52 Exterior Angles Of A Polygon Answer Key

Exterior Angles of a Polygon: What You Actually Need

The sum of exterior angles of any convex polygon is 360 degrees. This is true whether the polygon has 3 sides or 52 sides or 500. When someone posts a 52 exterior angles of a polygon answer key online, they're almost always dealing with a regular polygon problem where you divide 360 by 52 to get approximately 6.923 degrees per exterior angle. The corresponding interior angle works out to roughly 173.077 degrees. Here's how you actually calculate it without overthinking. Take 360, divide by the number of sides. That's your exterior angle for a regular polygon. For 52 sides: 360 ÷ 52 = 6.923076...° You can leave it as a fraction — 360/52 reduces to 90/13 — which is the exact answer. Most answer keys will show 6.92° or 6.9° depending on how strict the rounding is. The interior angle is what most people actually need but forget to ask for. Interior + exterior = 180° on a straight line. So 180 - 90/13 = 2340/13 173.077°. Or use the interior angle formula directly: (n-2) × 180 / n = 50 × 180 / 52 = 9000/52 = 2340/13. Same result.

I remember grading a set of worksheets where a student calculated the exterior angle of a regular 52-gon and got 6.92°, but then squared it because they misread the question as asking for area-related computation. The answer key had the right number but the method section was completely wrong. This happens more often than you'd think with large-n polygons. The numbers look intimidating and students second-guess themselves into making arithmetic errors. One thing most answer keys don't mention explicitly: this only works for convex polygons. If your 52-gon is concave, the exterior angle definition gets messy because some angles point inward and the 360° rule still holds for the signed sum, but individual exterior angles can exceed 180° depending on how you define them. Stick to convex unless your problem specifies otherwise. That's the standard assumption in nearly every textbook and worksheet. Another edge case that trips people up. If the polygon is regular but you're given the apothem or side length and asked to find something else, the exterior angle doesn't change — it's always 360/n for a regular polygon regardless of size. I've seen students try to incorporate the side length into the exterior angle calculation when it has nothing to do with it. It doesn't.

For practical purposes, if you're looking at an answer key for a homework assignment or test prep, here's what the core values should be: Sum of exterior angles: 360° (for any convex polygon)
Each exterior angle (regular 52-gon): 90/13° or approximately 6.92°
Each interior angle (regular 52-gon): 2340/13° or approximately 173.08°
Sum of interior angles: 9000° If your answer key shows different numbers, check whether they're using radians instead of degrees. 360° = 2 radians, so each exterior angle in radians is 2/52 = /26 0.1208 rad. Some advanced geometry courses work in radians and the answer key will look wrong if you're expecting degrees.

There's also the issue of significant figures. If your teacher or textbook is strict about this, 360 has two or three significant figures depending on convention, and 52 has two. That means your final answer should probably be reported as 6.9°, not 6.923°. Most answer keys ignore sig figs in geometry, but if you're in a science-adjacent math class, it matters. A quick note on where these problems commonly appear. You'll see 52-gon exterior angle questions in competition math prep, especially contests that like using less common polygon sides to make calculator use mandatory or to force students to work with fractions rather than clean decimals. It's a deliberate design choice — 360/52 doesn't simplify to a nice number, so students who just punch it into a calculator and round prematurely will lose points on multi-step problems. If you need a downloadable reference sheet with these calculations laid out, most educator resource sites like Kuta Software, Math-Aids, or CommonCoreSheets have polygon angle worksheets you can print. Search for "exterior angles polygon worksheet" and filter by difficulty. The 52-gon specific version is rare because it's an unusual side count, but the same worksheets will have fill-in-the-blank problems where you plug in 52 yourself.

The main takeaway: the math is straightforward and the formulas don't change based on how many sides you have. The difficulty with a 52-gon isn't the concept, it's the arithmetic and the tendency to overcomplicate it because the number looks unusual.