Here are the step-by-step solutions to find the surface area for each pyramid.
1. Square Pyramid (Base 8 m, Slant Height 11.7 m)
*
Step 1: Find the Perimeter ($p$) of the base.
The base is a square with side length 8 m.
$p = 4 \times 8 = 32$ m.
*
Step 2: Find the Area ($B$) of the base.
$B = 8 \times 8 = 64$ m².
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Step 3: Calculate Lateral Area ($LA$).
Formula: $LA = \frac{1}{2} \times l \times p$
$LA = \frac{1}{2} \times 11.7 \times 32$
$LA = 11.7 \times 16 = 187.2$ m².
*
Step 4: Calculate Total Surface Area ($SA$).
Formula: $SA = LA + B$
$SA = 187.2 + 64 = 251.2$ m².
2. Square Pyramid (Base 5 in, Slant Height 7.4 in)
*
Step 1: Find the Perimeter ($p$) of the base.
The base is a square with side length 5 in.
$p = 4 \times 5 = 20$ in.
*
Step 2: Find the Area ($B$) of the base.
$B = 5 \times 5 = 25$ in².
*
Step 3: Calculate Lateral Area ($LA$).
Formula: $LA = \frac{1}{2} \times l \times p$
$LA = \frac{1}{2} \times 7.4 \times 20$
$LA = 7.4 \times 10 = 74$ in².
*
Step 4: Calculate Total Surface Area ($SA$).
Formula: $SA = LA + B$
$SA = 74 + 25 = 99$ in².
3. Square Pyramid (Base 16 ft, Height 3 ft)
*Note: We are given the vertical height, so we must first calculate the slant height.*
*
Step 1: Find the Slant Height ($l$).
Imagine a right triangle inside the pyramid.
- One leg is the height ($h = 3$).
- The other leg is half the base width ($16 / 2 = 8$).
- Use the Pythagorean theorem ($a^2 + b^2 = c^2$):
$3^2 + 8^2 = l^2$
$9 + 64 = l^2$
$73 = l^2$
$l = \sqrt{73} \approx 8.54$ ft.
*
Step 2: Find the Perimeter ($p$) of the base.
$p = 4 \times 16 = 64$ ft.
*
Step 3: Find the Area ($B$) of the base.
$B = 16 \times 16 = 256$ ft².
*
Step 4: Calculate Lateral Area ($LA$).
Using the approximate slant height of 8.54:
$LA = \frac{1}{2} \times 8.54 \times 64$
$LA = 8.54 \times 32 \approx 273.28$ ft².
*(If using the rounded value 8.5 from your notes: $LA = 0.5 \times 8.5 \times 64 = 272$ ft²)*.
*
Step 5: Calculate Total Surface Area ($SA$).
Using the more precise calculation:
$SA = 273.28 + 256 = 529.28$ ft².
*(Using the rounded value 8.5: $SA = 272 + 256 = 528$ ft²)*.
Final Answer:
1. 251.2 m²
2. 99 in²
3. Approximately 529.3 ft² (or 528 ft² if using the rounded slant height of 8.5)
Parent Tip: Review the logic above to help your child master the concept of surface area of pyramids worksheet.
