Calculate pyramid volume, total surface area, lateral area, and slant heights for square and rectangular pyramids with step-by-step steps.
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A pyramid is a polyhedron formed by connecting a polygonal base and a point called the apex. Each base edge and the apex form a triangular lateral face.
V = \frac{1}{3}A_{\text{base}} h = \frac{1}{3}lwh, \quad s_l = \sqrt{h^2 + (w/2)^2}, \quad s_w = \sqrt{h^2 + (l/2)^2}Volume is one-third base area times vertical height. Lateral surface area is the sum of the four triangular side faces calculated from slant heights.
Scenario: Find volume and surface area of a square pyramid with base 6×6 and height 4
Base Area: A_base = 6 × 6 = 36
Volume: V = ⅓ × 36 × 4 = 48
Slant height: s = √(4² + 3²) = √(16 + 9) = 5
Lateral area: 4 triangles = 4 × (½ × 6 × 5) = 60
Total Surface Area: A_total = 36 + 60 = 96
V = (1/3) × Base Area × Height.
Apply the Pythagorean theorem to the right triangle inside: slant height = √(height² + (half_base)²).
A square pyramid has 5 faces (1 square base + 4 triangular sides), 8 edges, and 5 vertices.
A tetrahedron is a triangular pyramid where all four faces (including the base) are triangles.
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