1. **State the problem:** We need to find the volume and surface area of the triangular pyramid with edges 21 cm, 18.5 cm, and base side 18 cm. 2. **Identify the shape:** The volume formula for a triangular pyramid (tetrahedron) is $$V = \frac{1}{3} \times \text{Base Area} \times \text{Height}$$. Since only edges are given, we assume the triangular base has side 18 cm, and the other edges 21 cm and 18.5 cm form the apex height position. 3. **Calculate the base area:** Assume the base is an equilateral triangle with side $$a = 18$$ cm. Base area $$A = \frac{\sqrt{3}}{4} a^2 = \frac{\sqrt{3}}{4} \times 18^2 = \frac{\sqrt{3}}{4} \times 324 = 81\sqrt{3} \text{ cm}^2$$. 4. **Calculate the height of the pyramid:** We use edges 21 cm (from apex to one vertex) and 18.5 cm (from apex to another vertex) to find height. Let the height of the pyramid be $$h$$. Since exact location of apex relative to base is not fully defined, we approximate height using the Pythagorean theorem assuming apex is directly above centroid of base: Centroid to vertex distance for an equilateral triangle is $$\frac{2}{3} \times$ height of triangle $$= \frac{2}{3} \times \frac{\sqrt{3}}{2} \times 18 = 6\sqrt{3}$$ cm. Assuming edges 21 cm and 18.5 cm meet at apex, we take an average height estimate: Height $$h \approx \sqrt{21^2 - (6\sqrt{3})^2} = \sqrt{441 - 108} = \sqrt{333} \approx 18.25$$ cm. 5. **Calculate volume:** $$V = \frac{1}{3} \times 81\sqrt{3} \times 18.25 = 27 \times 18.25 \times \sqrt{3} \approx 27 \times 18.25 \times 1.732 \approx 851.5 \text{ cm}^3$$. 6. **Calculate surface area:** Surface area = Base area + area of 3 triangular faces. Base area = $$81\sqrt{3} \approx 140.3$$ cm². Approximate lateral faces using triangle sides: - Two faces with edges (18 cm, 21 cm, unknown height) - One face with edges (18 cm, 18.5 cm, unknown height) Using Heron's formula for each with semi-perimeter $$s$$ and sides $$a, b, c$$: Assuming lateral faces are roughly triangles with sides 18 cm, 21 cm, and 18.5 cm. Semi-perimeter $$s = \frac{18 + 21 + 18.5}{2} = 28.75$$ cm. Area $$= \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{28.75(28.75-18)(28.75-21)(28.75-18.5)} = \sqrt{28.75 \times 10.75 \times 7.75 \times 10.25} \approx \sqrt{24615} \approx 157$$ cm². Assuming 3 faces with roughly similar area, total lateral area $$\approx 3 \times 157 = 471$$ cm². Total surface area $$\approx 140.3 + 471 = 611.3$$ cm². **Final answers:** Volume of pyramid $$\approx 851.5$$ cm³. Surface area of pyramid $$\approx 611.3$$ cm².