In this tutorial, you’ll learn the basic facts about triangles, 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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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.
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. (These are called degenerate triangles).
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.
This states that the sum of any two sides of a triangle must be greater than or equal to the remaining side.
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.
You can classify a triangle either by side length or internal angle.
Another topic that we’ll briefly cover 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.
There are many methods available 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
Pythagoras’ theorem uses trigonometry to discover the longest side (hypotenuse) of a right triangle (right angled triangle in British English). 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
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.
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
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. You can use this 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.
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.
Below are some frequently asked questions about triangles.
How Many Degrees Are There in a Triangle?
The interior angles of all triangles add up to 180 degrees.
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
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?
This formula gives the square on a side opposite an angle, knowing the angle between the other two known sides. For a triangle, with sides a,b and c and angles A, B and C the three formulas 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 to Figure Out the Sides of a Triangle if I Know All the Angles?
You need to know at least one side, otherwise you can’t work out the lengths of the triangle. There’s no unique triangle that has all angles the same. Triangles with the same angles are similar but the ratio of sides for any two triangles is the same.
How to Work Out the Sides of a Triangle if I know All the Sides?
Use the cosine rule in reverse.
The cosine rule states:
c2 = a2 + b2 – 2abCos C
Then, by rearranging the cosine rule equation, you can work out the angle
C = Arccos ((a2 + b2 – c2) / 2ab)
B = Arccos ((a2 + c2 – b2) / 2ac)
The third angle A is (180 – C – B)
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.
You can implement the cosine rule in Excel using the ACOS Excel function to evaluate arccos. This allows the included angle to be worked out, knowing all three sides of a triangle.
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