Triangles represent the foundations of geometry. They introduce us to concepts related to algebra, geometry, area, perimeter, theorems, and angles. This foundation allows mathematicians to understand and explore more advanced concepts of geometry such as trigonometry.
Triangles also introduce a whole new world of terminology to students which can be overwhelming at times. As a result, it’s common for people to struggle when learning geometry in class.
This article aims to provide a clear, and simple explanation of triangles, Pythagorean theorem, and area and perimeter.
What is a right triangle?
A right triangle is a triangle that contains a 90° angle. Right triangles are one of the most important triangles in math as they are used in Pythagorean theory as well as trigonometry.
Types of triangles
There are three main triangles in math: equilateral, isosceles, and scalene. An equilateral triangle has equal side lengths, an isosceles triangle has two equal side lengths, and a scalene triangle has no side lengths which are equal.
Although there are three main triangles, only isosceles and scalene triangles can be right triangles. An isosceles will have a 90° angle with two accompanying 45° angles. A scalene, however, is the most used triangle in geometry questions, its sides are not of equal length and nor are its angles apart from one which is 90°.
A final piece of terminology to understand is what the sides of a triangle are known as. The only thing you must understand and remember is that the longest side is called the hypotenuse. In Pythagorean theorem, this is denoted as “c”. The other two sides are noted as “a” or “b” – but these are interchangeable.
Pythagorean Theorem
What is the Pythagorean Theorem?
Now that you understand the basic properties of triangles, it’s important to understand the Pythagorean theorem. First established by Pythagoras 2000 years ago in ancient Greece, this theorem discovered that:
When a triangle has a 90°, if you extend each side into squares, the largest square has the same area as the two smaller squares combined.
This can be expressed as a2 + b2= c2
Pythagorean Triple
Typically, in the Pythagorean theorem, when the length is calculated, it usually contains a decimal. There are, however, examples of so-called perfect triangles which yield a whole number as the answer. If you memorize these, it will allow you to quickly answer a question if you see the following numbers.
Common Pythagorean triples are:
- 3, 4, 5
- 6, 8, 10
- 9, 12, 15
What you will notice is the triple 3, 4, 5 is the derivative Pythagorean triple that others stem from. What this means is, if you had a question where two lengths were 18 and 24, and you had to determine the length of the third side, the answer would be 30. This is because 3, 4, 5 can be derived from 18, 24, 30 by multiplying the original by 6.
Examples and solutions
So far this has been very theoretical, but math is a practical subject so it’s important to go through some examples to make sure you fully understand right triangles.
How to find a length?
To find the perimeter, you must first find the lengths.
Essentially this is what Pythagorean theory is all about. If you only know two sides of a given right triangle, the theory allows you to work out the third length. Below are two examples, if you understand how to calculate these, then you can calculate any Pythagorean exam question.
- Find the hypotenuse
To find the hypotenuse we use the standard, given form of Pythagorean theory: a2 + b2= c2
Let us input the numbers into the equation. Remember – we are trying to find the hypotenuse which is “c”.
If a = 13 cm, and b = 9 cm, then:
132 + 92= c2
Square the numbers and add them together:
169 + 81 = c2
250 = c2
Finally, square root this to get “c”:
√250 = c
This tells you the answer and hypotenuse:
c = 15.81138830…
Remember, as this is centimeters, your answer should be to the correct decimal places:
c = 15.81cm
It’s as easy as that!
- Find length “b”.
Unlike the previous question, we know the hypotenuse which means we must adapt the Pythagorean theorem to find “b”.
If a2 + b2= c2, then b2= c2 – a2
If a = 6 cm, and c = 18 cm, then:
182 – 62 = b2
Square the numbers and add them together:
324 – 36 = b2
288 = b2
Finally, square root this to get “b”:
√288 = b
This tells you the answer and hypotenuse:
b = 16.97 cm
How to find the perimeter of a triangle?
Finding the perimeter of a triangle is extremely simple. The perimeter formula consists of adding up all the sides of a triangle. You may have to use the Pythagorean theorem to find the lengths, but once you know all the lengths, you simply add them together.
For example, if we look at the previous question, the perimeter is 6 cm + 18 cm + 16.97 cm which equals 40.97 cm.
How to find the area of a triangle?
Finding the area of a right triangle is a little more complex than finding its perimeter. For a right triangle, the area is calculated by multiplying the width and length and dividing it by 2.
For example, with the previous question, the area is calculated as such:
Area = (6 x 16.97) ÷ 2
Area = 50.91 cm2
Still easy enough, but do remember that because it is an area, the unit is cm2 whilst the perimeter is just cm.
Still Struggling?
This article covered a lot of information. We explored the Pythagorean theorem, understood the properties of triangles and calculated some lengths, perimeters, and areas. If you’re still confused, that’s completely fine! There are many tools out there which can help you including websites, YouTube tutorials, workbooks, and tutoring.
H3: Math Tutoring
You find it challenging to learn on your own and prefer to work with an instructor? If this is the case, tutors are the way to go. You will gain a greater understanding of the subject by working with a tutor because you will be in a small class or even 1:1.
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Tutorax offers each student individualized support to help them consolidate their learning and develop study strategies. The tutor’s job is to raise the student’s confidence and academic motivation as well as help him/her improve their grades.
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