

# Dimensions: length, width, height, and depth



Length is the distance along the longest straight line that can be drawn from one end of an object to the other. Width is the distance along the shortest straight line that can be drawn from one end of an object to the other. Height is the distance perpendicular to the base of an object. Depth is the distance perpendicular to the length and width of an object.

For example, consider a rectangular box. The length is the distance between opposite ends, the width is the distance between opposite sides, the height is the distance from the base to the top, and the depth is the distance from the front to the back.

Let's consider a rectangular box with dimensions:
- Length: 10 cm
- Width: 5 cm
- Height: 3 cm

The volume of the box can be calculated using the formula:

$$
Volume = Length \times Width \times Height
$$

In this case, the volume would be:

$$
Volume = 10 \times 5 \times 3 = 150 \text{ cubic centimeters}
$$

## Exercise
Calculate the volume of the following rectangular box:
- Length: 8 cm
- Width: 4 cm
- Height: 2 cm

### Solution
$$
Volume = 8 \times 4 \times 2 = 64 \text{ cubic centimeters}
$$

# Surface area: calculation and applications



For cubes and rectangular prisms, the surface area can be calculated using the formula:

$$
Surface\ Area = 2(Length \times Width + Width \times Height + Height \times Length)
$$

For cylinders and cones, the surface area can be calculated using the formula:

$$
Surface\ Area = \pi(Radius \times Perimeter + 2 \times Radius \times Height)
$$

For spheres and ellipsoids, the surface area can be calculated using the formula:

$$
Surface\ Area = 4 \times \pi \times Radius^2
$$

Let's consider a cube with dimensions:
- Length: 5 cm
- Width: 5 cm
- Height: 5 cm

The surface area of the cube can be calculated using the formula:

$$
Surface\ Area = 2(5 \times 5 + 5 \times 5 + 5 \times 5) = 60 \text{ square centimeters}
$$

## Exercise
Calculate the surface area of the following rectangular prism:
- Length: 8 cm
- Width: 4 cm
- Height: 2 cm

### Solution
$$
Surface\ Area = 2(8 \times 4 + 4 \times 2 + 2 \times 8) = 88 \text{ square centimeters}
$$

# Symmetry: types and properties



There are several types of symmetry:
- Rotational symmetry: the shape remains the same when it is rotated around a specific axis.
- Reflectional symmetry: the shape remains the same when it is reflected across a specific plane.
- Radial symmetry: the shape remains the same when it is rotated around a specific point.

Symmetry has important properties that can be exploited in various applications, such as in design and engineering.

# Transformations: scale, rotation, and translation



Scaling is the process of changing the size of an object without changing its shape. It can be performed by multiplying the dimensions of the object by a scaling factor.

Rotation is the process of changing the orientation of an object without changing its shape. It can be performed by rotating the object around a specific axis.

Translation is the process of changing the position of an object without changing its shape. It can be performed by moving the object along a specific direction.

# Volume: calculation and applications



The volume of a shape can be calculated using the formula:

$$
Volume = Length \times Width \times Height
$$

The volume has important applications in various fields, such as in engineering, physics, and architecture.

# Cubes and rectangular prisms



A cube is a three-dimensional shape with equal sides and six faces. A rectangular prism is a three-dimensional shape with equal sides and six faces.

Cubes and rectangular prisms are important in various applications, such as in packaging, construction, and engineering.

# Cylinders and cones



A cylinder is a three-dimensional shape with two circular bases and two circular lateral faces. A cone is a three-dimensional shape with a circular base and a circular lateral face.

Cylinders and cones are important in various applications, such as in engineering, architecture, and manufacturing.

# Spheres and ellipsoids



A sphere is a three-dimensional shape with equal radius in all directions. An ellipsoid is a three-dimensional shape with different radii in different directions.

Spheres and ellipsoids are important in various applications, such as in engineering, physics, and astronomy.

# Tessellations and regular polygons



A tessellation is the process of covering a surface with copies of a shape that fit together without gaps or overlaps. Regular polygons are polygons that have equal sides and angles.

Tessellations and regular polygons are important in various applications, such as in design, art, and mathematics.

# Comparing and contrasting different 3D shapes



Comparing and contrasting 3D shapes can help us understand their similarities and differences, which can be useful in various applications, such as in engineering, design, and architecture.

# Real-world applications and design considerations



The properties of 3D shapes can be exploited in various applications, such as in engineering, design, and architecture. For example, understanding the properties of materials and their behavior under different loads can help in the design of structures and machines.

Design considerations can include factors such as aesthetics, functionality, and practicality. For example, the shape of a chair can be designed to be comfortable and ergonomic for the user, or the shape of a building can be designed to be energy-efficient and sustainable.