Regular Polygon: Both Radii
Vertices sit on the circumcircle and the sides touch the incircle. Divide the side by 2·sin(180°/n) for the circumradius and by 2·tan(180°/n) for the inradius. As n grows the two converge and the shape becomes a circle.
Your numbers
Circumradius
6.533cm
Inradius
6.036cm
Perimeter
40cm
Area
120.71cm²
Formula
Circumradius = Side length ÷ (2 × sin(180 ÷ Number of sides))
The gap between the two radii tells you how large n must be to approximate a circle. At n = 36 they differ by under 0.4%.
What to enter
| Input | Default | Accepted range |
|---|---|---|
| Number of sidesHow many sides the polygon has, counting from three. | 8 | 3 ~ 100 |
| Side length (cm)One side of a regular polygon — they are all the same. | 5 | 0 and up |
Step by step
Quick reference table
Results when only Number of sides changes and everything else stays put.
| Number of sides | Circumradius (cm) | Inradius (cm) | Perimeter (cm) |
|---|---|---|---|
| 4 | 3.536 | 2.5 | 20 |
| 6 | 5 | 4.33 | 30 |
| 8 | 6.533 | 6.036 | 40 |
| 12 | 9.659 | 9.33 | 60 |
| 16 | 12.815 | 12.568 | 80 |
What each result means
| Result | At default values |
|---|---|
| Circumradius (cm)Radius of the circle through every vertex, wrapping the shape. | 6.533 |
| Inradius (cm)Radius of the circle touching all sides, sitting inside the shape. | 6.036 |
| Perimeter (cm)The total length around the outside of a shape. | 40 |
| Area (cm²)The space a shape covers — length squared. | 120.71 |
Common mistakes
The gap between the two radii tells you how large n must be to approximate a circle. At n = 36 they differ by under 0.4%.
Glossary
- Number of sides
- How many sides the polygon has, counting from three.
- Side length
- One side of a regular polygon — they are all the same.
- Circumradius
- Radius of the circle through every vertex, wrapping the shape.
- Inradius
- Radius of the circle touching all sides, sitting inside the shape.
- Perimeter
- The total length around the outside of a shape.
- Area
- The space a shape covers — length squared.
Frequently asked questions
Q. How is Regular Polygon: Both Radii calculated?
Circumradius = Side length ÷ (2 × sin(180 ÷ Number of sides)) — Vertices sit on the circumcircle and the sides touch the incircle. Divide the side by 2·sin(180°/n) for the circumradius and by 2·tan(180°/n) for the inradius. As n grows the two converge and the shape becomes a circle.
Q. Can you walk through an example?
With Number of sides 8, Side length 5cm, the answer is Circumradius 6.533cm.
Q. What do I need to enter?
Enter Number of sides, Side length. The result recalculates as you type, and an empty box counts as zero.
Q. How much does the answer move if I change a number?
Changing only Number of sides moves the answer to Number of sides 4 → Circumradius (cm) 3.536 and Number of sides 16 → Circumradius (cm) 12.815. The table below lays out five steps.
Q. How are the numbers rounded?
Money is shown to the nearest whole unit, percentages to one decimal place and everything else to two. What you see is rounded; the calculation itself carries the unrounded value forward.
Q. Anything to watch out for?
The gap between the two radii tells you how large n must be to approximate a circle. At n = 36 they differ by under 0.4%.
Related calculators
Keep your length units consistent across inputs — mixing centimetres and metres throws results far off.