Polygon Calculator
Free polygon calculator: general regular n-gon area, perimeter, apothem and angles.
Enter any side count 3–24 plus side length for polygon area, perimeter and angles.
- Grade level
- 6–10
- Curriculum band
- Difficulty
- Beginner
- For this topic
- Time to master
- 1 week
- With daily practice
- Tools on page
- 10 on page
- Calculator + practice
Calculator
General n-gon from n + side
Polygon Calculator Summary
Breakdown
| Measure | Value |
|---|---|
| Sides (n) | 7 |
| Side (s) | 4 |
| Apothem | 4.153043 |
| Area | 58.142599 |
| Perimeter | 28 |
| Interior | 128.571429 deg |
| Exterior | 51.428571 deg |
Substitution: n=7 s=4
Results
Area
58.142599sq units
P a / 2
28units
n s
4.153043units
center to side
128.571429 deg
per vertex
Closed-form geometry on the dimensions given. Invalid or impossible shapes report no area rather than a guess.
Statistics
Live area and perimeter from your current dimensions.
Area
58.142599
square units
Perimeter
28
units
Apothem
4.153043
units
Interior angle
128.571429 deg
Visualizations
Measures compared
Area scale
Formula
Any regular n-gon is n center triangles; angle sums give interior and exterior directly.
Alternative Form
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| n | Sides | 3–24 | |
| s | Side | units | Edge |
Explanation
Interior sum is (n-2)180; dividing gives per-vertex; exterior is the supplement 360/n.
Derivation
Apothem derivation matches the regular-polygon page.
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Record the given dimensions.
Validate positivity and shape constraints.
Apply the area formula.
Apply the perimeter formula.
Compute auxiliary measures.
State the finished sentence.
Worked Examples
See how the calculator works with different scenarios.
Scenario
Seven sides of 4.
Inputs
Calculation
A=58.14; P=28
Result
A=58.14
Interpretation
Seven triangles.
Understanding Your Result
Your Result: Area: 58.142599 sq units
Typical range: Keep units consistent: lengths in, square units out for area. Doubling all lengths quadruples area but only doubles perimeter.
What This Means
Area measures the enclosed region (square units); perimeter measures the boundary length. Both update live from the same dimensions.
Why It Matters
Flooring, fencing, cutting stock and land measurement all bill from these two numbers — one for cover, one for edge.
Professional Interpretation
Report area with square units and perimeter with linear units; round to instrument precision plus a waste allowance.
Recommendations
- •Keep one unit across all inputs
- •Add 5-10 percent waste to material orders
- •Verify triangle inequality where applicable
- •Round only the final display
Learn — The Complete Guide
Polygon Calculator computes area and perimeter from direct dimensions using closed-form geometry.
The core concept
Interior sum is (n-2)180; dividing gives per-vertex; exterior is the supplement 360/n.
- Area is quadratic; perimeter is linear
- One unit in, matching units out
- Feasibility before arithmetic
Derivations — why the formulas work
Where the formula comes from
- 1Start from unit squares
- 2Integrate or dissect into rectangles/triangles
- 3Simplify to the closed form used here
Where you meet this in real life
Learn this first — prerequisites in order
- 1Measure lengths accurately
- 2Square and root fluently
- 3Convert length units
Careers that use this daily
- •Carpenter — stock and cuts
- •Landscaper — beds and edging
- •Surveyor — parcels and areas
Why it helps
- •Instant area plus perimeter
- •Constraint checks catch impossible shapes
- •Labeled diagram builds intuition
- •Copy-ready breakdown table
Best practices
- •Sketch and label first
- •Measure twice in one unit
- •Check feasibility before ordering
- •Add waste to orders
Limitations — what this tool does not do
- •Single shape per run
- •Flat Euclidean plane only
- •Exact inputs assumed
Curriculum map — where this topic lives in school
| Level | Grades | Standards | Skills |
|---|---|---|---|
| Middle school | 6–8 | 6.G, 7.G | Area and perimeter · Scale and units |
| High school | 9–10 | G-GMD, G-MG | Circles and polygons · Modeling with geometry |
Timeline — how this idea grew up
c. 300 BCE
Euclid organizes geometry
Elements proves area formulas from axioms.
250 BCE
Archimedes and pi
Circle area squeezed between polygons.
