In a regular octagon ABCDEFGHIJ what would be the measure of ABC?

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dshepard37's profile pic

dshepard37 | High School Teacher | eNotes Newbie

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A regular Octagon means all sides and angles are equal.  Use the formula ((n-2)x180)/n where n is the number of sides to find your answer.  

((8-2)x180)/8  =  (6X180)/8  = 1080/8 = 135

The measure of ABC and every other angle inside the octagon is 135. 

 

Jyotsana's profile pic

Jyotsana | Student, Grade 10 | (Level 1) Valedictorian

Posted on

n=side of the shape

((n-2)*180)/n)

((8-2)*180)/8=135

The measure of the angle is 135 degree.

itssnigdha's profile pic

itssnigdha | Student, Grade 11 | (Level 2) Honors

Posted on

use this formula:-

{(2n-4)/n}x90, where n=no. of sides of a polygon..

In an octagon, n=8

therefore,

{(2x8-4)/8}x90 =  135 degrees..

hence measure of each interior angle of a regualar octagon= 135 degrees...

krishna-agrawala's profile pic

krishna-agrawala | College Teacher | (Level 3) Valedictorian

Posted on

I believe you are asking for the measure of angle ABC in your question above.

ABCDEFGHIJ has actually 10 vertices therefore it will be a polygon with 10 sides rather than an octagon, which has eight sides.

The method of finding out the measure of an angle of a polygon is same irrespective of number of sides in the polygon.

The sum of all the angles in a polygon with n sides is given by (2n-4) right angles.

Thus the sum of 3 angles of a triangle = 2*3 - 4 = 2 right angles.

Thus the sum of 4 angles of a rectangle = 2*4 - 4 = 4 right angles.

Thus the sum of 5 angles of a pentagon = 2*5 - 4 = 6 right angles.

Thus the sum of 6 angles of a hexagon = 2*6 - 4 = 8 right angles.

Thus the sum of 8 angles of a octagon = 2*8 - 4 = 12 right angles.

Thus the sum of 10 angles of a 10 sided polygon = 2*10 - 4 = 16 right angles.

In the given question the figure is a polygon with 10 sides. Therefore the sum of its ten angles will be 16 right angles.

As this is a regular polygon every angle will be equal.

Therefore the measure of each angle = (16*90)/10 =144 degrees.

Therefore sum of angle ABC = 144 degrees.

If we assume the figure to be a regular octagon, the measure of any one angle = (12*90)/8 = 135 degrees

 

neela's profile pic

neela | High School Teacher | (Level 3) Valedictorian

Posted on

A regular octagon is a convex plane shape, having 8 equal sides, 8 vertices , eight angles and 20 diagonals.

Consider a point O inside the octagon. Then Join OA,OB,OC, .....OG and OH. Now you have 8 triangles , OAB,OBC,OCD,....OFG,OGH and OGA and the total angles of these 8 triangles is 180*8 =1440 degree or 8*2=16 right angle. If you remove the 360 degree around the centre O , you get,1440-360= 1080 dgrees or 16-4=12 rigt angles. Therefore, the sum of the angles at the vertices of a regular octagon is 1080 degrees or 12 right angles. Each angle at the vertices of a regular octagon is, therefore, 1080/8= 135 degrees.

Measure of ABC:

We tell about the  angles, sides and  area of the triangle ABC.

Angles:

Since ABC is a triangle formed of the consecutive  of AB and BC of a regular Octogon, angle ABC= 135 degree and angle ACB=angle BAC = (80-135)/2 = 22.5 degree each.

The sides of  the triangle ABC:

From the triangle OAB, angle OAB= 360/8= 45 degrees.

Consider the triangle OAC, the angle AOC = (360/8)*2=90 degrees. Therefore, Triangle OAC is right angled at A. Therefore, by Pythagorus theorem, AC= sqrt (OA^2+OC^2)= (sqrt2)*r, where r is the radius of the circumcircle of the octagon.

AB =r*sqrt(2 - sqrt2) , where r is the radius of the circumcirle of the octagon a derived result for triangle for the isoscelus triangle where AB=AC and BC= sqrt(2-sqrt2) and angle ABC=135 deg using Pythagorus theorem.

Area of the triangle ABC:

ABC is an isosceles triangle bouded by AB=BC=r*sqrt(2-sqrt2)and BC= (sqrt2)*r, Therefore the  area of the triangle is (r^2/4) (2-sqrt2) , using Heron's formula.

 

 

 

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