Solid Geometry: Prism and Pyramid

Prism

The bases of a prism are congruent polygons, and its lateral faces are parallelograms. In a right prism, the lateral edges are perpendicular to the base.

• Volume: V = S · H (S is the base area).
• Lateral surface area of a right prism: lateral surface area = base perimeter · H. Total surface area = lateral surface area + 2 · base area.
• Rectangular parallelepiped (a, b, c): V = abc, diagonal d = √(a² + b² + c²), total surface area 2(ab + bc + ac).
• Cube (edge a): V = a³, S = 6a², diagonal d = a√3.
• Area of a regular triangle a²√3 / 4.

Pyramid

The base of a pyramid is a polygon, and its lateral faces are triangles sharing a common vertex.

Quantity	Formula
Volume	V = (1/3) · S · H
Lateral surface area of a regular pyramid	(1/2) · P · l (P is the base perimeter, l is the slant height)
Total surface area	lateral surface area + base area
Volume of a frustum	V = (H/3) · (S₁ + S₂ + √(S₁S₂))

• In a regular pyramid, the base is a regular polygon, and the altitude falls on the center of the base.
• The slant height, the altitude, and the distance from the center of the base to a side (r) form a right triangle: l² = H² + r².
• Do not forget the 1/3 in the pyramid volume: it is one third of the prism with the same base and height.
• The lateral surface area needs the slant height, not the altitude.

1. Identify the type of solid: right prism, regular pyramid
2. Find the base area and the height
3. If the slant height is needed, apply the Pythagorean theorem
4. Substitute into the volume or surface area formula

Examples

• Regular quadrilateral pyramid: base side 6, height 4. The distance from the center of the base to a side is 3, slant height l = √(9 + 16) = 5. Lateral surface area (1/2) · 24 · 5 = 60, total surface area 36 + 60 = 96.
• Parallelepiped 3, 4, 12: d = √(9 + 16 + 144) = 13.
• Pyramid with base area 36 and height 9: V = 36 · 9 / 3 = 108.