# The measure of one angle of an octagon is twice that of the other seven angles. What is the measure of each angle?

Help would be greatly appreciated.

-

1
– Parth Feb 2 at 14:48

2
It is up to you what you accept, however, my post is not really an answer. It is a comment that can't be posted as a comment. Clayton and amWhy have given actual answers. – robjohn♦ Feb 2 at 19:50

## 3 Answers

Images don't work in comments, so I am posting this even though it is not really an answer.

Using Clayton's answer, and because Valentine's Day is just a couple of weeks away, it seemed appropriate to post this image of the octagon:

$\hspace{3.5cm}$

-
Thanks for the display picture! – Parth Feb 2 at 14:46
Prediction #0129384: This will get 100+ upvotes. – Parth Feb 2 at 14:50

1
+1 I want THIS heart for my valentine gravatar - (background blue?) ;-) Than it be made so that all the lengths of the sides are congruent? (hint: challenge!) – amWhy Feb 2 at 14:55
@amWhy: I started out making all the sides congruent, but as the angles are fixed (and therefore all the sides must be parallel to those as drawn), the bottom two sides need to be a different length. – robjohn♦ Feb 2 at 15:01

1
Would the downvoter care to comment? I explained why I was posting a non-answer, although I realize that that doesn't make it an answer. – robjohn♦ Feb 2 at 18:14
show 1 more comment

Hint: An octagon has $1080^\circ$, and you have the equation $$2\theta+\theta+\cdots+\theta=9\theta=1080^\circ.$$

-

+1 magic $9\theta$. – Babak S. Feb 3 at 3:34

The sum of the interior angles of an octagon is $1080^\circ$.

There are eight interior angles, one twice the measures of all others:

$7 \theta + (2\theta) = 1080^\circ$

Now solve for $\theta$ (which is the measure of the 7 equal angles),

and then compute $2\theta$, the measure of the angle that is twice the size of the other 7 interior angles.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 7, "mathjax_display_tex": 1, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.3, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9438839554786682, "perplexity_flag": "middle"}