A 100-sided polygon is called a hectogon (also hecatontagon), and a regular one has 100 equal sides and angles.
If you’ve ever seen a “100-sided shape” mentioned in a worksheet, a math video, or a design app, you’re not alone. It sounds wild at first, then you realize it’s just a polygon with a lot of sides.
This article explains what the shape is called, what makes it regular or irregular, and the core number facts teachers like to test. You’ll also get a couple of quick calculations you can reuse any time a many-sided polygon shows up.
What Is A 100 Sided Shape In Plain Terms
A 100-sided shape in geometry is a polygon made from 100 straight line segments that meet end-to-end and close back on themselves. Each meeting point is a vertex (corner). Each segment is a side.
When people say “100-sided shape,” they usually mean a simple polygon: the sides don’t cross over each other. A self-crossing version exists too, but it’s a different topic and it behaves differently with angles and area.
Most school problems also assume the polygon is regular, meaning all sides match in length and all interior angles match in size. A regular 100-gon has a clean, balanced look. An irregular one can be lopsided, stretched, or have mixed angle sizes, while still keeping 100 sides.
What It’s Called And Why The Name Looks Odd
The most common name for a 100-sided polygon is hectogon. You may also see hecatontagon. Both point to the same side count.
“Hecto-” is a prefix tied to one hundred. “-gon” comes from a Greek root related to angles. Put them together and you get a one-word label for “one hundred angles,” which matches the idea of a one hundred-sided polygon.
Math naming has a mix of habits. For some side counts, one spelling wins in classrooms and textbooks. For others, you’ll run into two or three names and all of them get understood. If you write “100-gon” on a test, most teachers accept it, since it’s clear and unambiguous.
Regular And Irregular 100-Sided Polygons
A polygon can have 100 sides and still come in many shapes. The difference comes from side lengths and angle sizes.
Regular 100-gon
A regular 100-gon has:
- 100 equal sides
- 100 equal interior angles
- symmetry all the way around the center
In drawings, a regular 100-gon can look almost like a circle. That’s one reason it’s used in geometry talks about circles and limits: as you increase the side count of a regular polygon, the outline hugs a circle more tightly.
Irregular 100-gon
An irregular 100-gon still has 100 sides, yet it may have:
- different side lengths
- mixed interior angles
- corners that bunch up in one area and spread out in another
Irregular polygons show up in real tasks like mapping outlines, modeling parts, or tracing shapes from images. In those cases, the side count is just one descriptor, not the whole story.
Angle Facts You Can Calculate Fast
Angle questions are where a 100-sided shape turns from “cool trivia” into clear math. Two formulas do most of the work for a simple polygon.
Sum Of Interior Angles
The sum of interior angles of any simple n-sided polygon is:
(n − 2) × 180°
For n = 100, the sum is (100 − 2) × 180° = 98 × 180° = 17,640°.
Each Interior Angle In A Regular 100-gon
If the 100-gon is regular, each interior angle is the total sum split evenly across 100 angles:
[(n − 2) × 180°] ÷ n
For 100 sides, that gives 17,640° ÷ 100 = 176.4° per interior angle.
Exterior Angles
The exterior angles of any convex polygon (one exterior angle at each vertex, taken in the same turning direction) always add to 360°.
So a regular 100-gon has an exterior angle of 360° ÷ 100 = 3.6° at each corner. That tiny turn is another reason the outline feels so round.
Quick Comparison Table For Common Polygons And A 100-gon
It helps to anchor “100 sides” against shapes you already know. The table below lists regular interior angles for a few familiar polygons, then jumps to higher side counts.
| Sides (n) | Common Name | Regular Interior Angle |
|---|---|---|
| 3 | Triangle | 60° |
| 4 | Quadrilateral (Square) | 90° |
| 5 | Pentagon | 108° |
| 6 | Hexagon | 120° |
| 8 | Octagon | 135° |
| 10 | Decagon | 144° |
| 12 | Dodecagon | 150° |
| 20 | 20-gon | 162° |
| 50 | 50-gon | 172.8° |
| 100 | Hectogon (Hecatontagon) | 176.4° |
How Mathematicians Define A Polygon
Before you do area or angle work, it helps to know what “polygon” means in plain geometry language: a closed figure made of straight segments, joined so the segments don’t cross. If you want a formal definition and standard terms like side, vertex, convex, and concave, Britannica’s entry on polygon in mathematics lays it out clearly.
A 100-sided polygon can be convex or concave. Convex means every interior angle is less than 180°. Concave means at least one interior angle is greater than 180°. Regular polygons are convex by definition.
Perimeter And Area When You Know The Side Length
Perimeter is the easy win. If you know the length of one side, multiply by 100:
Perimeter = 100 × side length
Area depends on whether the 100-gon is regular.
