Calculator Inputs
Example Data Table
| Scenario | Inputs | Key output |
|---|---|---|
| Regular pentagon | Mode: Regular | Each interior angle = 108° |
| Missing interior angle | A=110°, B=95°, C=120°, D=100° | E = 115° (since total is 540°) |
| Missing exterior angle | A=70°, B=60°, C=80°, D=65° | E = 85° (since total is 360°) |
Examples assume a simple, convex pentagon.
Formulas Used
1) Sum of interior angles (pentagon)
For an n-sided polygon:
Suminterior = (n − 2) × 180°
For a 5-sided shape, (5 − 2) × 180° = 540°.
2) Regular pentagon angles
Each interior = 540° ÷ 5 = 108°
Each exterior = 360° ÷ 5 = 72°
Central angle = 360° ÷ 5 = 72°
3) Missing angle rules
If four interior angles are known:
Missing interior = 540° − (A + B + C + D)
If four exterior (turning) angles are known:
Missing exterior = 360° − (A + B + C + D)
How to Use This Calculator
- Select a mode: Regular, Missing interior, or Missing exterior.
- Choose degrees or radians to match your inputs.
- If you picked a missing-angle mode, enter four angles.
- Press Calculate to view results above the form.
- Use Download CSV or Download PDF for records.
1) Purpose of a five-sided angle check
This calculator helps you validate and complete pentagon angle sets without guesswork. Use it for geometry homework, CAD sketches, and inspection notes where a missing corner angle must be derived from measured values. It suits classroom practice and quick engineering sanity checks too daily.
2) Interior sum data for a pentagon
A simple pentagon always has the same interior total. With (n − 2) × 180° and n = 5, the sum is 540°. If you triangulate a pentagon, you get three triangles, explaining the fixed total.
3) Finding a missing interior angle
Enter four interior angles A–D, then compute the fifth as E = 540° − (A + B + C + D). For convex shapes, each interior angle should be between 0° and 180°. If E becomes ≤ 0° or ≥ 180°, recheck units, rounding, or measurement direction.
4) Exterior angles and turning totals
Standard exterior (turning) angles measure how much you rotate at each vertex while walking the outline. For any convex polygon, these turns add to 360°, one full rotation. The missing exterior is 360° − (A + B + C + D), assuming all angles use the same turning sense.
5) Interior–exterior relationship
When exterior angles are standard turning angles, each interior angle pairs with an exterior angle to form a straight line: interior + exterior = 180°. This converts a computed exterior into a corresponding interior. An exterior near 0° implies an interior near 180°.
6) Regular pentagon benchmark numbers
A regular pentagon is a useful reference case. Each interior angle equals 540° ÷ 5 = 108°. Each exterior and the central angle equal 360° ÷ 5 = 72°. If your average is far from 108°, the shape is not close to regular.
7) Degree and radian reporting
Select degrees for everyday measuring tools and radians for trigonometry work. Conversions follow rad = deg × π / 180 and deg = rad × 180 / π. For example, 108° ≈ 1.885 rad and 72° ≈ 1.257 rad.
8) Exports for documentation
After calculation, results appear above the form for quick review. Download CSV to paste into spreadsheets, or PDF to attach to reports. Use the precision selector to keep rounding consistent across teams, classrooms, or audit records. Saving the same unit and precision avoids mismatches during later recalculation.
Q1: Does the 540° interior total apply to every pentagon?
A: Yes for any simple (non‑self‑crossing) pentagon. Concave pentagons still sum to 540°, but one interior angle can be greater than 180°.
Q2: Why do exterior angles add to 360°?
A: Walking once around a convex polygon turns you exactly one full rotation. If you measure all exterior angles with the same turning direction, their total is 360°.
Q3: What exterior angle definition is used here?
A: The standard turning angle at each vertex, measured outside the polygon. For convex shapes it is positive, and interior + exterior equals 180°.
Q4: Can I enter angles in radians?
A: Yes. Choose radians, enter your values, and the calculator outputs and exports in radians. Internally it converts using π-based degree–radian formulas.
Q5: Why does a missing angle sometimes show as invalid?
A: In convex mode, a valid interior angle must be between 0° and 180°. If the computed value falls outside that range, your inputs are inconsistent or the pentagon is concave.
Q6: Can four angles define a unique pentagon shape?
A: Not uniquely. Angle sums can find a missing angle, but side lengths and geometry can still vary. For a unique shape, you need additional constraints.
Q7: Which export is better for reports?
A: Use CSV for spreadsheets and further calculations. Use PDF when you need a clean snapshot of the computed results card for sharing or printing.