Equilateral triangle.

A triangle with all three sides the same length is called an equilateral triangle. If all the sides are the same length, then all the angles must also be the same. In an equilateral triangle each angle equals 60° (180 ¸ 3).

Isosceles triangle.

An isosceles triangle has two sides the same length and two angles the same. The two angles at the 'feet' of these sides are equal, ie the angles that are not between the sides of the same length (see left).

Scalene triangle.

A scalene triangle has no sides or angles that are the same.

Right-angled triangle.

A right-angled triangle has one angle equal to 90°. the right-angle in the triangle is marked with a 'square' symbol. (see left).

A right-angled triangle can also be either isosceles or scalene.

The angles within a triangle must always add up to 180 degrees.

A + B + C= 180

So if you know the value of any two angles it is possible to work out the value of the third eg:

A = 180 - B - C

Example 1

So if A = 180 - B - C and B is 100° and C is 30°

then A = 180 - 100 - 30

So A = 50°

Right-angled triangle.

The right angle in a triangle is marked with a square rather than an arc (see C left). This tells you that the angle is a right angle, ie it is 90 degrees.

So in this case you only need to know one of the other angles, as C is given by its notation.

A = 180 - B - 90 this is the same as A = 90 - B and also B = 90 - A
This is because we know that 90 of the 180 degrees in the triangle are angle C. So the remaining 90 degrees must be made up of the other two.

Example 2

So A = 90 - B, where B is 36°

then A = 90 - 36

So A = 54 °

When a triangle has sides that are same length.

A triangle with all three sides the same length is called an equilateral triangle. If all the sides are the same length, then all the angles must also be the same.

In an equilateral triangle each angle is 60 degrees (180 ¸ 3)

When a triangle has two sides the same length it is called an isosceles triangle. The two angles at the 'feet' of these sides are also equal, ie the angles that are not between the sides of same length (see left)

The coloured marks on the diagrams left show that the sides and angles are the same.

Example 3

So if A = 180 - B - C and C is 72°

then A = 180 - B - 72

so A = 108 - B this is the same as B = 108 - A and as A + B = 108

If A + B = 108 and A and B are the same (as it is an isoscoles triangle)

then A = 108/2

so A = 54 °

In summary we can state that the angles in the isosceles triangle are A = 54 degrees, B = 54 ° (being the same as A) and C = 72 °

Sometimes you may need to work out the value of an angle within a word problem, without a picture of the triangle in question.

eg In a traingle ABC what is the value of A when C is 72 °? Angles A and B are equal.

This is the same question as example 3 above. It might help you to work out the angle by doing a rough sketch of the triangle and giving the angles letters and marking the angles that are the same.

Activity 1

If you want to practice finding the missing angle in a triangle, have a go at the questions below

1. In a traingle ABC, what is the value of A where B = 56° and C = 84°?
2. In a triangle DEF, what is the value of E where D = 25° and F = 37°?
3. In a right-angled triangle GHI, what is the value of G where H is the right angle and I is 48°?
4. In an isosceles triangle JKL, what is the value of J where K = 71°? K an L are equal.
5. In a triangle MNP, what is the value of P where M = 112° and N and P are equal?

Are you right? If not, have another look at the example earlier in this topic or have a look at some of the 'Resources You Can Use' that are recommended for this topic.

Finding the length of one side when you know the other two.

There is a relationship between the length of the sides of any right-angled triangle. This is often referred to as Pythagoras' theorem.

The theorem (or relationship) is named after Pythagoras, a Greek who lived around 500BC. Although the relationship was known long before Pythagoras lived, he is thought to be responsible for proving that the relationship ALWAYS exists for any size right-angled triangle.

Pythagoras' theorem.

The relationship can be described as:
"the square on the hypotenuse is equal to the sum of the squares on the other two sides"

This means that if the length of each side of the triangle is made into a square, the area of the square on the longest side is equal to the areas of the squares on the other two sides added together.

The hypotenuse.

Pythagoras' theorem can also be written as a formula.

a² + b² = c²

see left

Example 4

In a right-angled triangle, what is the length of side c if a = 3 and b = 4?