Area Of A Regular 100-gon
If a regular 100-gon has side length s, one clean method uses the apothem (the distance from the center to the midpoint of a side). The area formula is:
Area = (Perimeter × apothem) ÷ 2
To get the apothem from the side length, you can use a right triangle in the center. The central angle is 360° ÷ 100 = 3.6°, and half of that is 1.8°. That lets you relate the apothem to s with basic trig.
If you want a reference page that states the name and the “100 sides” fact in one line, Wolfram MathWorld’s entry on Hectogon is a handy check.
Area Of An Irregular 100-gon
For irregular shapes, there isn’t one plug-in formula from just a side length. In practice, people use one of these approaches:
- Split into triangles: draw diagonals from a chosen vertex and add triangle areas.
- Use coordinates: list vertices in order and apply the shoelace method to get area.
- Use software tools: CAD and graphing apps can compute area from points.
In classwork, you’ll usually be given more structure: equal sides, equal angles, a known radius, or an apothem. That extra information turns the question into one you can finish.
Table Of Core Formulas For A 100-Sided Polygon
These are the formulas that show up most in lessons and exam questions. Keep them together and you can handle most 100-gon problems without hunting through notes.
| Quantity | General Formula (n Sides) | Value For n = 100 |
|---|---|---|
| Sum of interior angles | (n − 2) × 180° | 17,640° |
| One interior angle (regular) | [(n − 2) × 180°] ÷ n | 176.4° |
| One exterior angle (regular, convex) | 360° ÷ n | 3.6° |
| Number of diagonals | n(n − 3) ÷ 2 | 4,850 |
| Perimeter (regular) | n × s | 100s |
| Area (regular, using apothem) | (Perimeter × a) ÷ 2 | (100s × a) ÷ 2 |
| Central angle (regular) | 360° ÷ n | 3.6° |
Diagonal Count And Why It Blows Up Fast
A diagonal connects two non-adjacent vertices. As the side count climbs, diagonals multiply quickly.
The diagonal formula is:
n(n − 3) ÷ 2
With 100 vertices, that’s 100 × 97 ÷ 2 = 9,700 ÷ 2 = 4,850 diagonals. That number alone explains why many diagrams of a 100-gon skip drawing every diagonal. The picture turns into a dense web.
How Close A Regular 100-gon Feels To A Circle
People often compare a regular 100-gon to a circle because the corners are so mild. Each exterior angle is only 3.6°, so the turn at each vertex is small.
You can make that “circle feel” more concrete by thinking in radii. If a regular 100-gon is inscribed in a circle, every vertex lies on the circle. The polygon sits inside the circle, and its perimeter is a little less than the circle’s circumference.
If the 100-gon is circumscribed around a circle (each side tangent to the circle), the polygon wraps around the circle, and its perimeter is a little more than the circle’s circumference. These two perimeters bracket the circle’s circumference, which is a classic stepping stone toward the idea of limits in geometry.
Where You Might See 100-Sided Shapes Outside Homework
Even if you never label a “hectogon” in daily life, many-sided polygons show up in places you’d recognize:
- Computer graphics: smooth curves get approximated by polygons when a model is rendered.
- 3D printing and CNC: curves can be stored as lots of short segments.
- Architecture drawings: round details may be drafted as many small straight edges.
- Game design: circles, wheels, and rings often start as regular polygons with high side counts.
In these settings, “100 sides” is one practical choice. More sides look smoother and cost more computing or file size. Fewer sides look more faceted and cost less.
Mistakes Students Make With 100-gon Questions
Most slip-ups fall into a few patterns. Catch them early and your answers clean up fast.
- Mixing interior and exterior angles: interior angles on a regular 100-gon are 176.4°, while exterior angles are 3.6°.
- Using the regular-polygon angle formula on an irregular shape: equal angles are not guaranteed unless the polygon is regular.
- Forgetting that diagonals don’t include sides: a diagonal can’t connect adjacent vertices.
- Dropping parentheses in (n − 2) × 180°: do the subtraction first, then multiply.
A Small Practice Set With Answers
Try these as a self-check. Each one uses a formula you already saw above.
- Find the sum of interior angles: (100 − 2) × 180° = 17,640°.
- Find one interior angle in a regular 100-gon: 17,640° ÷ 100 = 176.4°.
- Find the number of diagonals: 100(100 − 3) ÷ 2 = 4,850.
- Find the perimeter if each side is 2 cm: 100 × 2 cm = 200 cm.
If you can do these without pausing, you’re set for most classroom questions that mention a 100-sided polygon.
References & Sources
- Encyclopaedia Britannica.“Polygon | Definition, Examples, & Geometry.”Defines polygons and standard geometry terms used in this article.
- Wolfram MathWorld.“Hectogon.”States the name for a 100-sided polygon and provides a concise reference description.