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Hexahedron

Hexahedron

Any polyhedron with six faces, all of which are quadrilaterals. Note: A regular hexahedron is a cube.

Key Formula

Vcube=s3V_{\text{cube}} = s^3
Where:
  • VcubeV_{\text{cube}} = Volume of a regular hexahedron (cube)
  • ss = Length of one edge of the cube

Worked Example

Problem: A regular hexahedron (cube) has an edge length of 4 cm. Find its volume, surface area, and verify the number of faces, edges, and vertices.
Step 1: Identify the shape. A regular hexahedron is a cube — all six faces are congruent squares with side length 4 cm.
Step 2: Count faces, edges, and vertices. A cube has 6 faces, 12 edges, and 8 vertices. Verify with Euler's formula:
V−E+F=8−12+6=2✓V - E + F = 8 - 12 + 6 = 2 \checkmark
Step 3: Calculate the volume.
V=s3=43=64 cm3V = s^3 = 4^3 = 64 \text{ cm}^3
Step 4: Calculate the surface area. There are 6 square faces, each with area s2s^2.
SA=6s2=6×42=6×16=96 cm2SA = 6s^2 = 6 \times 4^2 = 6 \times 16 = 96 \text{ cm}^2
Answer: The cube has 6 faces, 12 edges, and 8 vertices. Its volume is 64 cm³ and its surface area is 96 cm².

Another Example

Problem: A rectangular box (cuboid) has dimensions 3 cm × 5 cm × 7 cm. Is it a hexahedron? Find its volume.
Step 1: Count the faces. A rectangular box has a top, bottom, front, back, left, and right face — that is 6 faces total, and each face is a rectangle (a type of quadrilateral). So yes, a cuboid is a hexahedron.
Step 2: Calculate its volume using length × width × height.
V=3×5×7=105 cm3V = 3 \times 5 \times 7 = 105 \text{ cm}^3
Answer: Yes, a cuboid is a hexahedron because it has exactly 6 quadrilateral faces. Its volume is 105 cm³.

Frequently Asked Questions

Is a cube the only type of hexahedron?
No. A cube (regular hexahedron) is the most symmetric type, but any polyhedron with six quadrilateral faces qualifies. A rectangular box (cuboid) is a hexahedron. So is a parallelepiped, where all six faces are parallelograms. There are also oblique and irregular hexahedra whose faces are general quadrilaterals.
How many edges and vertices does a hexahedron have?
A typical hexahedron like a cube or cuboid has 12 edges and 8 vertices. You can verify this with Euler's formula for polyhedra: V−E+F=2V - E + F = 2, giving 8−12+6=28 - 12 + 6 = 2.

Hexahedron vs. Cube

Every cube is a hexahedron, but not every hexahedron is a cube. A cube requires all six faces to be congruent squares and all angles to be right angles. A general hexahedron only requires six quadrilateral faces, which can be rectangles, parallelograms, or other quadrilaterals of different sizes.

Why It Matters

The hexahedron is one of the five Platonic solids (in its regular form as a cube) and appears throughout architecture, packaging, and engineering. Understanding that 'hexahedron' simply means 'six faces' helps you classify and compare 3D shapes systematically. Many real-world objects — boxes, rooms, bricks — are hexahedra, making this one of the most practically encountered polyhedra.

Common Mistakes

Mistake: Assuming 'hexahedron' means 'hexagon-based' because the prefixes sound similar.
Correction: The prefix 'hexa-' means six, but a hexahedron has six faces (quadrilaterals), not six-sided faces. A hexagon is a six-sided polygon — a completely different use of the same prefix.
Mistake: Thinking only cubes count as hexahedra.
Correction: A cube is the regular hexahedron, but cuboids, parallelepipeds, and other six-faced polyhedra with quadrilateral faces are all hexahedra. 'Regular' is what restricts the shape to a cube.

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