Identify a Polygon from Sides, Angles or Diagonals

Count sides when the boundary is available

A polygon’s name starts with its number of sides: three gives a triangle, four a quadrilateral, five a pentagon, six a hexagon, and eight an octagon. Count boundary segments between successive vertices, not the diagonals drawn inside the figure. A line drawn across a polygon does not automatically add a side to its outside boundary. Also check that the figure closes; a chain of segments with an open end is not a polygon.

Recover the side count from the angle sum

If a simple polygon’s interior angles sum to S degrees, solve S = (n − 2)180 for n: n = S/180 + 2. With a total of 1260°, this gives 7 + 2 = 9 sides, a nonagon. The result must be an integer at least three. A purported total of 1300° would not produce an integer side count, so either the measurements are approximate or the stated information is inconsistent with an exact simple polygon problem.

Use a regular exterior angle

For a regular convex polygon, equal exterior turning angles add to 360°. If each is 24°, the side count is 360/24 = 15. Its corresponding interior angle is 180° − 24° = 156°. Dividing 360 by an arbitrary interior angle would use the wrong quantity. When the polygon is irregular, one exterior angle does not determine the side count because the other exterior angles can differ. Confirm the equal-angle or regularity condition before using this shortcut.

Recover the count from a diagonal total

If a convex polygon has 20 diagonals, solve n(n − 3)/2 = 20. This becomes n² − 3n − 40 = 0, or (n − 8)(n + 5) = 0. The valid side count is n = 8; the negative root cannot describe a polygon. With a regular octagon and side length 2, the linked tool can then calculate perimeter 16 and interior angle 135°. The diagonal count established n, but the side length was still needed to set the polygon’s size. A different positive side length changes size while preserving the identified side count and regular interior angles.

Check whether the polygon is regular

A polygon has a closed boundary made from straight segments. A regular polygon has equal side lengths and equal interior angles. Equal sides alone do not guarantee regularity: a rhombus can have equal sides but unequal angles. Likewise, a rectangle can have equal angles without equal sides. The regular-polygon calculator takes one side length and a side count, so its area and angle results depend on the regularity assumption. Do not feed an average side length into that form and treat the result as the measured area of an irregular plot.

Keep the measurement types distinct

Side lengths and perimeter use ordinary length units; area uses square units; angles use degrees or radians. If every length in a similar figure is multiplied by k, perimeter multiplies by k and area by k², while angles remain unchanged. This gives a direct check on a calculation with a converted side length. Converting centimetres to metres before calculating should describe the same figure as converting the final square centimetres to square metres, provided the area conversion factor is squared rather than copied from the length conversion.

List enough information to determine the answer

A side count identifies a broad class of shapes, but it does not usually determine their size or area. A perimeter likewise allows many different irregular shapes. Label which conditions are given and which are assumptions, and sketch the boundary before substituting numbers. For coordinate data, preserve the order around the boundary; sorting vertices simply by x-coordinate can create crossing segments. A photograph or drawing is not automatically to scale. Calculator output describes the ideal geometry entered into its fields, so retain the underlying measurements and assumptions when reporting a result.

Related guides

Polygon Diagonals: Count Them or Find Their Length

Distinguish a polygon’s diagonal count from diagonal length, derive the count formula, and calculate rectangle and regular-polygon examples.

Polygon Area Formulas for Regular and Irregular Shapes

Find regular-polygon area from perimeter and apothem, derive a side-length formula, and choose decomposition for an irregular boundary.

Area from Coordinates: Ordered Vertices and Shoelace

Calculate polygon area from ordered coordinates, work a shoelace example, and check vertex order, orientation and self-intersecting boundaries.