Question

Find the area $A$ of a pyramid if height $h = 13 cm$ and volume $V = 182 cm$ .

Answer

$109.3698$

Explanation

STEP 1: find side $a$

To find side $a$ use formula:

V = \frac{a ^{2} \cdot h}{3}

After substituting $V = 182 cm$ and $h = 13 cm$ we have:

182 cm = \frac{a ^{2} \cdot ( 13 cm ) ^{4}}{3}

182 cm \cdot 3 = a^{2} \cdot (13 cm)^{4}

546 cm = 13 cm \cdot a^{2}

a^{2} = \frac{546 cm}{13 cm}

a^{2} \approx 13.369

a \approx 13.369

a \approx 3.6564

STEP 2: find base area $B$

To find base area $B$ use formula:

B = a^{2}

After substituting $a = 3.6564 cm^{0}$ we have:

B = 3.6564

B = 13.369

STEP 3: find slant height $s$

To find slant height $s$ use Pythagorean Theorem:

h^{2} + \frac{a ^{2}}{4} = s^{2}

After substituting $h = 13 cm$ and $a = 3.6564 cm^{0}$ we have:

(13 cm)^{2} + \frac{3.6564}{4} = s^{2}

169 cm^{2} + \frac{13.369}{4} = s^{2}

169 cm^{2} + 3.3422 = s^{2}

s^{2} = 172.3422 cm^{2}

s = 172.3422 cm^{2}

s = 13.1279 cm

STEP 4: find lateral surface $L$

To find lateral surface $L$ use formula:

L = 2 \cdot a \cdot s

After substituting $a = 3.6564 cm^{0}$ and $s = 13.1279 cm$ we have:

L = 7.3127 \cdot 13.1279 cm

L = 96.0008 cm

STEP 5: find total surface $A$

To find total surface $A$ use formula:

A = B + L

After substituting $B = 13.369 cm^{0}$ and $L = 96.0008 cm$ we have:

A = 13.369 + 96.0008 cm

A = 13.369 + 96.0008 cm

A = 109.3698

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