How to Calculate the Sides and Angles of Triangles
What Is a Triangle?
By definition, a triangle is a polygon with three sides.
Polygons are plane (flat, two-dimensional) shapes with several straight sides. Other examples include squares, pentagons, hexagons and octagons. The name originates from the Greek polús meaning "many" and gōnía meaning "corner" or "angle." So polygon means "many corners." A triangle is the simplest polygon, having only three sides.
In this tutorial, you'll learn about Pythagoras' theorem, the sine rule, the cosine rule and how to use them to calculate all the angles and side lengths of triangles when you only know some of the angles or side lengths. You'll also discover different methods of working out the area of a triangle.
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What Are the Different Types of Triangles?
Before we learn how to discover the sides and angles of a triangle, it is important to know the many different types of triangles. The classification of a triangle depends on two factors:
- The length of a triangle's sides
- The angles of a triangle's corners
Below is a graphic and table listing the different types of triangles along with a description of what makes them unique.
Types of Triangles
Type of Triangle
An isosceles triangle has two sides of equal length, and one side that is either longer or shorter than the equal sides. Angle has no bearing on this triangle type.
All sides and angles are equal in length and degree.
All sides and angles are of different lengths and degrees.
One angle is 90 degrees.
Each of the three angles measure less the 90 degrees.
One angle is greater than 90 degrees.
Using the Greek Alphabet for Equations
Another topic that must be covered before we delve into the mathematics of solving triangles is the Greek alphabet.
In science, mathematics, and engineering many of the 24 characters of the Greek alphabet are borrowed for use in diagrams and for describing certain quantities. For example, the characters θ (theta) and φ (phi) are often used for representing angles.
You may have also seen the character μ (mu) represent micro as in micrograms μg or micrometers μm. The capital letter Ω (omega) is the symbol for ohms in electrical engineering. And, of course, π (pi) is the ratio of the circumference to the diameter of a circle.
How Do You Find the Sides and Angles of a Triangle?
There are many methods available to the mathematician when it comes to discovering the sides and angles of a triangle. To find the length or angle of a triangle, one can use formulas, mathematical rules, or the knowledge that the angles of all triangles add up to 180 degrees.
Tools to Discover the Sides and Angles of a Triangle
- Pythagoras' theorem
- Sine rule
- Cosine rule
- The fact that all angles add up to 180 degrees
Before we delve into Pythagoras' theorem, the sine rule, and the cosine rule, it is important to state that all triangles have three corners with angles that add up to a total of 180 degrees. The angle between the sides can be anything from greater than 0 to less than 180 degrees. The angles can't be 0 or 180 degrees, because the triangles would become straight lines.
Degrees can be written using the symbol º. So, 45º means 45 degrees.
Triangles come in many shapes and sizes according to the angles of their corners. Some triangles, called similar triangles, have the same angles but different side lengths. This changes the ratio of the triangle, making it bigger or smaller, without changing the degree of its three angles.
Below, we will examine the many ways to discover the side lengths and angles of a triangle.
Pythagoras' Theorem (The Pythagorean Theorem)
Pythagoras' theorem uses trigonometry to discover the longest side (hypotenuse) of a right triangle. It states that for a right triangle:
The square on the hypotenuse equals the sum of the squares on the other two sides.
Written as a formula, Pythagoras' theorem is as follows:
c² = a² + b²
c = √(a² + b²)
The hypotenuse is the longest side of a right triangle, and is thus located opposite the right angle.
So, if you know the lengths of two sides, all you have to do is square the two lengths, add the result, then take the square root of the sum to get the length of the hypotenuse.
Example Problem Using the Pythagorean Theorem
The sides of a triangle are 3 and 4 units long. What is the length of the hypotenuse?
Call the sides a, b, and c. Side c is the hypotenuse.
a = 3
b = 4
c = Unknown
So, according to the Pythagorean theorem:
c² = a² + b²
So, c² = 3² + 4² = 9 + 16 = 25
c = √25
c = 5
Sine, Cosine, and Tan of an Angle
A right triangle has one angle measuring 90 degrees. The side opposite this angle is known as the hypotenuse (another name for the longest side). The length of the hypotenuse can be discovered using Pythagoras' theorem, but to discover the other two sides, sine and cosine must be used. These are trigonometric functions of an angle.
In the diagram below, one of the angles is represented by the Greek letter θ. Side a is known as the "opposite" side and side b is "adjacent" to the angle θ.
The vertical lines "||" around the words below mean "length of."
sine θ = |opposite side| / |hypotenuse|
cosine θ = |adjacent side| / |hypotenuse|
Tan θ = |opposite side| / |adjacent side|
Sine and cosine apply to an angle, any angle, so it's possible to have two lines meeting at a point and to evaluate sine or cos for that angle. However, sine and cosine are derived from the sides of an imaginary right triangle superimposed on the lines.
