How to Calculate Triangles From Points

A triangle is simply three points with lines between them.

A triangle is a simple geometric shape that conforms to mathematical principles discovered in antiquity. Triangles are made up of three points joined together by lines held at certain angles to one another that conform to the "sum of internal angles rule"; i.e. that the angles will add up to 180 degrees. By knowing the positions of the points in the triangle relative to one another, it is possible to calculate the properties of triangles solely from these points.

Things You'll Need

• Scrap paper
• Graph paper
• Pencil
• Ruler
• Calculator

Instructions

1. Make a note of the coordinates of your three points on your scrap paper. Decide on a scale for the graph paper that will allow you to draw measurable lines between the points. Plot your three points on your graph paper using your pencil. Draw a straight line between each of the points on your graph paper using your ruler. Name the sides of your newly drawn triangle a, b and c.

2. Determine the length of your sides using the Distance Formula, which is a variant of the Pythagorean Theorem. The Distance Formula says the length of a side, d, is equal to the square root of (x2 -- x1)^2 plus (y2 -- y1)^2. The values of x1, y1, etc. are the coordinates of the two points making up the line. Use your calculator and scrap paper to work out each length and note them on your graph paper.

3. Use the cosine rule to determine the size of the largest angle. The cosine rule states that a^2 = b^2 + c^2 -2bc cos A, where A is the largest angle and a, b and c are the lengths of the sides of the triangle. A must therefore equal cos^-1 (b^2 + c^2 -- a^2) / 2bc. Use your calculator to work out the angle and note it on your graph paper.

4. Use the sine rule to calculate one of the other angles. The sine rule states that a / sin A = b / sin B therefore B = sin^-1 (b sin A) / a. Now that you know the size of angle A as well as the lengths of the sides, you can plug this into the sine rule. Again, use the calculator to determine the size of angle B and note this on your graph paper.

5. Subtract A and B from 180 to find the size of the final angle, C. This is because, as a triangle is half of a rectangle, all the angles in a triangle must add up to 180 degrees. Use the calculator to find C if necessary and note the size of this angle on your graph paper also.

Tips & Warnings

• If you do not want to do any calculations, or your calculator does not have the sin, cos and tan angle functions on it, you can also measure the size of the angles in your triangle using a protractor. To do this, line the base line of the protractor up with one of the sides of the triangle and check where the adjoining line touches on the curve of the protractor. This will tell you the angle.

• A triangle cannot have angles that add up to more than 180 degrees, so if your triangle seems to have larger angles than this allows, you have either calculated your angles incorrectly, or you noted positions of your points wrong.

References

• Teacher's Choice: Triangle Given 3 Points
• Math Open Reference: How to Construct (Draw) a Triangle Given Two Sides and the Included Angle
• Henry Bottomley's Home Page: Area of a Triangle