Basic Math Examples

 $\begin{array}{r} h = & 20 \\ l = & 15 \\ w = & 20 \end{array}$

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 $w = 20$ , and the height $h = 20$ into the formula for surface area of a pyramid.

$15 \cdot 20 + 20 \cdot \sqrt{{(\frac{15}{2})}^{2} + {(20)}^{2}} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3

Simplify each term.

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Step 3.1

Multiply $15$ by $20$ .

$300 + 20 \cdot \sqrt{{(\frac{15}{2})}^{2} + {(20)}^{2}} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.2

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

$300 + 20 \cdot \sqrt{\frac{15^{2}}{2^{2}} + {(20)}^{2}} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.3

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

$300 + 20 \cdot \sqrt{\frac{225}{2^{2}} + {(20)}^{2}} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.4

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

$300 + 20 \cdot \sqrt{\frac{225}{4} + {(20)}^{2}} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.5

Raise $20$ to the power of $2$ .

$300 + 20 \cdot \sqrt{\frac{225}{4} + 400} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.6

To write $400$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$300 + 20 \cdot \sqrt{\frac{225}{4} + 400 \cdot \frac{4}{4}} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.7

Combine $400$ and $\frac{4}{4}$ .

$300 + 20 \cdot \sqrt{\frac{225}{4} + \frac{400 \cdot 4}{4}} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.8

Combine the numerators over the common denominator.

$300 + 20 \cdot \sqrt{\frac{225 + 400 \cdot 4}{4}} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.9

Simplify the numerator.

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Step 3.9.1

Multiply $400$ by $4$ .

$300 + 20 \cdot \sqrt{\frac{225 + 1600}{4}} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.9.2

Add $225$ and $1600$ .

$300 + 20 \cdot \sqrt{\frac{1825}{4}} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.10

Rewrite $\sqrt{\frac{1825}{4}}$ as $\frac{\sqrt{1825}}{\sqrt{4}}$ .

$300 + 20 \cdot \frac{\sqrt{1825}}{\sqrt{4}} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.11

Simplify the numerator.

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Step 3.11.1

Rewrite $1825$ as $5^{2} \cdot 73$ .

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Step 3.11.1.1

Factor $25$ out of $1825$ .

$300 + 20 \cdot \frac{\sqrt{25 (73)}}{\sqrt{4}} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.11.1.2

Rewrite $25$ as $5^{2}$ .

$300 + 20 \cdot \frac{\sqrt{5^{2} \cdot 73}}{\sqrt{4}} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.11.2

Pull terms out from under the radical.

$300 + 20 \cdot \frac{5 \sqrt{73}}{\sqrt{4}} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.12

Simplify the denominator.

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Step 3.12.1

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

$300 + 20 \cdot \frac{5 \sqrt{73}}{\sqrt{2^{2}}} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.12.2

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

$300 + 20 \cdot \frac{5 \sqrt{73}}{2} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.13

Cancel the common factor of $2$ .

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Step 3.13.1

Factor $2$ out of $20$ .

$300 + 2 (10) \cdot \frac{5 \sqrt{73}}{2} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.13.2

Cancel the common factor.

$300 + 2 \cdot 10 \cdot \frac{5 \sqrt{73}}{2} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.13.3

Rewrite the expression.

$300 + 10 \cdot (5 \sqrt{73}) + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.14

Multiply $5$ by $10$ .

$300 + 50 \cdot \sqrt{73} + 15 \cdot \sqrt{{(\frac{20}{2})}^{2} + {(20)}^{2}}$

Step 3.15

Divide $20$ by $2$ .

$300 + 50 \sqrt{73} + 15 \cdot \sqrt{10^{2} + {(20)}^{2}}$

Step 3.16

Raise $10$ to the power of $2$ .

$300 + 50 \sqrt{73} + 15 \cdot \sqrt{100 + {(20)}^{2}}$

Step 3.17

Raise $20$ to the power of $2$ .

$300 + 50 \sqrt{73} + 15 \cdot \sqrt{100 + 400}$

Step 3.18

Add $100$ and $400$ .

$300 + 50 \sqrt{73} + 15 \cdot \sqrt{500}$

Step 3.19

Rewrite $500$ as $10^{2} \cdot 5$ .

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Step 3.19.1

Factor $100$ out of $500$ .

$300 + 50 \sqrt{73} + 15 \cdot \sqrt{100 (5)}$

Step 3.19.2

Rewrite $100$ as $10^{2}$ .

$300 + 50 \sqrt{73} + 15 \cdot \sqrt{10^{2} \cdot 5}$

Step 3.20

Pull terms out from under the radical.

$300 + 50 \sqrt{73} + 15 \cdot (10 \sqrt{5})$

Step 3.21

Multiply $10$ by $15$ .

$300 + 50 \sqrt{73} + 150 \sqrt{5}$

Step 4

Calculate the approximate solution to $4$ decimal places.

$1062.6104$