Triangular Pyramid Surface Area 1A49E1

1. **Problem Statement:** Calculate the surface area of each triangular pyramid given the base edges and slant heights. 2. **Formula:** The surface area $SA$ of a triangular pyramid is the sum of the base area and the lateral area. For a triangular base with edges $a$, $b$, and $c$, and slant height $l$, the lateral area can be approximated by $\frac{1}{2} \times \text{perimeter} \times l$ if the slant height is consistent. 3. **Step-by-step for each pyramid:** **Top-left pyramid:** - Base edges: $7\,m$, $7\,m$ (assumed isosceles triangle) - Slant height: $10\,m$ - Base area $= \frac{1}{2} \times 7 \times \sqrt{7^2 - \left(\frac{7}{2}\right)^2} = \frac{1}{2} \times 7 \times \sqrt{49 - 12.25} = \frac{1}{2} \times 7 \times \sqrt{36.75} = \frac{1}{2} \times 7 \times 6.06 = 21.21\,m^2$ - Perimeter $= 7 + 7 + 7 = 21\,m$ - Lateral area $= \frac{1}{2} \times 21 \times 10 = 105\,m^2$ - Surface area $= 21.21 + 105 = 126.21\,m^2$ **Top-right pyramid:** - Base edges: $8\,ft$, $8\,ft$ - Slant height: $12\,ft$ - Base area $= \frac{1}{2} \times 8 \times \sqrt{8^2 - \left(\frac{8}{2}\right)^2} = \frac{1}{2} \times 8 \times \sqrt{64 - 16} = 4 \times \sqrt{48} = 4 \times 6.93 = 27.71\,ft^2$ - Perimeter $= 8 + 8 + 8 = 24\,ft$ - Lateral area $= \frac{1}{2} \times 24 \times 12 = 144\,ft^2$ - Surface area $= 27.71 + 144 = 171.71\,ft^2$ **Bottom-left pyramid:** - Base edges: $5\,ft$, $5\,ft$ - Slant height: $3\,ft$ - Another measure: $4.3\,ft$ (assumed height of base triangle) - Base area $= \frac{1}{2} \times 5 \times 4.3 = 10.75\,ft^2$ - Perimeter $= 5 + 5 + 5 = 15\,ft$ - Lateral area $= \frac{1}{2} \times 15 \times 3 = 22.5\,ft^2$ - Surface area $= 10.75 + 22.5 = 33.25\,ft^2$ **Bottom-right pyramid (1):** - Base edges: $2\,yd$, $2\,yd$, $1.7\,yd$ - Slant height: $4\,yd$ - Base semi-perimeter $s = \frac{2 + 2 + 1.7}{2} = 2.85\,yd$ - Base area $= \sqrt{s(s-2)(s-2)(s-1.7)} = \sqrt{2.85 \times 0.85 \times 0.85 \times 1.15} = \sqrt{2.37} = 1.54\,yd^2$ - Perimeter $= 2 + 2 + 1.7 = 5.7\,yd$ - Lateral area $= \frac{1}{2} \times 5.7 \times 4 = 11.4\,yd^2$ - Surface area $= 1.54 + 11.4 = 12.94\,yd^2$ **Bottom-left pyramid (2):** - Base edges: $10\,mm$, $10\,mm$, $8.7\,mm$ - Slant height: $7\,mm$ - Semi-perimeter $s = \frac{10 + 10 + 8.7}{2} = 14.35\,mm$ - Base area $= \sqrt{14.35(14.35-10)(14.35-10)(14.35-8.7)} = \sqrt{14.35 \times 4.35 \times 4.35 \times 5.65} = \sqrt{1927.5} = 43.91\,mm^2$ - Perimeter $= 10 + 10 + 8.7 = 28.7\,mm$ - Lateral area $= \frac{1}{2} \times 28.7 \times 7 = 100.45\,mm^2$ - Surface area $= 43.91 + 100.45 = 144.36\,mm^2$ **Bottom-right pyramid (2):** - Base edges: $6\,cm$, $6\,cm$, $5.2\,cm$ - Slant height: $9\,cm$ - Semi-perimeter $s = \frac{6 + 6 + 5.2}{2} = 8.6\,cm$ - Base area $= \sqrt{8.6(8.6-6)(8.6-6)(8.6-5.2)} = \sqrt{8.6 \times 2.6 \times 2.6 \times 3.4} = \sqrt{197.7} = 14.06\,cm^2$ - Perimeter $= 6 + 6 + 5.2 = 17.2\,cm$ - Lateral area $= \frac{1}{2} \times 17.2 \times 9 = 77.4\,cm^2$ - Surface area $= 14.06 + 77.4 = 91.46\,cm^2$