Area of a Triangle Using Sine

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram CDF Player or other Wolfram Language products.

Requires a Wolfram Notebook System

Edit on desktop, mobile and cloud with any Wolfram Language product.

Let the triangle have side lengths , and . Then the area is . A proof is outlined in the Details.

Contributed by: Enrique Zeleny (July 2018)
Open content licensed under CC BY-NC-SA


Snapshots


Details

Here is a derivation of the formula. Draw a perpendicular from the point to the side at . The triangle is now divided into two right triangles and . Let . Let the lengths of the two segments of be and .

Then, by trigonometry, , , . (*)

The area of the triangle is

(the base is and the height is )

()

(substituting from (*))

(factoring out )

(using the expansion of the sine of a sum in reverse)

. (adding the two angles at )




Feedback (field required)
Email (field required) Name
Occupation Organization
Note: Your message & contact information may be shared with the author of any specific Demonstration for which you give feedback.
Send