# Side Angle Side Formula

The side angle side formula also known as the SAS formula is used to calculate the height of the triangle with the help of trigonometry which in turn helps us to calculate the area of the triangle. As the name suggests, side angle side means that two sides and the angle between them is given. Let's explore more about the side angle side formula to calculate the area of a triangle.

## What is Side Angle Side Formula?

Below are steps listed to calculate the side angle side formula:

- The unknown side of the given triangle is found using the law of cosines.
- The smaller of the remaining two angles is found using the law of sines.
- Use the angle sum property of a triangle to calculate the third angle.

Thus, the side angle side formula is stated as follows:

Area of a triangle = (1/2) × side_{1} × side_{2} × sin (included angle)

Let's consider a triangle ABC as shown below:

The two sides given are 'a' and 'b' and the included angle between them is 'c'.

According to side angle side formula, the area of triangle ABC will be

Area of triangle ABC = (ab Sinc) / 2

Let's work on some problems to understand the side angle side formula.

**Break down tough concepts through simple visuals.**

## Examples Using Side Angle Side Formula

**Example 1:** What will be the area of a triangle whose sides are of length 5 cm and 10 cm and its included angle is 30 degrees?

**Solution:**

We know that the side angle side formula is given as:

Area of a triangle = (1/2) × side_{1} × side_{2} × sin (included angle)

Given: side_{1} = 5cm, side_{2} = 10cm, sin (included angle) = sin 30° = 1/2

Substituting the values,

Area = (1/2) × 5 × 10 × sin 30°

= (1/2) × 5 × 10 × (1/2)

= 12.5 cm^{2}

**Answer: **Thus, the area of the triangle is 12.5 cm^{2}

**Example 2:** In the triangle shown below, find all the dimensions using the side angle side formula.

**Solution:**

Given: Angle A = 49°, b = 5, c = 7

To find: a, Angle B, Angle C

Let's follow the steps of side angle side formula:

Step 1: To find the value of 'a' let's use the Law of Cosines

a^{2} = b^{2} + c^{2} − 2bc cosA

a^{2} = 5^{2} + 7^{2} − 2 × 5 × 7 × cos(49°)

a^{2} = 25 + 49 − 70 × cos(49°)

a^{2} = 74 − 70 × 0.6560...

a^{2} = 74 − 45.924... = 28.075...

a = √28.075...

a = 5.298..

**a = 5.30** (rounded upto 2 decimal places)

Step 2: To find the value of the smaller angle, we will use the law of sines

Angle B is smaller than Angle C as Angle B is opposite to the shorter side.

Thus, we will choose angle B and apply law of sines,

sin B / b = sin A / a

sin B / 5 = sin(49°) / 5.298...

sin B = (sin(49°) × 5) / 5.298...

sin B = 0.7122...

B = sin^{−1}(0.7122...)

**B = 45.4°** (rounded upto 1 decimal place)

Step 3: Now to find angle C, the angle sum property of the triangle can be used

C = 180° − 49° − 45.4°

**C = 85.6° **

**Answer: **Thus, we have calculated all the missing dimensions of the triangle.