In architecture, geometry is an integral part of the field. Architects use trigonometric functions to determine angles, areas, and volumes of shapes. They also rely on ratios and proportions for creating designs.

These functions include sinus (angle), cosinus(opposite angle), tangent (adjacent angle), secant (hypotenuse length). There are even inverse trigonometric functions such as cosecant which returns the opposite side of a triangle!

Trigonometry plays a big role in architectural drawing and design. It helps in determining dimensions, area, volume, and proportion relationships. Having a fundamental understanding of these concepts will make it easy to pick up new skills like designing your own room or drafting beautiful structures.

This article will go over some basics of trigonometry in architecture and how they can be used. We will also take a look at some examples using Adobe Illustrator.

Degrees, minutes, and seconds

Another fundamental concept in architecture is degree, minute, and second. These are very closely related terms that describe angles. A degree is defined as one half of a circle’s circumference, a minute is one fifth of an hour (60 seconds), and a second is one fifth of a minute (15 seconds).

So if you take one third of a circle’s diameter, which is two thirds of a circle’s length, then your angle would be one sixty-third or 2/3 radian. This is referred to as a radical angle. You can also find obtuse and acute angles using these definitions.

Acute angles have a tip at their end while obtuse angles have a broad base. As you may have noticed, there are many different types of angles!

Trigonometric ratios such as sine, cosine, tangent, and secant are used to determine the proportionality between two lengths or numbers. For example, let’s say we want to know what ratio the width of a room is compared to its height. We could use trigonometry to solve this problem.

We would need to know the rooms average width, it’s height, and then use those ratios to calculate the desired result.

Slicing a triangle into equal pieces

The next tool you will use to test angles is called bisection. This method works by splitting your angle in half, either higher or lower. If there are no restrictions, you could keep dividing your angle in half until it is smaller than a degree or greater than 180 degrees.

However, this cannot be done when solving triangles because we have already determined that two of the sides are known. To avoid false results, you must make sure that the third side is not longer than the other two.

Once this is guaranteed, you can divide your angle in half using the rule above. For example, if your angle was 120 degrees then start with one piece which is 60 degrees and another 30 degrees, making a total of 90 degrees.

Now, whether your high school math teacher told you about this or not, remember that opposite numbers have different answers! Because this tool creates a new pair of opposite sides, the process must be repeated until both pieces are the same length or the algorithm has run out of iterations.

Measuring angles

The first major use of trigonometric functions in architecture is to measure angles. You will almost certainly encounter this at least once while studying architectural concepts. This can be done using three main methods-tangent, secant, or inverse tangents.

The most basic way to determine an angle is by creating a line from one end point to another and measuring the difference between them and the other side. For example, if you have a line that goes up the wall and a segment that points away from the wall, then find the angle made by these two lines with each other.

This is called the angle formed by those two lines. In our case, it would be measured as 90 degrees, since we are looking at a vertical wall, and there is a horizontal segment.

By knowing the definition of angle, we know what ratio this sets off! We take the opposite of this ratio (inverse) and multiply it by the other side to get our final result. So, in our example, we would do 30 x 60 = 1800.

We now have our desired result! Simply round down for more precision.

Examples:

Given an oblique line joining A(100, 100) and B(200, 200), find the value of θ.

Solution: Use the formula given above and put in your values. 180 becomes 1 so we write .

Construction math

The second major way that tringonomy is used in architecture is construction mathematics. This includes things like calculating how many bricks you need to make up a wall, figuring out how much space an opening or window will take up, or determining how high a ceiling can be before it becomes impossible to fit everything in.

Construction mathematics uses ratios as well as proportions. Ratios are not equal, they are different, for example, saying there are three times as many inches in one piece of lumber as there are in another. Proportions refer to the size of something compared to something else, such as saying a door is twice as tall as the frame it fits into.

These concepts are mixed together to create complex equations that must have correct answers. When these answers are found they usually go towards creating your final project. For instance, if you were designing a house then the length and height of each room would come down to a calculation of proportionals in ratio and proportion.

This is because a room has an area and a volume, both determined by its lengths and heights.

Finding the area of a triangle

The most basic way to find the area of a triangle is using an equation that includes two sides and a hypotenuse, or third side. Each term in this expression contributes a part of the area, so it’s important to know what each one means before calculating the overall area.

The first term in the equation is called the “base” which is simply the length of a leg of the triangle. In our case, there are three legs, so we would use 3 for the base.

Next comes the “height”, which is the opposite edge of the triangle divided by two because you are taking the difference between the highest value and the lowest value. In our case, the top edge is 2 yards high, and the bottom edge is 0 yard, so we would have 1/2 as our height.

Now comes the tricky part – the hypotenuse! A hypotenuse is always longer than any other edge of a triangle, so it isn’t included as a separate term in the formula. Instead, you add all three lengths together and divide by two, just like with the heights.

So, our final area calculation looks like this:

Base + Height ÷ 2 = Area of Triangle

Remember, when solving equations, make sure your numbers match up. If they don’t, then you will get incorrect results. Check out this link for more tips on how to do this.

Finding the length of an angle

The next topic we will discuss is how to find the length of an angle. This can be done by using trigonometric functions such as sine, cosine, or tangent.

The sine function calculates the proportion of time it takes for a circle to go from one diameter to another. For example, if you take a triangle and draw a line down the middle, then use that lines diameter as the second leg, the sine of the included angle is equal to the ratio between the two other legs!

This means that the longer the third leg, the smaller the included angle gets, since there’s more space for the circle to expand before hitting the opposite side. Conversely, if the third leg is shorter than the first, the sine becomes larger because there’t enough area until it has to hit the second leg.

By taking the inverse of this value, we are able to get the fraction of the circle that the third leg represents, which gives us the desired angle size! To calculate the length of an angle, simply divide the number you got from the sine by the same factor as the radius.

Polar coordinates

Another way to describe angle is by using two coordinate axes that are not parallel to each other, but rather perpendicular to one another.

This method of defining angles comes from geometry, where you can define an angle as the number of degrees between two lines or points. In this case, the two lines are defined by the x-axis and y-axis, which are both considered vectors (or line segments).

The intersection of these two lines is then calculated via their equation, which is simply the ratio of the length of the shorter vector divided by the longer vector. This formula is repeated for every degree of the angle, creating what we refer to as polar coordinates.

Since radians and degrees are interchangeable ways to measure angles, it does not matter which one you use in your equations. It all depends on how you want to represent the angle!

For instance, if you wanted to find the total angle of a triangle, you would need to add up all three angles. Since there is no unique way to do so, people usually pick either radian or degree format, depending on personal preference.

Finding the center of a circle

The next topic we will look at is how to find the center of a circle. This can be done by using either trigonometric ratios or algebraic methods, depending on your level of math proficiency.

Trig is usually the easier way to go as most people are already familiar with it! If you have never learned trig before, no worries! You’re not going wrong doing this project because there is no formula for finding the radius and the center of a circle so you won’t need that info either.

To determine the center of a circle correctly, all you really need to know about triangles is that they each have one angle bigger than 90 degrees. In our case, we want the biggest angle which is called the hypotenuse. Therefore, our first step is to take the length of the side opposite the hypotenuse and add it to the other two sides. These three new lengths make up another triangle, and since we already knew one side of this triangle, we can use the Pythagorean theorem to calculate the rest!

The pythagorean theorem says that the sum of the square of the distances between any two points in a right-angle triangle equals the difference squared plus the product squared. In our case, the differences are just one side of our triangle (the longer side) and the product is the shorter side times the shorter side.