## Proving the Formula A = 1/2 ab sin(C)

#### Aligned To Common Core Standard:

**Trigonometry** - HSG-SRT.D.9

How Do You Prove the Area of a Triangle? The area of polygons is defined as the number of square units inside a polygon. A triangle is a three-sided polygon.
Triangle, the same shape of your pizza slice, is defined geometrically as a plane figure with three angles and three straight sides.
To calculate the area of a triangle, we first multiply the height by the base, and divide them by 2. the division by 2 indicates that the area of a triangle is one-half the area of a parallelogram. That means that a parallelogram can easily be divided into two triangles.
We know that the area of a parallelogram is equal to base times height (A = B x H).
The area of a triangle is one-half the area of a parallelogram. Thus, mathematically the area of a triangle is written as:
A = 1/2 b x h
Or A = (b x h)/2
Where: **b is the base and h is the height**, the height and base must lie perpendicularly to each other.

### Printable Worksheets And Lessons

- Area of a Triangle
Step-by-step Lesson - Find the area of a triangular area. This
has many applications in the real world.

- Guided Lesson
- This is more theoretical than anything they have seen before.
Might be too abstract for some students.

- Guided Lesson Explanation
- This all comes down to knowing how to manipulate three theorems
and going from there.

- Practice Worksheet
- I slaved over this one. I took me forever to think up these problems.

- Matching Worksheet
- Two of these problems really throw kids. Remind them to read carefully.

#### Homework Sheets

I had a really difficult time finding any quality examples of this skill in test form.

- Homework 1 - Acute triangle XYZ, with x, y, z being the respective opposite sides to angle X, angle Y, angle Z, and altitude h, drawn from angle X to Y.
- Homework 2 - We know that area of triangle = 1/2 x base x height.
- Homework 3 - The formula of sin W = opposite / hypotenuse.

#### Practice Worksheets

It time to start using Flashcards so that you can memorize the basic rules of this skill.

- Practice 1 - Prove: The area of triangle CDE = 1/2 cd Sin E.
- Practice 2 - Given side v = 25, side w = 40, and side x= 31.22. Prove: Find the area of triangle by sin rule formula.
- Practice 3 - Given side u = 150, side v = 250, and side w= 200. Find the area of triangle by sin rule formula.

#### Math Skill Quizzes

These question go in several directions, but are of the same root skill.

- Quiz 1 - Acute triangle XYZ, with x, y, z, being the respective opposite sides to angle X, angle Y, angle Z, and altitude, h, drawn from angle Y to y.
- Quiz 2 - Given side a = 6, side b = 10, and side c= 8. Find the area of triangle by sin rule formula.
- Quiz 3 - Prove area of triangle = 1/2 lm sin N.

### Why Determining the Area of a Triangle is Important?

We have worked with this formula and we get the concept that is we can determine the length of the base and the height of a triangle we can determine the size of the surface. What is funny is that very few students ever realize that the area of a rectangle and triangle are related. If you look at both formulas width x height (rectangle) and 1/2 base x height (triangle), you quickly see that a triangle is just a diagonal cut of a rectangle. The applications of the triangle in industry is huge. They can be used to find the area of circles and polygons which has an immense number of applications in and of itself. This geometric shape is often the fundamental structure in most bridge projects. The unique weight to strength ratio of this shape helps it find its way into forming the foundation of any project or product that requires a solid base.