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Calculate volume and surface area of octagonal prisms using side length and height
Volume: V = 2×(1+√2)×a²×h
Surface Area: SA = 4×a×(1+√2)×(a+h)
where a is the side length and h is the prism height
An octagonal prism is a 3D solid with two parallel regular octagon bases (8-sided polygons) connected by 8 rectangular faces. Regular octagons have all sides equal length and all interior angles equal to 135°. Common examples include some columns, towers, and architectural features.
Volume = 2(1+√2) × s² × h, where s is the side length and h is the height. First calculate the octagon base area [2(1+√2) × s² ≈ 4.828s²], then multiply by the prism height. For example, s=5cm and h=10cm gives V ≈ 4.828 × 25 × 10 = 1207 cm³.
Area = 2(1+√2) × s² ≈ 4.828 × s², where s is the side length. This formula comes from dividing the octagon into 8 isosceles triangles or alternatively using the apothem method. The factor 2(1+√2) equals approximately 4.828427.
Surface Area = 2 × (octagon area) + 8 × (side rectangle area). Calculate: 2 × [2(1+√2)s²] + 8(s × h). This accounts for the two octagonal bases plus the 8 rectangular faces connecting them.
Octagonal prisms appear in architecture (columns, towers, gazebos), design (stop signs are flat octagons), engineering structures, decorative elements, packaging, and some optical prisms. The 8-sided symmetry provides structural strength and aesthetic appeal.
A regular octagonal prism has bases that are regular octagons (all sides and angles equal). An irregular octagonal prism has octagonal bases with unequal sides or angles. The formulas provided here apply only to regular octagonal prisms.
The space diagonal (longest interior line) can be found using the 3D distance formula combined with the octagon's dimensions. For a regular octagon with side s, the diameter is s(1+√2), so diagonal ≈ √[(s(1+√2))² + h²].
No, by definition a prism must have congruent (identical) parallel bases. If the top and bottom octagons differ in size, it would be an octagonal frustum or truncated pyramid, not a prism, and would require different formulas.