# Geometry Formulas Triangles

Welcome to the Math Salamanders' Geometry Formulas Triangles area.

Here you will find information about the properties of triangles and different triangle formulas and theorems.

## Geometry Formulas Triangles

### Geometry Formulas Triangles

Below are some of the triangle formulas and theorems for our Geometry Formulas Triangles page.

### Geometry Formulas Triangles - Angles in a Triangle

The angles in a triangle add up to 180°.

#### Missing Angle in a Triangle Example

Find the missing angle in the triangle below. We know that all three angles in the triangle must add up to 180°.

So we know that ? + 128 + 23 = 180.

This means that ? = 180 - 128 - 23 = 29°.

So the missing angle is 29°.

Example

### Geometry Formulas Triangles - Pythagoras' Theorem

#### Example 1) Find the missing side of this triangle. In this example, we need to find the hypotenuse (longest side of a right triangle).

So using pythagoras, the sum of the two smaller squares is equal to the square of the hypotenuse.

This gives us: $4^2 + 6^2 \; = \; ?^2$

Which means that $?^2 = 16 + 36 = 52$

So $? = \sqrt {52} = 7.21 \; to \; 2dp$

#### Example 2) Find the missing side of this triangle. In this example, we need to find the length of the base of the triangle, given the other two sides.

So using pythagoras, the sum of the two smaller squares is equal to the square of the hypotenuse.

This gives us $?^2 \; + \; 5^2 \; = \; 8^2$

So $?^2 \; = \; 8^2 \; - \; 5^2 \; = \; 64 - 25 \; = \; 39$

So $? \; = \; \sqrt {39} \; = \; 6.25 \; to \; 2dp$

### Geometry Formulas Triangles - Basic Triangle Trigonometry

#### Example 1) Find the length of the hypotenuse In the triangle below, we have been given the angle which is 35°

We have also been told the length of the adjacent side a, which is 8cm.

We need to find out the length of the hypotenuse h.

So as the letters a and h are the two letters involved, we need the cosine operation (CAH).

So the formula we need is $\cos (θ) \; = \; {a \over h}$

So $\cos (35°) = {8cm \over ?}$

This means that $? \; = \; {8cm \over \cos (35° ) }$

So $? \; = \; 9.77cm \; to \; 2dp.$

#### Example 2) Find the length of the missing side In the triangle below, we have been given the angle, which is 48°.

We have also been told the adjacent side a, which is 11m.

We need to find out the length of the opposite side o.

So as the letters o and a are the two letters involved, we need the tangent operation (TOA).

The formula we need is: $\tan (θ) \; = \; {o \over a}$

So $\tan (48°) \; = \; {? \over 11}$

So $? \; = \; 11 \times \tan (48°) \; = \; 12.22m \; to \; 2dp$

#### Example 3) Find the length of the missing side. In this triangle, we need to find the length of the opposite side of the triangle.

We have been given the angle and the hypotenuse.

So as the letters o and h are used, we need the sine operation (SOH).

The formula we need is: $\sin (θ) \; = \; {o \over h}$

So $\sin (42° ) \; = \; {? \over 4.2}$

So $? \; = \; 4.2 \times \sin (42° ) \; = \; 2.81cm \; to \; 2dp$

### Sine & Cosine Rule Examples

Example 1) Find the missing side. In this example, we have been given 2 of the angles, but only one side measurement.

The rule we need is the sine rule, as 2 angles are involved.

So using the sine rule gives us:

${12 \over \sin (123° )} \; = \; { ? \over \sin (27° )}$

So this means that:

$? \; = \; { 12 \times \sin (27° ) \over \sin (123° )}$

So $? \; = \; 6.50cm \; to \; 2dp.$

Example 2)

Find the missing angle in the triangle below. Give your answer to 1 decimal place. In this example, we have been given three sides and we need to find the angle.

The rule we need is the cosine rule which involves 3 sides and one angle.

It is easier to use the cos A formula, as it is the angle we want to find.

$\cos (A) \; = \; {b^2 \; + \; c^2 \; - \; a^2 \over 2bc}$

Using ? as our angle A, this gives us: $\cos (?) \; = \; {10^2 \; + \; 11^2 \; - \; 6^2 \over 2 \times 10 \times 11}$

So $\cos (?) \; = \; {185 \over 220} \; = \; {37 \over 44}$

This means that: $? \; = \; \cos^{-1} {37 \over 44}$

This gives us: $? \; = \; 32.8° \; to \; 1dp$

### More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

### Geometry Cheat Sheets

How to Print or Save these sheets

Need help with printing or saving?
Follow these 3 easy steps to get your worksheets printed out perfectly!

How to Print or Save these sheets

Need help with printing or saving?
Follow these 3 easy steps to get your worksheets printed out perfectly!