1. **State the problem:** We are given a pyramid with base edge length $5$ ft and slant height $13$ ft. We need to find the lateral area, surface area, and volume. 2. **Identify given values and variables:** - Perimeter of base, $P = 2$ (given, but this seems inconsistent with base edge 5 ft; we will calculate $P$ based on base edge) - Slant height, $l = 3$ (given, but the image shows slant height $13$ ft; we will use $l=13$ ft as per the label) - Base edge length = $5$ ft - Base area, $B = ?$ - Apothem, $a = ?$ 3. **Calculate perimeter $P$ of the base:** Assuming the base is a square (common for pyramids), perimeter is $$P = 4 \times 5 = 20$$ 4. **Calculate base area $B$:** For a square base, $$B = 5 \times 5 = 25$$ 5. **Calculate lateral area:** Formula for lateral area of a pyramid: $$\text{Lateral Area} = \frac{1}{2} P l$$ Substitute values: $$= \frac{1}{2} \times 20 \times 13 = 10 \times 13 = 130$$ 6. **Calculate surface area:** Surface area is lateral area plus base area: $$\text{Surface Area} = \text{Lateral Area} + B = 130 + 25 = 155$$ 7. **Calculate volume:** Volume formula for pyramid: $$V = \frac{1}{3} B h$$ We need height $h$, not slant height. Use Pythagoras theorem in the triangle formed by height, apothem, and slant height. Calculate apothem $a$ (distance from center to midpoint of base edge): $$a = \frac{5}{2} = 2.5$$ Use Pythagoras theorem: $$l^2 = h^2 + a^2$$ $$13^2 = h^2 + 2.5^2$$ $$169 = h^2 + 6.25$$ $$h^2 = 169 - 6.25 = 162.75$$ $$h = \sqrt{162.75} \approx 12.76$$ Calculate volume: $$V = \frac{1}{3} \times 25 \times 12.76 = \frac{25 \times 12.76}{3} = \frac{319}{3} \approx 106.33$$ **Final answers:** - Lateral Area = 130 - Surface Area = 155 - Volume = 106.33