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How to find areas of triangles using the formula: A = ½ab×sinC.

This topic is part of the TCS FREE high school mathematics 'How-to Library', and will help you to find areas of triangles using the formula: A = ½ab×sinC.
(See the index page for a list of all available topics in the library.) To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.


Consider a triangle with two sides 'a' and 'b', with angle 'C' between them:

Let side 'a' be the base, then the perpendicular height, 'h' is measured from the top vertex of the triangle, making a 90° angle with the base:

Because 'h' creates a right angled triangle, with 'h' opposite to angle 'C', then 

    sinC = h/b

so:  h = bsinC

If you know the base and perpendicular height of a triangle, then you can find the area using this formula:

Area  =  ½ × base × perpendicular height

So in this case, with base = 'a' and perpendicular height = 'bsinC', we have

Area  =  ½ × a × b × sinC

NOTE: This formula works for any triangle, even when angle 'C' is greater than 90º


Maths Helper Plus can create a worked solution including a labelled diagram for finding the area of a triangle using the formula:  A = ½ab×sinC

Step 1 Download the free support file... We have created a Maths Helper Plus document to complement this topic. You can use it to practice the steps described below, and as a starting point for solving your own problems.

File name:  'Triangle Area 2.mhp'   File size: 7kb
Click here to download the file.

If you choose 'Open this file from its current location', then Maths Helper Plus should open the document immediately. If not, try the other option: 'Save this file to disk', then run Maths Helper Plus and choose the 'Open' command from the 'File' menu. Locate the saved file and open it. If you do not yet have Maths Helper Plus installed on your computer, click here for instructions.


Step 2  Display the parameters box

Press the F5 key to display the parameters box:


Click on the 'A' edit box with the mouse, then enter one of the known sides of your triangle. 

Similarly, click on the 'B' edit box and enter the other known side.

Click on the 'C' edit box, and enter the angle between the two known sides, in degrees.


Click the 'Update' button to refresh the diagram and calculations.


Step 3  Adjust the scale of the labelled diagram

If the labelled diagram is too large for the screen, then the scale needs to be reduced. Briefly press the F10 key enough times until the entire diagram is visible.


You can make the diagram bigger by holding down 'Ctrl' while you press F10.


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