Non-Right Triangles

Areas of non right triangles

1/2 Sine and Heron's Formulas

1/2 Sine formula

We use this formula when the information we know are 2 sides and the angle between them!

We call this the SAS (side-angle-side) case.

For the non-right triangle

with 2-sides and the angle between them known... (SAS)

Note: a, b, and C are just labels for the 2 sides and angle between them. They don't have to be labeled as 'a', 'b', or 'C' in the diagram!

Multiple Choice

Your turn!

Find the area.

• 52.5 m²

• 64.8 m²

• 81.4 m²

• 130m²

Example 2 solution

Remember: a, b, and C are just labels for the 2 sides and angle between them. They don't have to be labeled as 'a', 'b', or 'C' in the diagram!

Make sure your calculator is in "degree" mode and not "radian" mode.

Multiple Choice

One more before we move on!

What is the area of this triangle?

• 14.494 meters squared

• 108.184 meters squared

• 54.092 meters squared

• 54.35 meters squared

Heron's formula (SSS) side-side-side

When we know all 3 sides, we use Heron's formula.

Heron's formula uses a value 's' that is the semi-(or half) perimeter of the triangle you must calculate first.

Example 5 Solution

For a triangle with sides: 6, 11, 16

Find 's' first by adding all 3 sides and dividing by 2.

Then plug in the area formula:
A= $\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$  

Note: the order you choose 'a' 'b' and 'c' does not matter.

Multiple Choice

Your turn!

Find the area. Round to the nearest tenth.

• 27.0 m²

• 35.1 m²

• 37.9 m²

• 47.3 m²

Example 6 Solution

Since we know all 3 sides, use Heron's rule!

1st, find the semi-perimeter, 's'.

Then plug in the area formula.

Multiple Choice

Let's put your skills to practice!

Find the area of the triangle. Round to the nearest tenth.

• 10.8 meters squared

• 27.8 meters squared

• 52.3 meters squared

• 43.9 meters squared

It looks like you're ready to tackle the homework!