Find apothem from sides, radius, area, or perimeter values. Review formulas, examples, and downloadable results. Simple layout keeps important geometry outputs easy to review.
| Polygon | Sides (n) | Side Length | Apothem | Perimeter | Area |
|---|---|---|---|---|---|
| Equilateral Triangle | 3 | 6 | 1.7321 | 18 | 15.5885 |
| Square | 4 | 6 | 3 | 24 | 36 |
| Regular Pentagon | 5 | 8 | 5.5055 | 40 | 110.1106 |
| Regular Hexagon | 6 | 10 | 8.6603 | 60 | 259.8076 |
From side length: apothem = side / (2 × tan(π / n))
From circumradius: apothem = circumradius × cos(π / n)
From perimeter: side = perimeter / n, then apothem = side / (2 × tan(π / n))
From area and perimeter: apothem = (2 × area) / perimeter
From area and sides: apothem = √(area / (n × tan(π / n)))
Related formulas: perimeter = n × side, area = (1 / 2) × perimeter × apothem, circumradius = apothem / cos(π / n)
A regular polygon apothem is the distance from the center to the midpoint of any side. It is also perpendicular to that side. This value matters in geometry, drafting, design, and measurement tasks. A reliable apothem calculator helps students, teachers, engineers, and analysts solve shape problems quickly. It also reduces manual trigonometry errors.
The apothem connects several important regular polygon formulas. It helps calculate area, side relationships, perimeter checks, and radius comparisons. When a polygon has equal sides and equal angles, the apothem creates congruent right triangles inside the shape. Those triangles make it easier to use tangent, cosine, and area rules. That is why apothem calculations appear in geometry homework, CAD planning, tiling layouts, and technical models.
This page supports multiple input methods for advanced use. You can find the apothem from side length and number of sides. You can also use circumradius and number of sides. Another option uses perimeter and number of sides. An area and perimeter mode is included for direct area solving. There is also an area and sides mode for inverse geometry work. These options make the tool useful for classroom problems and real project checks.
After calculation, the page returns the apothem, side length, perimeter, area, circumradius, central angle, exterior angle, and interior angle. This gives more than one answer. It also helps verify whether the chosen input set is consistent. Export buttons let you save results for reports, assignments, or audits.
Use this calculator when checking floor patterns, polygonal covers, machine parts, signs, icons, and mathematical diagrams. It is useful when converting between radius based and side based dimensions. The example table, formula section, and step guide also make the page easier to understand. For fast and accurate regular polygon apothem work, a structured calculator saves time and improves confidence.
Because each mode is tied to standard geometry relations, the calculator is suitable for study and quick professional review. It keeps the workflow simple, yet it still exposes the main derived values that matter when comparing regular polygon dimensions in real tasks.
The apothem is the perpendicular distance from the polygon center to the midpoint of any side. In regular polygons, every side shares the same apothem value.
This tool works with any regular polygon that has equal sides and equal angles. Examples include triangles, squares, pentagons, hexagons, and larger regular n-gons.
The minimum valid side count is 3. A regular polygon must have at least three sides, so values below 3 are not accepted.
Different problems provide different known values. Some questions give side length, some give radius, and others provide area or perimeter. Multiple modes make the calculator more useful.
Yes. The direct relation is apothem = 2 × area ÷ perimeter. This is one of the fastest regular polygon formulas when both measurements are known.
The apothem reaches the midpoint of a side. The circumradius reaches a vertex. Both start at the center, but they end at different points on the polygon.
When you use area, perimeter, and side count together, the values may not perfectly describe the same regular polygon. The note warns you when the inputs do not closely match.
Yes. After a successful calculation, you can export the result table as a CSV file or a PDF file for later use.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.