1600s
Formulas standardize
Algebra turns dissections into plug-in equations.
A little history
- • Heron formula is nearly 2000 years old.
- • Pi notation standardized in the 1700s.
Exam-day checklist
- • Label the diagram first.
- • Write the formula before numbers.
Myths vs facts
Practical Tips
- •Measure twice in one unit
- •Sketch before calculating
- •Check feasibility first
- •Add waste to materials
- •Track square vs linear units
Common Mistakes
Assumptions
This calculator makes the following assumptions to provide estimates:
Limitations
Be aware of these limitations when using this calculator:
Frequently Asked Questions
Answers to common questions about this calculator.
How many sides can I enter?
What are interior and exterior angles?
How do I floor a gazebo?
Related Calculators
Explore more calculators related to this topic.
Related Guides
Deepen your understanding with these educational resources.
Glossary
Technical terms used in mortgage calculations explained in plain language.
- Area
- Two-dimensional extent enclosed by a boundary, in square units.
- Perimeter
- Total distance around the boundary, in linear units.
- Apothem
- Distance from center to midpoint of a side in a regular polygon.
- Diagonal
- Segment joining non-adjacent vertices.
- Height
- Perpendicular distance from base to opposite side or vertex.
- Semi-perimeter
- Half the perimeter; used in Heron formula.
Authoritative References
Trusted sources and standards referenced by this calculator.
Applicable Standards
Diagrams & Illustrations
Visual intuition for every core idea. Select a card to enlarge it.
Did You Know?
Short intuition builders — great for class warm-ups.
Why pi appears
A circle encloses the most area per perimeter of any shape — nature uses it for bubbles, pipes and tree trunks.
Hexagons tile perfectly
Bees use hexagons because they tile with minimal wax per honey stored.
Pythagoras everywhere
Diagonals, heights and apothems all reduce to right triangles.
Compare Side by Side
Add sides to approach the circle.
| n | Area | Perimeter | Interior |
|---|---|---|---|
| 7 | 58.142599 | 28 | 128.571429 deg |
Recent Calculations
Your last runs on this page — stored only in this browser, never uploaded.
7-gon s=4: A=58.142599, P=28
Enter any side count 3–24 plus side length for polygon area, perimeter and angles.
Study Plan
A week-by-week path from prerequisites to mastery. Your checklist progress saves in this browser.
Prerequisites
- ▸Length measurement
- ▸Squares and roots
- ▸Units and conversion
Mastery progress — 0%
- I pick the right formula per shape
- I validate feasibility
- I track units correctly
- I estimate before computing
- 1
Week 1 · ~15 min/day
Rectangles and triangles
Goals
- Apply A=lw and Heron fluently
- Distinguish height from slant side
- Track square vs linear units
Practice
Solve 10 mixed shapes on this page.
- 2
Week 2 · ~20 min/day
Circles and polygons
Goals
- Use pi correctly
- Apply apothem formulas
- Size fencing from perimeter
Practice
Measure two rooms and a round object.
Exam tips
- •Draw and label before substituting.
- •Show the formula with numbers plugged in.
Practice Problems
0/4 solvedTry each problem first, peek at the hint only if stuck, then reveal the solution and mark it solved.
Compute area and perimeter with the defaults.
Convert the area to square feet.
Double every length. What happens?
Enter an impossible triangle (1, 2, 3).
Printable Worksheet
Generate a fresh problem set for class or homework. Print it, or copy the answer key separately.
- 1.18 × 47 =
- 2.8 ÷ 11 =
- 3.38 × 26 =
- 4.35 ÷ 4 =
- 5.21 − 16 =
- 6.2 + 26 =
- 7.14 ÷ 12 =
- 8.23 ÷ 3 =
- 9.49 ÷ 7 =
- 10.22 − 17 =
- 11.41 + 18 =
- 12.47 − 35 =
Formula Cheat Sheet
Every formula on this page in one printable table.
| Formula | Meaning | Example |
|---|---|---|
| A = P a / 2, a = s/(2 tan(pi/n)) | Headline area | Plug in defaults |
| P = sum of sides | Headline perimeter | Add every edge |
| d = sqrt(a^2+b^2) | Right-triangle diagonal | 3-4-5 gives 5 |
Was This Calculator Helpful?
Page Information
2026-09-29
3.1.0-math
Free to use for educational and professional purposes.