Question

1. calculate the surface area of each square based pyramid.

a.
b.

1. determine the surface area of the square pyramid shown below. express your solution in square centimeters
2. calculate the volume of each pyramid shown below:

a.
b.

Explanation:

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Problem 4a: Square Pyramid Surface Area

Step1: Find base area

Base is square: $s=17\,\text{m}$
$A_{\text{base}} = s^2 = 17^2 = 289\,\text{m}^2$

Step2: Find one triangular face area

Slant height $l=39\,\text{m}$, base $s=17\,\text{m}$
$A_{\text{face}} = \frac{1}{2} \times s \times l = \frac{1}{2} \times 17 \times 39 = 331.5\,\text{m}^2$

Step3: Total lateral surface area

4 congruent triangular faces:
$A_{\text{lateral}} = 4 \times 331.5 = 1326\,\text{m}^2$

Step4: Total surface area

Add base and lateral areas:
$A_{\text{total}} = 289 + 1326 = 1615\,\text{m}^2$

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Problem 4b: Square Pyramid Surface Area

Step1: Find base area

Base is square: $s=24\,\text{in}$
$A_{\text{base}} = s^2 = 24^2 = 576\,\text{in}^2$

Step2: Find one triangular face area

Slant height $l=45\,\text{in}$, base $s=24\,\text{in}$
$A_{\text{face}} = \frac{1}{2} \times s \times l = \frac{1}{2} \times 24 \times 45 = 540\,\text{in}^2$

Step3: Total lateral surface area

4 congruent triangular faces:
$A_{\text{lateral}} = 4 \times 540 = 2160\,\text{in}^2$

Step4: Total surface area

Add base and lateral areas:
$A_{\text{total}} = 576 + 2160 = 2736\,\text{in}^2$

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Problem 5: Square Pyramid Surface Area (convert to cm²)

Step1: Convert units to cm

$28\,\text{mm} = 2.8\,\text{cm}$, $38\,\text{mm}=3.8\,\text{cm}$

Step2: Find base area

Base is square: $s=2.8\,\text{cm}$
$A_{\text{base}} = s^2 = 2.8^2 = 7.84\,\text{cm}^2$

Step3: Find one triangular face area

Slant height $l=3.8\,\text{cm}$, base $s=2.8\,\text{cm}$
$A_{\text{face}} = \frac{1}{2} \times s \times l = \frac{1}{2} \times 2.8 \times 3.8 = 5.32\,\text{cm}^2$

Step4: Total lateral surface area

4 congruent triangular faces:
$A_{\text{lateral}} = 4 \times 5.32 = 21.28\,\text{cm}^2$

Step5: Total surface area

Add base and lateral areas:
$A_{\text{total}} = 7.84 + 21.28 = 29.12\,\text{cm}^2$

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Problem 6a: Rectangular Pyramid Volume

Step1: Find base area

Base is rectangle: $l=13\,\text{cm}$, $w=6\,\text{cm}$
$A_{\text{base}} = l \times w = 13 \times 6 = 78\,\text{cm}^2$

Step2: Calculate volume

Height $h=14\,\text{cm}$, use $V=\frac{1}{3}A_{\text{base}}h$
$V = \frac{1}{3} \times 78 \times 14 = 364\,\text{cm}^3$

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Problem 6b: Square Pyramid Volume

Step1: Find base area

Base is square: $s=21\,\text{m}$
$A_{\text{base}} = s^2 = 21^2 = 441\,\text{m}^2$

Step2: Calculate volume

Height $h=29\,\text{m}$, use $V=\frac{1}{3}A_{\text{base}}h$
$V = \frac{1}{3} \times 441 \times 29 = 4263\,\text{m}^3$

Answer:

4a. $\boldsymbol{1615\,\text{m}^2}$
4b. $\boldsymbol{2736\,\text{in}^2}$

1. $\boldsymbol{29.12\,\text{cm}^2}$

6a. $\boldsymbol{364\,\text{cm}^3}$
6b. $\boldsymbol{4263\,\text{m}^3}$