Basic Math Examples

 $\begin{array}{r} h = & 6 \\ l = & 4 \\ w = & 3 \end{array}$

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 $l = 4$ , the width $w = 3$ , and the height $h = 6$ into the formula for surface area of a pyramid.

$4 \cdot 3 + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(6)}^{2}} + 4 \cdot \sqrt{{(\frac{3}{2})}^{2} + {(6)}^{2}}$

Step 3

Simplify each term.

Tap for more steps...

Step 3.1

Multiply $4$ by $3$ .

$12 + 3 \cdot \sqrt{{(\frac{4}{2})}^{2} + {(6)}^{2}} + 4 \cdot \sqrt{{(\frac{3}{2})}^{2} + {(6)}^{2}}$

Step 3.2

Divide $4$ by $2$ .

$12 + 3 \cdot \sqrt{2^{2} + {(6)}^{2}} + 4 \cdot \sqrt{{(\frac{3}{2})}^{2} + {(6)}^{2}}$

Step 3.3

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

$12 + 3 \cdot \sqrt{4 + {(6)}^{2}} + 4 \cdot \sqrt{{(\frac{3}{2})}^{2} + {(6)}^{2}}$

Step 3.4

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

$12 + 3 \cdot \sqrt{4 + 36} + 4 \cdot \sqrt{{(\frac{3}{2})}^{2} + {(6)}^{2}}$

Step 3.5

Add $4$ and $36$ .

$12 + 3 \cdot \sqrt{40} + 4 \cdot \sqrt{{(\frac{3}{2})}^{2} + {(6)}^{2}}$

Step 3.6

Rewrite $40$ as $2^{2} \cdot 10$ .

Tap for more steps...

Step 3.6.1

Factor $4$ out of $40$ .

$12 + 3 \cdot \sqrt{4 (10)} + 4 \cdot \sqrt{{(\frac{3}{2})}^{2} + {(6)}^{2}}$

Step 3.6.2

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

$12 + 3 \cdot \sqrt{2^{2} \cdot 10} + 4 \cdot \sqrt{{(\frac{3}{2})}^{2} + {(6)}^{2}}$

Step 3.7

Pull terms out from under the radical.

$12 + 3 \cdot (2 \sqrt{10}) + 4 \cdot \sqrt{{(\frac{3}{2})}^{2} + {(6)}^{2}}$

Step 3.8

Multiply $2$ by $3$ .

$12 + 6 \sqrt{10} + 4 \cdot \sqrt{{(\frac{3}{2})}^{2} + {(6)}^{2}}$

Step 3.9

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

$12 + 6 \sqrt{10} + 4 \cdot \sqrt{\frac{3^{2}}{2^{2}} + {(6)}^{2}}$

Step 3.10

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

$12 + 6 \sqrt{10} + 4 \cdot \sqrt{\frac{9}{2^{2}} + {(6)}^{2}}$

Step 3.11

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

$12 + 6 \sqrt{10} + 4 \cdot \sqrt{\frac{9}{4} + {(6)}^{2}}$

Step 3.12

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

$12 + 6 \sqrt{10} + 4 \cdot \sqrt{\frac{9}{4} + 36}$

Step 3.13

To write $36$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$12 + 6 \sqrt{10} + 4 \cdot \sqrt{\frac{9}{4} + 36 \cdot \frac{4}{4}}$

Step 3.14

Combine $36$ and $\frac{4}{4}$ .

$12 + 6 \sqrt{10} + 4 \cdot \sqrt{\frac{9}{4} + \frac{36 \cdot 4}{4}}$

Step 3.15

Combine the numerators over the common denominator.

$12 + 6 \sqrt{10} + 4 \cdot \sqrt{\frac{9 + 36 \cdot 4}{4}}$

Step 3.16

Simplify the numerator.

Tap for more steps...

Step 3.16.1

Multiply $36$ by $4$ .

$12 + 6 \sqrt{10} + 4 \cdot \sqrt{\frac{9 + 144}{4}}$

Step 3.16.2

Add $9$ and $144$ .

$12 + 6 \sqrt{10} + 4 \cdot \sqrt{\frac{153}{4}}$

Step 3.17

Rewrite $\sqrt{\frac{153}{4}}$ as $\frac{\sqrt{153}}{\sqrt{4}}$ .

$12 + 6 \sqrt{10} + 4 \cdot \frac{\sqrt{153}}{\sqrt{4}}$

Step 3.18

Simplify the numerator.

Tap for more steps...

Step 3.18.1

Rewrite $153$ as $3^{2} \cdot 17$ .

Tap for more steps...

Step 3.18.1.1

Factor $9$ out of $153$ .

$12 + 6 \sqrt{10} + 4 \cdot \frac{\sqrt{9 (17)}}{\sqrt{4}}$

Step 3.18.1.2

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

$12 + 6 \sqrt{10} + 4 \cdot \frac{\sqrt{3^{2} \cdot 17}}{\sqrt{4}}$

Step 3.18.2

Pull terms out from under the radical.

$12 + 6 \sqrt{10} + 4 \cdot \frac{3 \sqrt{17}}{\sqrt{4}}$

Step 3.19

Simplify the denominator.

Tap for more steps...

Step 3.19.1

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

$12 + 6 \sqrt{10} + 4 \cdot \frac{3 \sqrt{17}}{\sqrt{2^{2}}}$

Step 3.19.2

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

$12 + 6 \sqrt{10} + 4 \cdot \frac{3 \sqrt{17}}{2}$

Step 3.20

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.20.1

Factor $2$ out of $4$ .

$12 + 6 \sqrt{10} + 2 (2) \cdot \frac{3 \sqrt{17}}{2}$

Step 3.20.2

Cancel the common factor.

$12 + 6 \sqrt{10} + 2 \cdot 2 \cdot \frac{3 \sqrt{17}}{2}$

Step 3.20.3

Rewrite the expression.

$12 + 6 \sqrt{10} + 2 \cdot (3 \sqrt{17})$

Step 3.21

Multiply $3$ by $2$ .

$12 + 6 \sqrt{10} + 6 \sqrt{17}$

Step 4

Calculate the approximate solution to $4$ decimal places.

$55.7123$