*Yuanxin (Amy) Yang Alcocer*Show bio

Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.

Instructor:
*Yuanxin (Amy) Yang Alcocer*
Show bio

Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.

Knowing the geometry of objects and things in the real world is a very useful skill. Architects use geometry to aid them in designing buildings and cities that work. Engineers use geometry to build working components in a bigger machine.

Look around at the different ways that companies package their products. Some companies package their products so they can maximize the number of products that fit into a certain space, while others package their products so they are very bulky. You know how sometimes you have a big box and you think there's something big inside, but when you open it, you find it has lots of packaging and a tiny actual product inside? This is where stores like Costco and IKEA are masters at using the geometry of space to maximize the number of items a particular space can hold.

The geometry of space is so much more than mathematical calculations. The geometry of space is about how everything fits together. For example, if you have a packing box, it is the geometry of space that determines just how many items can fit inside the box. It is also the geometry of space that lets you fit more items in a box if they are placed in a certain way.

So what is this **geometry of space**? It involves three dimensions: a length, a width, and a height. Every single object in space has all three of these dimensions. But not all objects are simple block shapes with just one length, one width, and one height. Some objects vary in length, width, and height in different parts of the object. For example, the honeycomb pattern of a beehive has different lengths at different parts of the hexagon that is each honey-containing hole.

To define objects in space, all three dimensions are used. For simple block shapes, a simple *l* × *w* × *h* description is all that is needed. For more complicated objects, more complex mathematical functions are used. For example, a sphere can be written mathematically with this function.

The height of the cylinder can be defined by setting limits to the *z* variable. For this example, the cylinder has a height of 4. The *x*, *y*, and *z* variables represent each of the three dimensions in space. The *x* and *y* variables typically represent the length and width of objects, while the *z* variable represents the height of the object.

Any problem that requires calculating space uses the geometry of space. For example, problems that ask you to find out how many packages fit inside another larger package use the geometry of space. Toys that have moving parts that allow you to turn the toy from a box shape into a robot use geometry of space to figure out where and how the parts need to fit together to get from one shape to another.

Let's look at an example problem.

You have a box that measures 2 feet by 1 foot by 6 inches tall. How many products fit inside if each product package measures 3 inches by 6 inches by 1 inch tall?

Two different ways of solving this problem can be used. One way is by visualizing the problem, drawing it out, and seeing how the packages fit inside the box.

Another way is by using algebra. Because the box has one side that is 6 inches tall and the product package does as well, the packages will be packed so these sides match. Then all that is needed is to determine how many packages fit the other two sides and then multiply them together. Placing the 3-inch sides toward the 2-foot side of the larger box, you can fit 8 boxes that way. Placing the 1-inch side toward the 1-foot side gives you 12 boxes that way. Multiplying the 8 and the 12 together gives a total of 96 boxes in the big box.

Let's review.

The **geometry of space** is defined by the three dimensions that all objects have in the real world.

The geometry of space is used in architecture and engineering to make sure all parts of a bigger object fit together as needed. Problems involving geometry of space most often deal with how much space various objects take up.

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