Easy Ways to Calculate Polygon angles : Easy Ways to Calculate Polygon angles
Learning Objective : Learning Objective Understand that interior angles of polygons follow a set pattern of rules
Be able to work out the missing angles in any triangle
Interior angles of a triangle : Interior angles of a triangle In any triangle, the interior angles (those inside) add up to 180°
In this triangle a + b + c = 180°
If a = 65° and b = 50° c = ? a b c
Try these… : Try these… 70° x 95° x 30° 65° 42° x
Exterior angles of a triangle : Exterior angles of a triangle On this triangle opposite, the interior angle are labelled, a, b & c.
The exterior angles (those outside) are labelled d, e & f. a b f d e c
Exterior angles of a triangle : Exterior angles of a triangle If I was to walk from d down the sides of the triangle, past e and f, how far have I travelled?
360º
Therefore, the sum of the exterior angles of a triangle add up to 360º. a b c d e f
Exterior angles of a triangle : Exterior angles of a triangle If angle a was 35º what is angle d?
If angle b is 55º what is angle e?
If angle c is 90º what is angle f?
Once you have done this, add up angles d, e & f…what does this add up to? a b c d e f
Exterior Angles : Exterior Angles An exterior angle of a triangle is an angle, such as angle 1 in the figure, that is formed by a side of the triangle and an extension of a side.
Note that the measure of the exterior angle 1 is the sum of the measures of the two remote interior angles 3 and 4. To see why this is true, note that 1 2 3 4
Interior Angles of a polygon… : Interior Angles of a polygon… You know now that all the interior angles of a triangle add up to 180°.
You can use this knowledge to make working out interior angles in other polygons really easy. The quadrilateral below has been split into 2 triangles with the red line.
If we have 2 triangles making up this shape, the sum (total) of the interior angles = 2 x 180 = 360°.
The same trick applies to other polygons, for example: : The same trick applies to other polygons, for example: Here the sum of the interior angles = 3 x 180 = 540o. There is a pattern emerging here…look at the number of sides on the first picture and compare it to the number of triangles.
What’s the pattern???? : What’s the pattern???? First Picture: No. of sides = 4, No. of triangles = 2
Second Picture: No. of sides = 5, No. of triangles = 3
A polygon with 6 sides has 4 triangles, one with 7 sides has 5 triangles and so on. We can create a formula from this which will quickly tell us the number of triangles which will be found in a polygon of any number of sides.
Formula… : Formula… The Sum of a Polygon's interior angles = The number of sides minus 2 multiplied by 180.
Using the shorthand letter n to represent the number of sides we get the formula:
Sum of interior angles = (n - 2) x 180, or 180(n – 2).
The brackets are there to make sure that you subtract 2 from n before multiplying by 180.
Last thing….almost : Last thing….almost If you have a regular polygon (pentagon, hexagon etc.) all the interior angles will be the same!
So to find one interior angle you just divide the sum of the interior angles (which you now know how to calculate) by the number of sides!
Defo last thing.. : Defo last thing.. Exterior Angles
The interior angle of a polygon and the exterior angle both add up to 180°
The sum of all the exterior angles of any polygon = 360°.