Find the Surface Area pyramid (5)(5)(5)

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

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

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

Step 3

Simplify each term.

Multiply $5$ by $5$ .

$25 + 5 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(5)}^{2}} + 5 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(5)}^{2}}$

Apply the product rule to $\frac{5}{2}$ .

$25 + 5 \cdot \sqrt{\frac{5^{2}}{2^{2}} + {(5)}^{2}} + 5 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(5)}^{2}}$

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

$25 + 5 \cdot \sqrt{\frac{25}{2^{2}} + {(5)}^{2}} + 5 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(5)}^{2}}$

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

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

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

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

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

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

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

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Multiply $25$ by $4$ .

$25 + 5 \cdot \sqrt{\frac{25 + 100}{4}} + 5 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(5)}^{2}}$

Add $25$ and $100$ .

$25 + 5 \cdot \sqrt{\frac{125}{4}} + 5 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(5)}^{2}}$

Rewrite $\sqrt{\frac{125}{4}}$ as $\frac{\sqrt{125}}{\sqrt{4}}$ .

$25 + 5 \cdot \frac{\sqrt{125}}{\sqrt{4}} + 5 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(5)}^{2}}$

Simplify the numerator.

Rewrite $125$ as $5^{2} \cdot 5$ .

Factor $25$ out of $125$ .

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

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

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

Pull terms out from under the radical.

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

Simplify the denominator.

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

$25 + 5 \cdot \frac{5 \sqrt{5}}{\sqrt{2^{2}}} + 5 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(5)}^{2}}$

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

$25 + 5 \cdot \frac{5 \sqrt{5}}{2} + 5 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(5)}^{2}}$

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

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

$25 + \frac{5 (5 \sqrt{5})}{2} + 5 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(5)}^{2}}$

Multiply $5$ by $5$ .

$25 + \frac{25 \sqrt{5}}{2} + 5 \cdot \sqrt{{(\frac{5}{2})}^{2} + {(5)}^{2}}$

Apply the product rule to $\frac{5}{2}$ .

$25 + \frac{25 \sqrt{5}}{2} + 5 \cdot \sqrt{\frac{5^{2}}{2^{2}} + {(5)}^{2}}$

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

$25 + \frac{25 \sqrt{5}}{2} + 5 \cdot \sqrt{\frac{25}{2^{2}} + {(5)}^{2}}$

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

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

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

$25 + \frac{25 \sqrt{5}}{2} + 5 \cdot \sqrt{\frac{25}{4} + 25}$

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

$25 + \frac{25 \sqrt{5}}{2} + 5 \cdot \sqrt{\frac{25}{4} + 25 \cdot \frac{4}{4}}$

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

$25 + \frac{25 \sqrt{5}}{2} + 5 \cdot \sqrt{\frac{25}{4} + \frac{25 \cdot 4}{4}}$

Combine the numerators over the common denominator.

$25 + \frac{25 \sqrt{5}}{2} + 5 \cdot \sqrt{\frac{25 + 25 \cdot 4}{4}}$

Simplify the numerator.

Multiply $25$ by $4$ .

$25 + \frac{25 \sqrt{5}}{2} + 5 \cdot \sqrt{\frac{25 + 100}{4}}$

Add $25$ and $100$ .

$25 + \frac{25 \sqrt{5}}{2} + 5 \cdot \sqrt{\frac{125}{4}}$

Rewrite $\sqrt{\frac{125}{4}}$ as $\frac{\sqrt{125}}{\sqrt{4}}$ .

$25 + \frac{25 \sqrt{5}}{2} + 5 \cdot \frac{\sqrt{125}}{\sqrt{4}}$

Simplify the numerator.

Rewrite $125$ as $5^{2} \cdot 5$ .

Factor $25$ out of $125$ .

$25 + \frac{25 \sqrt{5}}{2} + 5 \cdot \frac{\sqrt{25 (5)}}{\sqrt{4}}$

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

$25 + \frac{25 \sqrt{5}}{2} + 5 \cdot \frac{\sqrt{5^{2} \cdot 5}}{\sqrt{4}}$

Pull terms out from under the radical.

$25 + \frac{25 \sqrt{5}}{2} + 5 \cdot \frac{5 \sqrt{5}}{\sqrt{4}}$

Simplify the denominator.

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

$25 + \frac{25 \sqrt{5}}{2} + 5 \cdot \frac{5 \sqrt{5}}{\sqrt{2^{2}}}$

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

$25 + \frac{25 \sqrt{5}}{2} + 5 \cdot \frac{5 \sqrt{5}}{2}$

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

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

$25 + \frac{25 \sqrt{5}}{2} + \frac{5 (5 \sqrt{5})}{2}$

Multiply $5$ by $5$ .

$25 + \frac{25 \sqrt{5}}{2} + \frac{25 \sqrt{5}}{2}$

Step 4

Simplify terms.

Combine the numerators over the common denominator.

$25 + \frac{25 \sqrt{5} + 25 \sqrt{5}}{2}$

Add $25 \sqrt{5}$ and $25 \sqrt{5}$ .

$25 + \frac{50 \sqrt{5}}{2}$

Cancel the common factor of $50$ and $2$ .

Factor $2$ out of $50 \sqrt{5}$ .

$25 + \frac{2 (25 \sqrt{5})}{2}$

Cancel the common factors.

Factor $2$ out of $2$ .

$25 + \frac{2 (25 \sqrt{5})}{2 (1)}$

Cancel the common factor.

$25 + \frac{2 (25 \sqrt{5})}{2 \cdot 1}$

Rewrite the expression.

$25 + \frac{25 \sqrt{5}}{1}$

Divide $25 \sqrt{5}$ by $1$ .

$25 + 25 \sqrt{5}$

Step 5

Calculate the approximate solution to $4$ decimal places.

$80.9017$

Find the Surface Area pyramid (5)(5)(5)

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