Start with base area times perpendicular height
A pyramid’s volume is one third of its base area times its perpendicular height: V = Gh/3. With a rectangular base 6 × 4 cm and height 9 cm, the base area is 24 cm² and volume is 72 cm³. The same volume relation works for an offset apex when height is measured perpendicular to the base plane. It does not mean that the same base and height determine the face slopes or exterior surface. Use the general-base method if the apex position is unspecified.
Perpendicular height is not the slant along a face
The h arrow ends at the base center only because this model explicitly declares a centered apex. L and W follow perpendicular base edges. For a 6 × 4 cm base and h 9 cm, the two face-altitude families differ.
| Quantity | Centered worked example | Meaning |
|---|---|---|
| Base L × W | 6 × 4 = 24 cm² | Rectangle area G |
| Perpendicular h | 9 cm | Normal to base, not apex-to-corner length |
| Volume Gh/3 | 24 × 9/3 = 72 cm³ | Also valid for offset apex with same base and perpendicular h |
| Face altitude across width | √(9²+2²) = √85 cm | About 9.21954 cm |
| Face altitude across length | √(9²+3²) = √90 cm | About 9.48683 cm |
| Closed centered surface | 24 + 6√85 + 4√90 cm² | About 117.265 cm²; requires centered apex |
Read diagram directions and numbered surface cues
- h: Perpendicular height. Perpendicular apex-to-base-center height; centered apex is explicitly declared.
- L: Base length. First rectangular base edge.
- W: Base width. Perpendicular rectangular base edge.
- 1 ✓ Included: Triangular side faces.
- 2 ✓ Included: Base polygon.
Solid outlines are schematic. Arrow endpoints identify the measurement; letters map to the table. ✓ included; × excluded; ? undetermined. Hidden edges use dashes.
Unspecified apex: preserve volume, leave surface undetermined
Only the base and apex reference planes are shown. This drawing does not assert an outline, centered apex or face slope.
| Known data | Valid result | Still unknown |
|---|---|---|
| Base area G 24 cm²; perpendicular h 9 cm | Volume 72 cm³ | Total surface and each face slant |
| Known V 80 cm³; G 30 cm² | h = 3V/G = 8 cm | Apex projection and surface |
| Centered square side 6 cm; face slant 5 cm | h = √(5²−3²) = 4 cm; V = 48 cm³ | Face slant must end at side midpoint, not a corner |
Read diagram directions and numbered surface cues
- G: Base area. Known area of the base plane; outline and apex projection are unspecified.
- h: Perpendicular height. Perpendicular distance from apex reference plane to base plane; never a slanted edge.
- 1 ? Undetermined: Exterior surface; apex position and face altitudes unspecified.
Solid outlines are schematic. Arrow endpoints identify the measurement; letters map to the table. ✓ included; × excluded; ? undetermined. Hidden edges use dashes. General-volume reference planes only: no centered apex or face slopes are asserted. Exterior surface is undetermined.
Declare a centered apex for surface and slant
The right rectangular or square method explicitly puts the apex directly above the base center. With base 6 × 4 cm and height 9 cm, its face altitudes are √(9²+2²) and √(9²+3²). The closed surface is about 117.265 cm². Moving the apex projection can change those face altitudes while preserving volume 72 cm³. Regular-polygon surface methods also require a centered apex; their common face slant is based on the base apothem, not the distance to a vertex.
Use a square face slant correctly
A centered square pyramid with base side 6 cm and face slant 5 cm has perpendicular height √(5²−3²) = 4 cm. Its volume is 6²×4/3 = 48 cm³ and its closed surface is 96 cm². The face slant is the altitude within a triangular face; it is not the lateral edge from apex to a base corner. For regular triangular, square or hexagonal bases, enter the side count and side length under the explicitly centered regular-pyramid model.
Solve the supported inverse and check units
With known base area 30 cm² and volume 80 cm³, perpendicular height is 3×80/30 = 8 cm. Conversely, base area 30 cm² and height 8 cm yield volume 80 cm³. A base area has square units, height has length units and the result is cubic. General-base total surface remains undetermined; adding a perimeter alone does not locate the apex or supply each face altitude. The primary answer follows your requested volume, height or surface status, with an auditable model and reusable contextual report.