We hear geometry-related words all the time: ‘what’s your angle?’ and ‘everyone should eat three square meals a day!’ and ‘she ran circles around me!’, often with little thought to how fundamental those shapes are to the discipline called geometry.

So ubiquitous is geometry to our daily lives that it is downright strange for students to dread having to calculate the area and volume of a shape or the circumference of a circle – let alone the lengths and degrees of arcs.

Still, there is a substantial phobia of anything algebraic, be it solving an equation or performing calculus. Yes, those are related to geometry too.

Geometry doesn’t exist as a standalone mathematical discipline, it is intimately intertwined with every branch and facet of maths, from algebra to trigonometry, and extending all the way to the highest of math specialities, the aforementioned calculus.

Did you know that geometry is intrinsic to computer coding, ciphering and communication, too?

Did you also know that, if not for the continuous application of geometry, the world as we know it would screech to a halt?

Before such a doomsday scenario can come to pass, let’s take a close look at the history of geometry and the resultant equations that high school math students struggle with to this day.

No need to worry, though. As always, your Superprof is here with handy tips and tricks for you to learn geometry most effortlessly!

The Bases of Geometry: Where Studies Began

We find many right angles and perpendicular lines in Babylon's hanging gardens
This artist's rendition of what Babylon's Hanging Gardens must have looked like suggests a great deal of geometrical calculation was used Source: Wikipedia Credit: Maarten van Heemskerck

Euclid of Alexandria is widely considered to be the father of geometry; nobody would argue against the vast contributions he made. However, he wasn’t the world’s first geometer.

Nor, for that matter, was Pythagoras, he who gave us the memorable solution for calculating the hypotenuse of right triangles – and, by extension, a way to solve any number of equations that involve a perpendicular line and a base.

Food for thought: the Great Pyramid of Giza was built around 2650BC. Pythagoras lived from around 570BC to about 495BC and Euclid was born around 300BC.

We must accept that it took some knowledge of geometry to design and construct those magnificent polyhedra (a geometrical term for pyramids); builders had to have calculated the area of the base and its corresponding height and a variety of other factors.

Likewise, we must accept that Pythagoras walked the earth about 200 years before Euclid was born because ancient records indicate it. Putting two and two together...

It hardly takes a mathematician to conclude that Euclid is clearly not the first to train his brain to all things geometric.

However, he was the first to formally outline geometric precepts into a single volume, a book of thirteen chapters that forms the basis of what we call today Euclidean geometry.

That is the type of geometry we learn in school, in case you were wondering.

Naturally, if Euclidean is a type of geometry, that suggests that there are other types, right?

Indeed, once you master the Pythagorean Theorem and the formulas for the area and perimeter of shapes, you may consider furthering your geometric studies in:

  • Non-Euclidean Geometry: also called elliptic geometry, concerns itself with familiar theorems applied to 3D shapes
  • Analytic Geometry studies geometric figures using a system of coordinates
  • Differential Geometry uses differential equations to solve a variety of unresolved postulates
  • Topology deals with properties of spaces; their connectedness and compactness
  • Computational Geometry is used in image processing, auto-CAD and medical imaging, among others
  • Algebraic Geometry studies geometry using concepts of commutative algebra
    • this is the type of geometry used in string theory – a theory used to address fundamental questions in physics

Clearly, geometry is far from being a reviled part of your maths curriculum – a useless part, you might aver, seeing as we have tools to measure an acute angle without having to calculate it.

Of course, you are perfectly right; what’s the point of learning how to calculate the area of a right triangle or a parallelogram when there are apps to do it for you…

Unless you consider basic geometry to be a door, swinging open to let you in on further mysteries of the earth and space.

That being the case, you really need to master all of the bases of geometry.

Geometry provides proofs for the polygon and parallel lines
Geometry provides a formula for calculating any shape from an isosceles triangle to polygons Image by Gerd Altmann from Pixabay
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Basic Geometry Equations and Examples

As mentioned before, one great mind (Euclid) did not simply wake up one day with a burning desire to write a book about things he thought were true with absolutely no evidence or concurrence; a book revered still today, whose theories have been proved over and over again.

