Geometry often gets a bad rap, seen by many as a dry and uninspiring subject. However, beneath its surface lies a fascinating world of shapes, angles, and relationships that can be both engaging and rewarding. Far from being just a collection of formulas, geometry is a key to understanding the physical world, influencing everything from architecture to art.
In recent years, popular culture has started to embrace math, bringing it into the spotlight. Shows like The Big Bang Theory have showcased how mathematical concepts can intertwine with storytelling, making the subject more relatable. While the equations might not always steal the show, they highlight the intriguing connections between math and everyday life. Now, it’s your turn to dive into the realm of basic geometry equations. Whether you’re looking to strengthen your foundational skills or venture into Cartesian geometry, we’ve got you covered.
To help you on your journey, we’ve compiled a list of essential geometry equations along with clear examples. With these tools at your disposal, mastering the fundamentals of geometry will not only be efficient but also enjoyable. So, grab your tools, and let’s get started!
Key Geometry Equations
- Perimeter of a Square = 4(Side)
- Perimeter of a Rectange = 2(Length + Width)
- Area of a Square = Side^2
- Area of a Rectangle = Length x Width
- Area of a Triangle 0.5 x Base x Height
- Area of a Trapezoid = 0.5 x (Base1 + Base2) x Height
- Area of a Circle = A = π×r2
- Circumference of a Circle = 2πr
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.
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Calculating Triangles
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!
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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...
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Calculating Quadrilaterals
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.
Calculating Polygons
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!
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Calculating Circles
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: ?r2, or pi * r2
Surely you know/remember that pi (?) has a value of 3.1415...
The less-renown formula concerning circles, the one for calculating circumferences is: 2 * ? * 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!
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