hexahedron

Hexahedron in Math: Definition and Properties

Definition

In mathematics, a hexahedron is a three-dimensional geometric shape that consists of six faces, twelve edges, and eight vertices. It is also known as a cuboid or a rectangular prism. The term "hexahedron" is derived from the Greek words "hexa" meaning six and "hedra" meaning face.

Knowledge Points

A hexahedron encompasses several important concepts in geometry, including:

1. Faces: A hexahedron has six faces, which are all rectangles.
2. Edges: It has twelve edges, where each edge connects two vertices.
3. Vertices: A hexahedron has eight vertices, which are the corners of the shape.
4. Diagonals: The hexahedron has three types of diagonals - face diagonals, space diagonals, and body diagonals.
5. Surface Area: The total surface area of a hexahedron can be calculated by summing the areas of all six faces.
6. Volume: The volume of a hexahedron is obtained by multiplying the length, width, and height of the shape.

Formula or Equation

The formula for calculating the surface area of a hexahedron is given by: Surface Area = 2lw + 2lh + 2wh

The formula for calculating the volume of a hexahedron is given by: Volume = lwh

Here, l represents the length, w represents the width, and h represents the height of the hexahedron.

Application of the Formula

To apply the formula for surface area, measure the length, width, and height of the hexahedron. Substitute these values into the surface area formula and calculate the result.

To apply the formula for volume, measure the length, width, and height of the hexahedron. Substitute these values into the volume formula and compute the result.

Symbol for Hexahedron

There is no specific symbol exclusively used for representing a hexahedron. However, the term "hexahedron" itself can be used to refer to this geometric shape.

Methods for Hexahedron

There are various methods to work with hexahedrons, including:

1. Determining the length, width, and height of a hexahedron using given information.
2. Calculating the surface area and volume of a hexahedron using the provided formulas.
3. Identifying the diagonals of a hexahedron and calculating their lengths.
4. Exploring the relationships between the faces, edges, and vertices of a hexahedron.

Solved Examples

1. Example 1: Find the surface area and volume of a hexahedron with length 5 cm, width 3 cm, and height 4 cm.

   Solution: Surface Area = 2lw + 2lh + 2wh = 2(5)(3) + 2(5)(4) + 2(3)(4) = 30 + 40 + 24 = 94 cm²

   Volume = lwh = (5)(3)(4) = 60 cm³

2. Example 2: Given a hexahedron with a surface area of 120 cm² and a length of 6 cm. Find the width and height of the hexahedron.

   Solution: Surface Area = 2lw + 2lh + 2wh 120 = 2(6)(w) + 2(6)(h) + 2(w)(h) Simplifying the equation, we get: 60 = 6w + 6h + wh

   Since we have only one equation and two variables, we cannot determine the exact values of width and height without additional information.

Practice Problems

1. Find the surface area and volume of a hexahedron with length 8 cm, width 5 cm, and height 3 cm.
2. Given a hexahedron with a surface area of 180 cm² and a height of 10 cm. Find the length and width of the hexahedron.

FAQ

Q: What is a hexahedron? A: A hexahedron is a three-dimensional geometric shape with six faces, twelve edges, and eight vertices. It is also known as a cuboid or a rectangular prism.

Q: How do you calculate the surface area of a hexahedron? A: The surface area of a hexahedron can be calculated by summing the areas of all six faces using the formula: Surface Area = 2lw + 2lh + 2wh.

Q: How do you find the volume of a hexahedron? A: The volume of a hexahedron is obtained by multiplying the length, width, and height of the shape using the formula: Volume = lwh.

Q: Can a hexahedron have equal length, width, and height? A: Yes, a hexahedron can have equal length, width, and height, resulting in a cube.