His was perhaps more of a compilation of previously held, maybe even previously proven ideas.

Nobody is saying that Euclid found a way to profit from other geometers’ work; there is no doubt he too put forth great effort in solving some of geometry’s most persistent mysteries.

However, we should also give him credit for taking the time to compile all of these theorems into a book that is widely held to be the most important book of science in the history of humankind.

The language is archaic, originally translated from Greek into Latin, and then, as the text made its way into Europe, rather roughly translated into French and German and, finally, English, it expounds on certain simple truths.

A line segment may be continued as a straight line indefinitely - Euclid

You may consider that postulate self-evident – why wouldn’t such a line be able to continue indefinitely? But then, back when the world was new and there were no such things as laser levels, scanners or theodolites, such things were not known and had to be established as axioms.

As the discipline developed, so too did the number of ways to prove new postulates through calculation.

We have a companion article filled with the simplest to the more complex geometric equations just waiting for you to discover...

Online Resources to Further Your Geometry Studies

By now, we’ve hopefully established that geometry is an ancient discipline, one full of existing uses and laden with the possibility for future innovations.

Somehow, against all odds, you now find yourself interested in a career involving geometry and you want to be able to bust out formulas to calculate geometric constructions as casually as you eat breakfast.

Lucky for you, there is plenty of help to be had in cyberspace.

Besides reference standards like Britannica and Wikipedia, and beyond any school support site you may subscribe to and anything your school has put online, there are hordes of other pages that address geometry at any level – from basic geometry to university-level materials.

For one, it would be quite handy to have a maths dictionary to look up all of the terms that define your geometry problems, such as ‘angle bisector’ and ‘interior angle’.

You may even want to consult the Math Is Fun dictionary to get a concise definition of what a cosine represents.

The site mentioned above is truly a great resource but the one that really covers all things math is called Homeschool Math.

Targeted to American students who are homeschooled, it covers anything you could ever want insofar as supplemental geometry information: definitions and equations, formulae and constructions.

You can even find geometry worksheets to practise geometry independent of anything you get in school!

You may check that most helpful resource for further links to geometry study websites; we also invite you to discover other online resources we compiled.

Grasp trigonomic functions better with the help of a tutor
Working with a geometry tutor can help you sort through geometry word problems; a common stumbling block for budding geometers Image by Daniela Dimitrova from Pixabay

How You Can Find The Best Geometry Tutor

Even the best students occasionally need a helping hand.

When all else fails and you simply can’t find your way through all of the geometric shapes you’re confronted with, you may just need someone to sit down with you, maybe offer a different perspective on your particular area of difficulty.

Whether for ongoing support, meaning somebody works with you regularly or you just need a boost – someone who will present geometry concepts in a new way, a geometry tutor would surely be your best bet.

You may ask your teacher or perhaps a fellow student if s/he might know someone well-versed in plane geometry or coordinate geometry – whatever your particular stumbling block is, that could help you a bit.

You may also check with your local library. Often, they host homework help sessions one or two afternoons per week with qualified tutors.

If neither of those options suits you, you could go on the search for a tutor near you – just beware that, if you type that phrase into your favourite search engine, you’ll be overwhelmed with the listings it returns!

If you’re after efficiency and convenience – and, of course, stellar results in your tutor search and in geometry, you should bypass all other searches and head straight to Superprof.

Superprof has more than 22,000 geometry tutors scattered across the UK; surely one is close to you? Or you could always opt for online lessons…

You might think that such a tutor would be out of your price range but, with an average rate of £11 per hour of instruction, you could hardly go wrong, especially when you consider that most Superprof tutors offer their first hour of instruction for free!

Geometry is as vital today as it was when the pyramids were built; more so because much of our lives, from the GPS guiding us to our destinations to the radars keeping air traffic flowing.

That is why keen minds like yours are needed to continue making great advances in this mathematical discipline we call geometry.

Keep your compass and straightedge handy; you’re going to need them!

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Daniel