Polygon explorer

What the explorer shows

A polygon is a flat, closed shape with straight sides. A regular polygon has all sides equal and all angles equal. On this page the Sides stepper chooses 3 to 12 sides, and the tool draws the regular polygon with that many sides. Angles marks every interior angle with its size, Lines of symmetry draws each mirror line, and Split into triangles draws lines from one corner to all the others.

The readout gives the shape's name, from equilateral triangle and square up to regular dodecagon, the size of each interior angle, the total of all the angles and the number of lines of symmetry. A regular polygon with n sides always has n lines of symmetry.

Why the triangles matter

Lines from one corner split any polygon with n sides into n − 2 triangles. The angles of each triangle add up to 180°, so the angles of the polygon add up to (n − 2) × 180°. In a regular polygon every angle is the same, so each one is that total divided by n. Seeing the triangles turns the formula into something a child can explain instead of something to remember, and it works for irregular polygons too.

Common mix-ups

• Thinking more sides means smaller angles. The angles get bigger as sides are added, creeping toward 180° but never reaching it.
• Calling any six-sided shape a regular hexagon. A hexagon has six sides; it is regular only when all its sides and angles are equal.
• Counting n triangles instead of n − 2. The two sides next to the starting corner do not make triangles of their own.
• Thinking lines of symmetry always go through corners. In shapes with an even number of sides, half the lines go through the middles of opposite sides.

A 10-minute polygon session

• Start at 3 sides. Ask the child to name each shape as you step up to 6.
• Turn on Split into triangles and count them for each shape. Ask what the pattern is.
• Use the pattern to predict the angle sum for 10 sides, then check.
• Press Challenge. The numbers hide, so the child reasons with the formula before pressing Check.
• Finish with Lines of symmetry on 5 and 6 sides and describe the difference.

Where it fits by grade

• Grade 2: recognize and draw triangles, quadrilaterals, pentagons and hexagons by their sides and angles (2.G.A.1).
• Grade 4: lines of symmetry (4.G.A.3).
• Grades 7 and 8: angle facts (7.G.B.5) and the angle sum of a triangle (8.G.A.5), which extends to polygons.
• England: Year 5 distinguishes regular and irregular polygons, Year 6 finds unknown angles in regular polygons, and Key Stage 3 uses the triangle angle sum to find the angle sum of any polygon.

Learning more

Angles are a skill on Eurekami's skill map, and Daily Practice brings angle facts back when they start to fade. Lessons draw diagrams live from the numbers in each question, and when a polygon problem is tricky the AI tutor asks guiding questions so the student does the reasoning.