In the second diagram below, you can imagine a right angled triangle superimposed on the purple triangle, from which the opposite, adjacent, hypotenuse sides can be determined.
Over a range 0 to 90 degrees, sine ranges from 0 to 1, and cos ranges from 1 to 0.
Remember, sine and cosine only depend on the angle, not the size of the triangle. So if the length a changes in the diagram below when the triangle changes in size, the hypotenuse c also changes in size, but the ratio of a to c remains constant. They are similar triangles.
Sine and cosine are sometimes abbreviated to sin and cos.
The Sine Rule
The ratio of the length of a side of a triangle to the sine of the angle opposite is constant for all three sides and angles.
So, in the diagram below:
a / sine A = b / sine B = c / sine C
Now, you can check the sine of an angle using a scientific calculator or look it up online. In the old days before scientific calculators, we had to look up the value of the sine or cos of an angle in a book of tables.
The opposite or reverse function of sine is arcsine or "inverse sine", sometimes written as sin-1. When you check the arcsine of a value, you're working out the angle which produced that value when the sine function was operated on it. So:
sin (30º) = 0.5 and sin-1(0.5) = 30º
The Sine Rule Should Be Use If ...
The length of one side and the magnitude of the angle opposite is known. Then, if any of the other remaining angles or sides are known, all the angles and sides can be worked out.
The Cosine Rule
For a triangle with sides a, b, and c, if a and b are known and C is the included angle (the angle between the sides), C can be worked out with the cosine rule. The formula is as follows:
c2 = a2 + b2 - 2abCos C
The Cosine Rule Should Be Used If ...
- You know the lengths of two sides of a triangle and the included angle. You can then work out the length of the remaining side using the cosine rule.
- You know all the lengths of the sides but none of the angles.
Then, by rearranging the cosine rule equation:
C = Arccos ((a2 + b2 - c2) / 2ab)
The other angles can be worked out similarly.
How to Get the Area of a Triangle
There are three methods that can be used to discover the area of a triangle.
The area of a triangle can be determined by multiplying half the length of its base by the perpendicular height. Perpendicular means at right angles. But which side is the base? Well, you can use any of the three sides. Using a pencil, you can work out the area by drawing a perpendicular line from one side to the opposite corner using a set square, T-square, or protractor (or a carpenter's square if you're constructing something). Then, measure the length of the line and use the following formula to get the area:
Area = 1/2ah
"a" represents the length of the base of the triangle and "h" represents the height of the perpendicular line.
The simple method above requires you to actually measure the height of a triangle. If you know the length of two of the sides and the included angle, you can work out the area analytically using sine and cosine (see diagram below).
Use Heron's formula. All you need to know are the lengths of the three sides.
Area = √(s(s - a)(s - b)(s - c))
Where s is the semiperimeter of the triangle
s = (a + b + c)/2
Three Ways of Working Out the Area of a TriangleClick thumbnail to view full-size
If you've made it this far, you've learned numerous helpful methods to discover different aspects of a triangle. With all this information, you may be confused as to when you should use which method. The table below should help you identify which rule to use depending on the parameters you have been given.
Which Rule Do I Use?
Rule to Use
Triangle is right and I know length of two sides.
Use Pythagoras's Theorem to work out remaining side and sine rule to work out angles.
Triangle is right and I know the length of one side and one angle
Use the trigonometric identities sine and cosine to work out the other sides and sum of angles (180 degrees) to work out remaining angle.
I know the length of two sides and the angle between them.
Use the cosine rule to work out remaining side and sine rule to work out remaining angles.
I know the length of two sides and the angle opposite one of them.
Use the sine rule to work out remaining angles and side.
I know the length of one side and all three angles.
Use the sine rule to work out the remaining sides.
I know the lengths of all three sides
Use the cosine rule in reverse to work out each angle. C = Arccos ((a² + b² - c²) / 2ab)
FAQs About Triangles
Below are some frequently asked questions about triangles.
What Is the Hypotenuse of a Triangle?
The hypotenuse of a triangle is its longest side.
What Do the Sides of a Triangle Add up to?
The sum of the sides of a triangle depend on the individual lengths of each side. Unlike the interior angles of a triangle, which always add up to 180 degrees, the sum of the sides of a triangle must be calculated after determining the length of each of the sides.
How Do You Calculate the Area of a Triangle?
