As a wide-ranging discipline with countless applications, math is inherently practical. Though there is no shortage of math in everyday life, one area that dominates our daily existence is geometry. After all, each day we encounter a wide range of geometric shapes, such as riding in cylindrical subways or rectangular buses, crossing rivers over arched bridges, and working and living in rectangular buildings.

And for teachers, low on time and pressed for thoughtful, engaging lesson ideas, geometry in architecture is a great topic. After all, shapes in structural design are omnipresent (but easily overlooked because they’re so common), and best of all, practical. There are countless hands-on projects that you can do with this subject.

Let’s take a look at three key shapes, their strengths, and how they are used in architecture today.

### Triangle

Triangles possess a number of key advantages that make them ideal for both architects and curious students: these shapes are incredibly common, structurally sound, and easy to apply and use in everyday life.

The strength of a triangle derives from its shape, which spreads forces equally between its three sides. No matter what type of triangle is used in a structure (isosceles, scalene, or equilateral), triangles are stable, as they are inherently rigid, the three sides mutually reinforcing each other. As one thoughtful Redditor explained, the angles of a triangle will deform and buckle before the sides give way. Simply put, there’s no way to deform a triangle without destroying it in the process.

This can be an excellent experiment for curious students. While gumdrop bridges have traditionally been a student’s introduction into architecture, this lesson plan takes the concept several steps further, forcing teams of students to think from the perspective of both a city planner and a civil engineer. Not only do students sketch out the design, they must also budget enough (imaginary) funds to buy sufficient gumdrops and toothpicks to span a river.

Yet another good suggestion is to have your students stress test structures that are reinforced with triangular trusses. This experiment, courtesy of Discovery Channel’s Mythbusters, has students build various shapes and evaluate their strength with real weights. When conducting this experiment, students should pay attention to two things: first, how much each structure can take before failure, and secondly, how each structure breaks apart. Do the sides give way first? Or do the angles bend and deform until the material can’t take the strain anymore? Remember, this distinction will be important in reinforcing the unique qualities of triangles, and why they’re far stronger than other shapes.

### Arches

Yet another shape with a stellar reputation is the humble arch. A fixture in architecture and design, arches have been in use for many millennia and across many cultures, from Roman aqueducts carrying millions of gallons of water daily to traditional Chinese bridges crossing raging rivers.

The reason for their widespread use is an appealing balance of utility and strength. Properly designed and constructed, an arch will spread the weight of a load evenly throughout each of its components, which radiates down to its abutments, or the embedded ground supports. The abutments will then press into the ground; because every action will generate an equal and opposite reaction (Newton’s Third Law), the ground will push back and create a resistance. Thus, this phenomena helps an arch bear loads that are many times its own weight.

Yet arches aren’t without their weaknesses. For one, arches face a natural limit: the greater the degree of curvature (the longer and larger the arch), the greater the tension on the structure; in other words, build too long of an arch and it’ll be too weak to support anything. Because of this, arches were long limited by the strength of their building materials: take one look at the mighty Roman aqueducts, and you’ll notice that rather than a few long arches, they consist of many smaller arches, tightly packed together. Today, however, arches can now be longer and wider because of advances in building materials (China’s hybrid Chaotianmen Bridge has a single arch with an impressive length of 1,741 meters, or 5,711 feet).

One thing to note: though arches are beyond the scope of kindergarten-8th grade math, they are worth a brief lesson for a number of reasons. First, they’re very common throughout architecture, and are likely the subject of many questions and musings from children. Secondly, arches are an excellent opportunity to introduce an interdisciplinary dimension to math, given that a number of ancient civilizations and cultures mastered their use.

And lastly, even if lessons involving arches are too advanced for your students, this shape is a good segue for the next, and arguably most important, shape: circles.

### Circle

Aside from its mystical connotations, circles are an exceedingly useful shape for architects and interior designers alike. Interestingly, of the three shapes, circles may be the only one whose value comes from its aesthetic qualities, and not because of any utilitarian advantages.

In a recent study, a team of designers and neuroscientists discovered that human brains are hardwired to prefer curved, rounded shapes, rather than sharp, angular ones. Scientists showed images of circular and rectangular rooms to participants, and at the same time, measured their brain activity; they found that looking at curves activated an area in the brain, the anterior cingulate cortex (ACC), which is heavily involved in governing our emotions.

Conversely, a Harvard study from 2007 found that the opposite was true: upon seeing sharp, pointed angles, participants experienced significant activity in the infamous amygdala, an area of the brain associated with fear processing and fight-or-flight response. Though researchers speculated that this was perhaps due to an unconscious association with threats (for instance, sharp edges on a cliff), there isn’t a clear consensus.

Regardless, the positive qualities of circles have been long known to architects and engineers, evidenced in structures ranging from the domed, regal roof of the Parthenon to the curved, winding Guggenheim Museum (designer Frank Lloyd Wright incorporated the use of the circle into many of his structures).

Luckily, thanks to events like Pi Day, circles are also extremely easy to teach. On their own time, students can access this simple, online game to deepen their understanding of circles, check out the first million digits of Pi, listen to musical renderings of Pi, or even come up with their own song for Pi Day.

Teachers, however, can turn to more complicated fare. Take a look at this lesson plan directly from the Guggenheim, which provides an interesting look at the architectural applications of circles. There’s an interdisciplinary dimension to this lesson (describe the strange feeling of being in a totally circular building without walls at 90-degree angles), but also a pure math angle, as well.

For a greater challenge, task your students to use the dimensions provided by this blueprint of the Guggenheim in order to calculate the following:

• Using the scale provided, measure and then calculate the dimensions of the Guggheim.
• Afterwards, use Pi to calculate the area of the central, circular structure–and for your more advanced students, the total area of the museum.
• An even more ambitious project is to have students use the dimensions to create, to scale, another, larger representation of the Guggenheim’s floor plans–or perhaps a circular structure of their own.

In conclusion, it’s no exaggeration to say that our society is built on shapes, from river-spanning bridges to skyscrapers. Because of this, it is both important and easy to teach students about the geometric foundations of the modern world. After all, inspiration for lessons abounds.