Recent trends in architecture have incorporated geometry into the design process. Architects now routinely use mathematical tools to understand proportions, create shapes, organize space, and determine functionality for buildings or spaces.

This article will discuss some of these applications and how you can begin using them. While not every architectural project requires advanced mathematics, it is important to be familiar with this tool set so that you are able to take advantage of them when they are needed.

If you’re already very comfortable working with math, then you're ahead of the game! But if you’re not, don't worry—this article will be easy to follow even if you've never done any before.

There are many ways to apply basic geometric concepts in your designs, and this article will focus mostly on solid, proportion, and ratio-based concepts. We'll also look at circle relationships and some simple coordinate systems like Cartesian (x, y) and polar (r,theta).

Compute the area of a shape

Area is an important geometric measure that determines how space is used by a shape. The area can be calculated using two different methods, but either one will work for this article. To calculate the area of a parallelogram, you would use the formula A = b x c, where b is the base and c is the height. For a triangle, you could use the simpler form A = 1/2bh, where b is the length of a side and h is the height.

In architecture, areas are very frequently needed to determine things like the size of a room or whether there’s enough space for a door to open fully. In these cases, we refer to the first equation as the rectangle rule and the second as the triangle rule.

The rectangle rule applies when the bottom edge of your box meets a horizontal line or wall. Because the top of the box is fixed, the area decreases as you go down due to the way the slope changes. The area is determined only by the lengths of the sides of the box so it can be hard to estimate exactly unless you have a good sense of geometry.

The triangle rule works similarly to the half-angle theorem we learned about earlier. If a lower corner makes a right angle (90 degrees) with the floor, then the other two angles add up to 180 degrees.

Estimate the number of seats in a restaurant

When looking to estimate how many tables and chairs you will need for your business, there is an easy way to do it. By figuring out the amount of space per table and per seat, then multiplying those together, you have your total square footage!

This process can be done by yourself or with help from professionals. You just need to know some basic geometry and how spaces are organized in buildings.

The easiest way to understand this is to think about what size rooms people typically have in their homes. A room like that is called a pre-defined area or closed shape. For example, a kitchen has defined areas for both a stove and a refrigerator as well as a table where you could eat food.

These types of rooms are important geometrically because they give you a basis for comparison when estimating other open shapes suchas tables and chairs at restaurants.

If you want to learn more about the basics of architecture, start looking into building styles and proportions.

Find the shortest distance between two points

In architecture, geometry is everywhere you look. Some examples are designing doors and windows, figuring out where to place furniture or sculptures, creating layouts for rooms, and determining how tall an apartment building should be. All of these things relate to each other and combine into one beautiful space!

Architects use several types of geometry when drawing and designing buildings. They may draw lines, shapes, and polygons on paper, computer software, or both. These geometric forms connect with other parts of their design as they explore different styles, concepts, and textures.

The concept of “form” ties together all of the above. As architects, we pay close attention to how form relates to function. Functions include things like what something does (for example, being a door) and why it exists (to frame another element or shape).

Form also includes how something looks. We make sure that its proportions are aesthetically pleasing. This is called formalism. It is our way of expressing ourselves through the work we do.

Geometric forms come in many shapes and sizes. Many think about proportionality when thinking about formalisms but there is more than just size involved. What colors a line or polygon uses, whether it is circular or square shaped, and even if it has curves can influence perception of balance and harmony.

Some say that nature uses geometry in ways to inspire awe and wonder.

Find the midpoint of a line

The Midpoint Of A Line is an interesting concept that can be used to do some cool things. If you have a coordinate pair that defines a line, you can find the middle point by taking the average of those two numbers.

The middle point will always lie on the line, but it will take up a bit more space than either of the coordinates alone. This way, the middle point represents the whole length of the line.

Geometric examples of finding the midpoint of a line include finding the mean (average) height of someone so that you can determine what size house they would need or determining where the centerline of a highway carpool lane should start.

These types of applications are important because they give you information about the shape of the area being studied. By figuring out the mean width of a road, for example, you know how wide the carpool lane needs to be to fit all cars within that distance of the centerline.

This geometry tool can also help answer other questions like “how tall is this person?” by calculating their mean height. Architects use these tools frequently to see how much space different parts of a building require.

Find the center of a circle

A circle’s circumference is defined as the distance that it surrounds, or bounds, its inner radius. The diameter is therefore twice the inner radius, so to find the circumference of a circle, you just need to multiply the diameter by 2!

This process can be extended indefinitely to determine the circumference of an arbitrary shape. Given a set of points that define the perimeter of a shape, you can use geometry to calculate the area using this formula: Area = (Length * Width).

By taking the derivative of this equation with respect to both length and width, we are left only with equations for the perimeters of circles and squares.

Calculate the volume of a 3D shape

The most basic way to calculate the volume of an object is using the formula V = πy²*x∗z, where x, y, and z are dimensions of the object. For example, if you want to know how much space a cylinder would take up in a room, you would need to find its height, diameter, and depth.

The length of the diameter is already included in the equation, so we can drop that term. Now, all you have to do is multiply the remaining two variables together and you’ve got your volume!

To illustrate this concept, let’s look at a circle and square. A circle has a constant radius, which makes it easy to compute, but what about a square with a variable side? You could use the area formula for that, but there’s an easier way.

Just like our volumetric formula, you start by finding the perimeter of the shape, then multiplying that by three to get the area.

Estimate the cost of building a new room

The next thing to do is determine how much space you have for creating an addition or what areas of your home can be extended. You will need to know where all your walls are!

It is very common to run into problems when trying to add onto your house. There may not be enough wall area, roofs could leak, spaces between walls cannot hold up against pressure, etc.

These issues can be addressed through careful planning and geometry before construction begins. By using these tools early on, the architect has them covered. They will make sure there are no surprises during the process.

Architects use computer software to create diagrams and calculations of your home. These diagrams and formulas help predict how many windows and doors would fit in a room, as well as figuring out if that window/door looks good with the rest of the room.

Geometric shapes like squares and rectangles are used frequently in architecture.

Find the closest distance between two shapes

In architecture, geometry is heavily used to determine how spaces relate to each other. For example, if you wanted to make a room with an open window, you would need to know where your wall or windowsill ends before you could begin building the next part of the space.

Geometry comes into play here because that edge or barrier must be determined. Sometimes this information is given by the architect or builder, but more commonly it has to be found using math.

There are many ways to find the closest distance between two shapes, so let’s look at one method. We will also use some geometric figures as examples. These can be done easily online or via software like Google Sketchup.

The figure below shows three boxes with different heights and widths. What we want to know is what size box these triangles resemble most closely.

The triangle in the middle is the only one with a 90 degree angle in all three sides. This means that it is equilateral, which is a word for having all equal lengths. It also means that it is a perfect cube, just like our first example. All three angles are exactly 180 degrees.

So why not scale up the box in the middle until it looks exact? Well, there is no way to make it bigger without making another box that is parallel to the others, which takes away from the effect of the design.