Given a² + b² = c²

so 3² + 4² = c²

9 + 16 = c²

25 = c²

c=Ö25

So c = 5

For more help with squares and square roots see the sub topic 'Powers and Roots' in the menu on the left of the screen.

Example 5

In a right-angled triangle, what is the length of side b if a = 5 and c = 13?

Given a² + b² = c²

so 5² + b² = 13²

25 + b² = 169

Now rearrange the formula so that b² is on its own on the left hand side. You can do this by taking away 25 from the left hand side. Anything that you do to one side, you must do to the other side so the equation still balances. So take 25 away from the right hand side too.

For more help with rearranging equations see the sub topic 'Formulae and algebra' in the menu on the left of the screen.

25 + b² - 25 = 169 - 25

b² = 169 - 25

b² = 144

b = Ö144

So b = 12

Activity 2

If you want to practice finding the length of a side of a right-angled triangle using Pythagoras' theorem, have a go at the questions below:

For questions 1-4 use the triangle ABC left as a guide. Produce all answers to 2 decimal places:

1. What is the length of c where a = 7 and b = 24?
2. What is the length of c where a = 5 and b = 11?
3. What is the length of a where b = 8 and c = 30?
4. What is the length of b where a = 2 and c = 7?
5. An isosceles triangle has a hypotonuse of 18 cm, what are the lengths of the other two sides?

Are you right? If not, have another look at the example earlier in this topic or have a look at some of the 'Resources You Can Use' that are recommended for this topic.

Relationships exist between the size of an angle and and the length of any two of the sides in a right-angled triangle. These relationships are called trigonometric ratios.

There are six of these ratios but the main three that you need to know are sine, cosine and tangent.

Trigonometric ratio.

sine A = a/c or opposite/hypotenuse - ie the length of the side opposite the angle divided by the length of the hypotenuse.

This is sometimes written as SOH - Sine, Opposite, Hypotenuse

cosine A = b/c or adjacent/hypotonuse - ie the length of the side adjacent (or next) to the angle divided by the length of the hypotenuse.

This is sometimes written as CAH - Cosine, Adjacent, Hypotenuse

tangent A = a/b or opposite/adjacent - ie the length of the side opposite the angle divided by the length of the side adjacent (or next) to the angle.

This is sometimes written as TOA - Tangent, Opposite, Adjacent

To help remember all three ratios, you can join up the initials as follows:

SOHCAHTOA (pronounced sock - a - toe - a)

To work out the sine, cosine and tangent of an angle, you will need to use either a scientific or graphical calculator, which have these functions.

If you do not have a scientific or graphical calculator you can use the Windows calculator (in accessories) and convert it to scientific mode by selecting 'View' then 'Scientific'.

For more information about scientific and graphical calculators see the sub topic 'Calculators and IT' in the menu on the left of the screen.

What is the length of side a, when c = 11cm and the angle A is 35°?

The first step is to identify wich of the ratios we can use. We need to find the length of the side Opposite and we know the length of the Hypotenuse. Looking at SOHCAHTOA we can use the sine ratio.

So sine A = a/c (this is usually written sin A = a/c)
Cosine and tangent are abbreviated to cos and tan.

So sin 35 = a/11

Rearrange the formula by multiplying both sides by 11.

sin 35 x 11 = a

You will need to use your calculator to work out the sine of 35°
To do this input 35 then press the sin key.
sin 35 = 0.5736 (4dp)
Have a go at this yourself to check that you are comfortable with the process.

So 0.5736 x 11 = a

Therefore a = 6.31cm (2dp)

For more help with rounding see the sub topic 'Decimals' in the menu on the left of the screen.
For more help with rearranging equations, see the sub topic 'Formulae and algebra' in the menu on the left of the screen.

Example 7

What is the length of side c, when b = 8 cm and angle A is 53°?

The first step is to identify which of the ratios we can use. We need to find the length of the Hypotenuse and we know the length of the Adjacent side. Looking at SOHCAHTOA we can use the cosine ratio.

So cos A = b/c

So cos 53 = 8/c

Rearrange the formula by multiplying both sides by c.

cos 53 x c = 8

Rearrange the formula by dividing both sides by cos 53.

