Calculator
Formula Used
When Circumradius Is Known
A = (n × R² × sin(360° ÷ n)) ÷ 2
s = 2 × R × sin(180° ÷ n)
a = R × cos(180° ÷ n)
When Inradius or Apothem Is Known
A = n × a² × tan(180° ÷ n)
s = 2 × a × tan(180° ÷ n)
R = a ÷ cos(180° ÷ n)
Other Useful Relations
Perimeter = n × s
Area = 1/2 × Perimeter × Apothem
Interior Angle = ((n - 2) × 180) ÷ n
Exterior Angle = 360 ÷ n
Central Angle = 360 ÷ n
How to Use This Calculator
- Enter the number of polygon sides.
- Enter the radius value you already know.
- Select circumradius or inradius from the list.
- Add a unit label if needed.
- Choose how many decimal places you want.
- Click the calculate button.
- Read the result shown above the form.
- Download the result as CSV or PDF.
Example Data Table
| Shape | Sides | Known Radius Type | Radius | Side Length | Perimeter | Area |
|---|---|---|---|---|---|---|
| Hexagon | 6 | Circumradius | 10 cm | 10.0000 cm | 60.0000 cm | 259.8076 cm2 |
| Octagon | 8 | Inradius | 12 m | 9.9411 m | 79.5291 m | 477.7484 m2 |
| Pentagon | 5 | Circumradius | 7 ft | 8.2290 ft | 41.1450 ft | 116.5044 ft2 |
Regular Polygon Area with Radius Guide
Why This Calculator Is Useful
A regular polygon area calculator with radius helps you solve geometry problems faster. It works for triangles, pentagons, hexagons, octagons, and many larger shapes. You only need the number of sides and a radius value. The tool then finds area, perimeter, side length, and angles. This saves time in schoolwork and technical tasks. It also reduces manual calculation errors. Clean outputs make checking answers much easier.
What Radius Means in a Regular Polygon
A regular polygon has equal sides and equal angles. In this shape, two radius values matter. The circumradius runs from the center to any vertex. The inradius, also called the apothem, runs from the center to the middle of a side. Both values can be used to find area. This calculator supports each option. That makes it useful for more problem types. It also supports better accuracy when only one radius is known.
How the Area Is Found
The page uses standard geometry formulas. If circumradius is known, the tool first derives side length and apothem. If inradius is known, it derives the circumradius and side length. After that, it computes perimeter and area. The calculator also shows the central, interior, and exterior angles. These extra values help students understand the full shape. They also help builders, drafters, and designers verify dimensions before work begins.
Best Uses for This Tool
Use this regular polygon area calculator with radius for homework, exam practice, layout planning, floor sketches, and design checks. The unit label field keeps results organized. Decimal control helps you match school or project requirements. CSV export is useful for spreadsheets. PDF export is useful for sharing and printing. Because the output appears instantly above the form, users can compare values quickly. This makes the calculator practical, simple, and efficient.
Frequently Asked Questions
1. What is a regular polygon?
A regular polygon is a shape with equal side lengths and equal interior angles. Examples include an equilateral triangle, square, regular pentagon, and regular hexagon.
2. What is the difference between circumradius and inradius?
The circumradius goes from the center to a vertex. The inradius goes from the center to the midpoint of a side. The inradius is also called the apothem.
3. Can I calculate the area with only the radius?
Yes, but you must also know the number of sides. The same radius value gives different areas for different regular polygons.
4. Why does the calculator ask for the radius type?
The formulas change depending on the known radius. Selecting the correct type ensures the area, side length, and perimeter are all computed correctly.
5. Does this calculator work for a triangle or square?
Yes. Any regular polygon with three or more sides is supported. That includes regular triangles, squares, pentagons, hexagons, and larger shapes.
6. What unit does the result use?
The calculator uses the unit label you enter. Length values stay in that unit. Area values are shown in squared units.
7. Why are angle values included?
Angles help verify the shape. They are useful in geometry learning, drafting, layout work, and many measurement checks.
8. Can I save the result for reports?
Yes. The result can be exported as CSV for data work or as PDF for printing, sharing, and record keeping.