\sqrt{4}}$ .

$12 + 4 \cdot \frac{\sqrt{109}}{\sqrt{4}} + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(5)}^{2}}$

Step 3.11

Simplify the denominator.

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Step 3.11.1

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

$12 + 4 \cdot \frac{\sqrt{109}}{\sqrt{2^{2}}} + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(5)}^{2}}$

Step 3.11.2

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

$12 + 4 \cdot \frac{\sqrt{109}}{2} + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(5)}^{2}}$

Step 3.12

Cancel the common factor of $2$ .

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Step 3.12.1

Factor $2$ out of $4$ .

$12 + 2 (2) \cdot \frac{\sqrt{109}}{2} + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(5)}^{2}}$

Step 3.12.2

Cancel the common factor.

$12 + 2 \cdot 2 \cdot \frac{\sqrt{109}}{2} + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(5)}^{2}}$

Step 3.12.3

Rewrite the expression.

$12 + 2 \cdot \sqrt{109} + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(5)}^{2}}$

Step 3.13

Divide $4$ by $2$ .

$12 + 2 \sqrt{109} + 3 \cdot \sqrt{2^{2} + {(5)}^{2}}$

Step 3.14

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

$12 + 2 \sqrt{109} + 3 \cdot \sqrt{4 + {(5)}^{2}}$

Step 3.15

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

$12 + 2 \sqrt{109} + 3 \cdot \sqrt{4 + 25}$

Step 3.16

Add $4$ and $25$ .

$12 + 2 \sqrt{109} + 3 \sqrt{29}$

Step 4

Calculate the approximate solution to $4$ decimal places.

$49.0361$