To calculate the area of a triangle, simply use the formula:
Area = 1/2ah
"a" represents the length of the base of the triangle. "h" represents its height, which is discovered by drawing a perpendicular line from the base to the peak of the triangle.
How Do You Find the Third Side of a Triangle That Is Not Right?
If you know two sides and the angle between them, use the cosine rule and plug in the values for the sides b, c, and the angle A.
Next, solve for side a.
Then use the angle value and the sine rule to solve for angle B.
Finally, use your knowledge that the angles of all triangles add up to 180 degrees to find angle C.
How Do You Find the Missing Side of a Triangle?
Assuming the triangle is right, use the Pythagorean theorem to find the missing side of a triangle. The formula is as follows:
c² = a² + b²
c = √a² + b²
What Is the Name of a Triangle With Two Equal Sides?
A triangle with two equal sides and one side that is longer or shorter than the others is called an isosceles triangle.
What Is the Cosine Formula?
There are a few variations of the cosine formula these are:
a2 = b2 + c2 - 2bc cos A
b2 = a2 + c2 - 2ac cos B
c2 = a2 + b2 - 2ab cos C
How Do I Calculate the Volume of a Triangle?
Since a triangle is a plane and two-dimensional object, it is impossible to discover its volume. A triangle is flat. Thus, it has no volume.
Triangular prisms, on the other hand, are three-dimensional objects with a determinable volume. To determine the volume of a triangular prism, you must discover the area of the base of the prism, then multiply it by the height. The formula is as follows:
V = bh
In the above formula, "V" represents volume, "b" represents the area of the base of the triangular prism, and "h" represents the height of the triangular prism.
How Many Degrees Are There in a Triangle?
The interior angles of all triangles add up to 180 degrees.
How Do You Measure Angles?
You can use a protractor or a digital angle finder. These are useful for DIY and construction if you need to measure an angle between two sides, or transfer the angle to another object.
This digital protractor or angle finder comes in very useful when constructing stuff from wood or metal. I also use it as a replacement for a bevel gauge for transferring angles e.g. when marking the ends of rafters before cutting. The rules are graduated in inches and centimetres and angles can be measured to 0.1 degrees.
Triangles in the Real World
A triangle is the most basic polygon and can't be pushed out of shape easily, unlike a square. If you look closely, triangles are used in the designs of many machines and structures because the shape is so strong.
The strength of the triangle lies in the fact that when any of the corners are carrying weight, the side opposite acts as a tie, undergoing tension and preventing the framework from deforming. For example, on a roof truss the horizontal ties provide strength and prevent the roof from spreading out at the eaves.
The sides of a triangle can also act as struts, but in this case they undergo compression. An example is a shelf bracket or the struts on the underside of an airplane wing or the tail wing itself.
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Questions & Answers
- Helpful 10
What is the formula for finding what an equilateral triangle of side a, b and c is?
Since the triangle is equilateral, all the angles are 60 degrees. However, the length of at least one side must be known. Once you know that length, since the triangle is equilateral, you know the length of the other sides because all sides are of equal length.Helpful 15
How do I find the missing side of a triangle when only its height is known?
Use Pythagoras's Theorem. Add the sine, cosine and tan relationships between angles and the hypotenuse of the triangle to work out the remaining side.Helpful 6
How do I find the value if all three sides of a scalene triangle are unknown?
If all the sides are unknown, you can't solve the triangle. You need to know at least two angles and one side, or two sides and one angle, or one side and one angle if the triangle is a right-angled triangle.Helpful 8
How would you solve this problem: The angle of elevation of the top of a tree from point P due west of the tree is 40 degrees. From a second point Q due east of the tree, the angle of elevation is 32 degrees. If the distance between P and Q is 200m, find the height of the tree, correct to four significant figures?
One angle is 40 degrees, the other angle is 32 degrees, therefore the third angle opposite the base PQ is 180 - (32 + 40) = 108 degrees.
You know one side of the triangle has length PQ = 200 m
A right angled triangle is formed between point P, the top of the tree and its base and also point Q, the top of the tree and its base.
The best way to solve is to find the hypotenuse of one of the triangles.
So use the triangle with vertex P.
Call the point at the top of the tree T
Call the height of the tree H
The angle formed between sides PT and QT was worked out as 108 degrees.
Using the Sine Rule, PQ / Sin(108) = PT/ Sin(32)
So for the right angled triangle we chose, PT is the hypotenuse.
Rearranging the equation above
PT = PQSin(32) / Sin(108)
Sin(40) = H / PT
So H = PTSin(40)
Substituting the value for the hypotenuse PT we calculated above gives
H = (PQSin(32) / Sin(108)) x Sin(40)
= 71.63 mHelpful 5
© 2016 Eugene Brennan