So c = 8/cos 53

This has the effect of putting the side you are looking for on its own on one side of the equation. If you are confident about rearranging formulae you could do both the steps above at the same time. If you are not so confident, do each stage separately. It may seem to take a long time but you can check that you are right.

You will need to use your calculator to work out the cosine of 53°
To do this input 53 then press the cos key.
cos 53 = 0.6018 (4dp)

So c = 8/0.6018

Therefore c= 13.29cm (2dp)

For more help with rounding see the sub topic, 'Decimals' in the menu on the left of the screen.
For more help with rearranging equations, see the sub topic 'Formulae and algebra' in the menu on the left of the screen.

Activity 3

If you want to practice using sine, cosine and tangent to find the length of sides, have a go at the questions below:

Use the triangle ABC left as a guide. Produce all answers to 2 decimal places:

1. What is the length of a where b = 7 and A = 60°?
2. What is the length of a where c= 25 and A = 18°?
3. What is the length of b where c = 12.7 and A= 38°?
4. What is the length of c where a = 2 and A = 27.5°?
5. What is the length of c where b = 32 and A = 50°?

Tip: remember to choose the appropriate ratio first.

Are you right? If not, have another look at the example earlier in this section or have a look at some of the 'Resources You Can Use' that are recommended for this topic.

Relationships exist between the size of an angle and and the length of any two of the sides in a right-angled triangle. These relationships are called trigonometric ratios.

There are six of these ratios but the main three that you need to know are sine, cosine and tangent.

Trigonometric ratio.

These are the same ratios as explained above in the section on finding the length of one side when you know one angle and one side (above).

sine A = a/c or opposite/hypotenuse - ie the length of the side opposite the angle divided by the length of the hypotenuse.

This is sometimes written as SOH - Sine, Opposite, Hypotenuse

cosine A = b/c or adjacent/hypotonuse - ie the length of the side adjacent (or next) to the angle divided by the length of the hypotenuse.

This is sometimes written as CAH - Cosine, Adjacent, Hypotenuse

tangent A = a/b or opposite/adjacent - ie the length of the side opposite the angle divided by the length of the side adjacent (or next) to the angle.

This is sometimes written as TOA - Tangent, Opposite, Adjacent

To help remember all three ratios, you can join up the initials as follows:

SOHCAHTOA (pronounced sock - a - toe - a)

To work out the sine, cosine and tangent of an angle, you will need to use either a scientific or graphical calculator, which have these functions.

If you do not have a scientific or graphical calculator you can use the Windows calculator (in accessories) and convert it to scientific mode by selecting 'View' then 'Scientific'.

For more information about scientific and graphical calculators see the sub topic 'Calculators and IT' in the menu on the left of the screen.

What is the the size of angle A when side a = 8cm, and side c = 15cm ?

The first step is to identify wich of the ratios we can use. We know the length of the side Opposite and we know the length of the Hypotenuse. Looking at SOHCAHTOA we can use the sine ratio.

So sine A = a/c (this is usually written sin A = a/c)
Cosine and tangent are abbreviated to cos and tan.

So sin A = 8/25

So sin A = 0.32

You will need to use your calculator to work out the value of A when the sine of A is 0.32
To do this input 35 then press the shift (or inverse) key and then press the sin key.
Have a go at this yourself to check that you are comfortable with the process.

So A = 18.66° (2dp)

For more help with rounding see the sub topic, 'Decimals' in the menu on the left of the screen.
For more help with rearranging equations, see the sub topic 'Formulae and Algebra' in the menu on the left of the screen.

Activity 4

If you want to practice using sine, cosine and tangent to find an angle when you know two sides, have a go at the questions below:

Use the triangle ABC left as a guide. Produce all answers to 2 decimal places:

1. What is the the size of angle A where b = 7 and c = 19?
2. What is the the size of angle A where a = 74 and c = 79?
3. What is the the size of angle A where a= 6.5 and c = 11?
4. What is the the size of angle A where b = 18 and c = 35?
5. What is the the size of angle A where a = 22.4 and b = 8.7?

Tip: remember to choose the appropriate ratio first.

Are you right? If not, have another look at the example earlier in this section or have a look at some of the 'Resources You Can Use' that are recommended for this topic.