Side Angle Side Formula

The side angle side formula which is also known as the SAS formula is used to calculate the area of the triangle using trigonometry rules. This formula which is based on the side angle side theorem helps us to calculate the area of a triangle. As the name suggests, side angle side represents the two sides and the angle between them. Let us explore more about the side angle side formula to calculate the area of a triangle.

What is Side Angle Side Formula?

The side angle side formula is the SAS area formula which means we can find the area of a triangle if the length of two sides of a triangle and its included angle is known. The SAS formula is expressed as:

Area of a triangle = (1/2) × side₁ × side₂ × sin (included angle)

The side angle side formula was created on the basis of the Side Angle Side Theorem. According to the Side Angle Side theorem, two triangles are said to be congruent if two sides and the angle that lies between these two sides are equal. The angle between the two sides is known as the included angle.

Derivation of Side Angle Side Formula

We know that the area of a triangle is = 1/2 × base × height. So, let us consider the following triangle to understand the derivation of the SAS formula using the steps given below.

• Step 1: If the two given sides are 'a' and 'b' and the included angle between them is 'c'.
• Step 2: If we draw a perpendicular 'p' from X to side YZ, then using the trigonometric ratio, we can write the value of p as, p = a × sin c considering p as the height, and applying the formula Sin c = p/a.
• Step 3: Since we know that the area of a triangle = 1/2 × base × height. Substituting the value of base as b and height as p, the area of the triangle will be = 1/2 × b × p.
• Step 4: Since p = a × sin c, the formula for the area of the triangle will be = 1/2 × b × a × sin c

Therefore, the side angle side formula or the area of the triangle using the SAS formula = 1/2 × a × b × sin c

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

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°?

Solution:

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

Area of a triangle = (1/2) × side₁ × side₂ × sin (included angle)

Given: side₁ = 5 cm, side₂ = 10 cm, 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 cm²

Answer: Thus, the area of the triangle is 12.5 cm²

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 us name side AB as c, and side AC as b

Let us follow the steps of the side angle side formula:

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

a² = b² + c² − 2bc cosA

a² = 5² + 7² − 2 × 5 × 7 × cos(49°)

a² = 25 + 49 − 70 × cos(49°)

a² = 74 − 70 × 0.6560...

a² = 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.