Surface area problems for pyramids and cones with labeled measurements.

Here are the step-by-step solutions for each problem on your homework sheet.

Problem 1: Square Pyramid

Goal: Find the total surface area.
Given:
* Base side ($s$) = $11$ in
* Slant height ($l$) = $20.7$ in

Step 1: Find the area of the square base.
$$Area = s \times s$$
$$11 \times 11 = 121 \text{ sq in}$$

Step 2: Find the lateral area (the 4 triangular sides).
The formula for the lateral area of a regular pyramid is $\frac{1}{2} \times \text{Perimeter} \times \text{Slant Height}$.
* Perimeter of base = $11 \times 4 = 44$ in
* Lateral Area = $\frac{1}{2} \times 44 \times 20.7$
* Lateral Area = $22 \times 20.7 = 455.4 \text{ sq in}$

Step 3: Add them together.
$$Total Surface Area = 121 + 455.4 = 576.4$$

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Problem 2: Square Pyramid

Goal: Find the total surface area.
Given:
* Base side ($s$) = $6$ yd
* Slant height ($l$) = $8.2$ yd
*(Note: The measurement $5.2$ yd is the vertical height inside the pyramid, which we do not need for surface area).*

Step 1: Find the area of the square base.
$$6 \times 6 = 36 \text{ sq yd}$$

Step 2: Find the lateral area.
* Perimeter of base = $6 \times 4 = 24$ yd
* Lateral Area = $\frac{1}{2} \times 24 \times 8.2$
* Lateral Area = $12 \times 8.2 = 98.4 \text{ sq yd}$

Step 3: Add them together.
$$Total Surface Area = 36 + 98.4 = 134.4$$

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Problem 3: Cone

Goal: Find the total surface area.
Given:
* Diameter = $8$ mm (so Radius $r = 4$ mm)
* Slant height ($l$) = $13.6$ mm

Step 1: Find the area of the circular base.
Formula: $\pi r^2$
$$\pi \times 4^2 = 16\pi \approx 50.27 \text{ sq mm}$$

Step 2: Find the lateral area (the curved part).
Formula: $\pi r l$
$$\pi \times 4 \times 13.6 = 54.4\pi \approx 170.90 \text{ sq mm}$$

Step 3: Add them together.
$$Total Surface Area = 50.27 + 170.90 = 221.17$$

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Problem 4: Square Pyramid

Goal: Find the total surface area.
Given:
* Base side ($s$) = $4$ m
* Slant height ($l$) = $11.1$ m
*(Note: The measurements $3.5$ m and $4$ m inside are heights/depths that are not needed for surface area).*

Step 1: Find the area of the square base.
$$4 \times 4 = 16 \text{ sq m}$$

Step 2: Find the lateral area.
* Perimeter of base = $4 \times 4 = 16$ m
* Lateral Area = $\frac{1}{2} \times 16 \times 11.1$
* Lateral Area = $8 \times 11.1 = 88.8 \text{ sq m}$

Step 3: Add them together.
$$Total Surface Area = 16 + 88.8 = 104.8$$

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Final Answer:
1. 576.4 in²
2. 134.4 yd²
3. 221.17 mm²
4. 104.8 m²