Some people might say that geometry is in no way a ‘sexy’ subject; really, as a general rule, calculating angles, volumes and areas is seldom considered enticing or fun.
Could the opposite be true?
Over the last 10 years, we’ve seen mathematics creeping into films and television shows; The Big Bang Theory is a prime example of such. Granted, equations are not central to the plot and, quite frankly, only the first few shows were math-heavy. After that, algebraic work popped up only occasionally.
Still, it is nice to see complex calculations playing out in a popular arena, and it’s even better that both male and female characters take part in tweaking the equations; a mere 20 years ago, cinematic mathematicians could only be male!
Now it’s your turn to master basic geometry equations and you want the most efficient way of doing so. Or maybe you’re a fan of Descartes and wish to take Cartesian geometry to the next level but you need a solid foundation, first.
Your Superprof wants to help you get a good grasp of fundamental geometrical formulas; grab your squares and compasses… we’re off!
The Basic Shapes
You might be tempted to think ‘circle’, ‘triangle’ or ‘square’ and you’d be absolutely correct.
Each of those geometric shapes fall into one of these four general categories:
- Triangles have three sides; the sides may be of equal length (equilateral triangle) or all different length (scalene triangle).
- A quadrilateral is any four-sided polygon. Those would be rectangles, squares, rhombuses, diamonds…
- the parallelogram, a shape that has 2 pairs of equal sides, is also a quadrilateral
- Polygons: literally ‘many sides’. These shapes can be triangles, hexagons, pentagons… all of those ‘gons’ are polygons. Essentially, anything that has straight sides is called a polygon.
- Circles are a class onto themselves because they have no straight lines
Their unique characteristics include:
- Squares have four equal sides and four right angles
- Rectangles have two pairs of equal sides
- A trapezoid has only one pair of parallel sides
- A trapezium has no sides of equal length
- Rhomboids: opposite sides and opposing angles are equal
- The isosceles triangle has two equal sides
- Right triangles have one 90-degree angle opposite of the hypotenuse
Each of these shapes has its own formula to calculate its perimeter, area and angles. Some you may be familiar with, such as the Pythagorean theorem while others are perhaps a bit less memorable.
Let’s take a look at them now.
Do you need help with your geometry studies? Perhaps you could find a geometry tutor…
Starting with the shapes of the fewest sides (but sometimes the most complicated formulas), we tackle geometric formulas head-on!
The simplest formula for the perimeter of any triangle is a+b+c, with each letter representing a side. It is beautiful in its simplicity and easy to work with, provided you know each side's length.
Let’s say your triangle has these measurements: a = 3 inches, b = 4 inches and c = 5 inches
Its perimeter would then be 3+4+5=12 inches.
Clearly, this is a triangle is neither equilateral nor isosceles; nor is it a right triangle. How would we calculate the perimeter if only two values, the bottom and one side, are given?
In such a case, we have to draw on Pythagoras’ theorem: a2+b2=c2. You remember that one, right?
First, draw a line from the triangle’s peak straight down to its base. This line, h, should be perpendicular to the base, thereby forming two 90-degree angles – one on each side of the line.
You now have two right triangles, one of which has a measurement for both a and b. From there, it is a simple matter to plug known values into the theorem (don’t forget to square them!) and find your missing value.
Let’s try it with a fictitious triangle:
a = unknown b = 5 c = 7
a2 * 52 = 72
a2 * 25 = 49 the unknown value must stand alone on one side of the equation
a2 = 49 – 25 move 25 to the other side of the equal sign, subtracting it from the given value of c
a2 = 24
Now you have to calculate the square root of 24 to find the value of 'a', which is 4.898. Once you've calculated the perimeter of one right triangle, you must calculate the second to get the dimensions of the original triangle.
Congratulations! You now know how to calculate the perimeter of any triangle!
Calculating Triangles’ Area
While perimeter calculation is a rather simple endeavour, figuring the area of a triangle is a bit more involved.
If values are given for all three sides, you may apply Heron’s Formula:
area = square root of [s(s-a)(s-b)(s-c)], with 's' being the semi-perimeter, that is (a+b+c)/2
It only looks complicated; remember that, when working with a formula, you only need to plug in known values to solve for the unknown. When thought of in that way, the Hero’s Formula, as it is also called, is pretty easy!
Now, for ‘area of triangles’ equations where one or more values are unknown.
If you know only the value of the triangle’s base and its height, you may apply: area = (½) * b * h
If only the length of two sides and the degree of the angle joining them are known, you would use trigonometry to find the missing values. The basic formula is:
Area = (½) * a * b * sin C
Keep in mind that lowercase letters signify line measurements while uppercase letters represent angles.
If you only know the values of sides a and c, you would plug them in and calculate sin B. Likewise, if you know b and c, you would employ sin A to get your triangle’s area.
Why not practise those for a while before moving on...
You may be able to figure the perimeter of a square or a rectangle in your sleep. Those formulae are P=4a (a represents the square’s sides) and P=2l + 2w, respectively.
Those area calculations should also come to you rather easily. For squares, it is A=a2 and for rectangles, it is A=l * w. Simple, right?
Things start getting complicated when we get into parallelograms and trapezoids; to solve both of those equations, you will need to know the height of the shape (h) an d the length of the base (b) – the line at the bottom.
Once you know those values, choose the appropriate formula for the shape:
b * h = area of parallelograms (½)(a+b) * h = area of trapezoids, where ‘a’ represents the side opposite of ‘b’.
Quadrilaterals may just be the easiest shapes to work with. If you need extra practice, there are plenty of resources online where you can find geometry worksheets and equations to solve.
Whether you are confronted with an apeirogon (a polygon with an infinite number of sides) or the more familiar hexagon, you need to know how to calculate its perimeter and area.
Luckily, apeirogons are only hypothetical; imagine having such a figure to calculate an area for!
If your polygon’s sides are all the same length, you can apply P=n * v, where ‘n’ is the number of sides and ‘v’ is the value of each side.
If said polygon’s side are not all the same length, you will have to add up those values to get its perimeter.
Calculating Areas of Polygons
There are several ways to realise the value of any polygon’s area, some of which involve calculations for triangles.
First, we tackle the equations for a regular polygon; one whose sides are all the same length. Before we can start any ciphering, we have to determine the polygon’s radius.
That involves drawing a circle inside the polygon in such a manner that the circle’s perimeter touches the polygon’s perimeter. This is called an inscribed circle. Once we know that radius’ value, we can apply this formula:
A = ½ * p * r
Formulae get more complicated the more sides the polygon has.
Let’s say the number of sides is represented by ‘n’ and sides by ‘s’. The radius, also called apothem, is designated ‘a’. Of course, ‘A’ represents ‘area’, yielding a formula that looks so:
A = ns/4 √ 4-s2
From here, the formulas get ever more complex. Do they leave you struggling with the basics of geometry? You can refer to our complete guide!
Circles involve neither angles nor lines and their perimeters are called ‘circumference’. However, their calculations do require at least a line segment which is instrumental to any formula for circles.
Oddly enough, it seems that the formula for calculating areas of circles is more renown than perhaps for any other geometric shape: πr2, or pi * r2
Surely you know/remember that pi (π) has a value of 3.1415...
The less-renown formula concerning circles, the one for calculating circumferences is: 2 * π * r
Bear in mind that these are formulae for calculating the area and perimeter of two-dimensional shapes; once they gain an additional dimension – they become 3-D shapes and merit a calculation of volume as well as area and perimeter.
Let’s not go off on a tangent, here; we’re quite happy to provide formulas for these basic geometric constructions...
But you don’t have to stop here; latch on to our beginner’s guide to geometry!