Shapes and Polygons

In Grade 6 you learned to name and sort 2D shapes. In Stage 8 the goal is to do this fluently — and to back up every answer with a reason from a shape's properties.

Remember the two ways to classify a triangle: by its sides (equilateral, isosceles, scalene) or by its angles (acute, right-angled, obtuse). A triangle can carry one name from each list.

For quadrilaterals, the family builds up:

• Square — 4 equal sides, 4 right angles, 4 lines of symmetry.
• Rectangle — opposite sides equal, 4 right angles, 2 lines of symmetry.
• Parallelogram — opposite sides equal and parallel, opposite angles equal.
• Rhombus — 4 equal sides, opposite sides parallel.
• Trapezium — exactly one pair of parallel sides.
• Kite — 2 pairs of equal adjacent sides.

A useful idea: a square is a rectangle, a rhombus and a parallelogram all at once — it follows the most rules.

Stage 8 builds your angle reasoning. There are two key facts for any polygon.

Interior angles. Split a polygon into triangles from one corner — an $n$-sided polygon gives $(n - 2)$ triangles, so: $\text{interior angle sum} = (n - 2) \times 180\degree.$

Exterior angles. At every corner, the exterior angle is the turn you make walking around the outside. The exterior angles of any polygon always add to $360\degree$.

For a regular polygon every angle is equal, which makes both rules quick:

• Each exterior angle $= 360\degree \div n$.
• Each interior angle $= 180\degree - \text{exterior angle}$ (they sit on a straight line together).

So a regular hexagon ($n = 6$) has each exterior angle $360\degree \div 6 = 60\degree$, and each interior angle $180\degree - 60\degree = 120\degree$.

You can always check with the interior sum: $(6 - 2) \times 180\degree = 720\degree$, and $720\degree \div 6 = 120\degree$. Both routes agree.

Two shapes are congruent when they are exactly the same — same shape and same size. If you cut one out, it would fit perfectly on top of the other.

The important idea: a shape can be turned, flipped or slid and still be congruent to the original. Moving a shape never changes its size, so it stays congruent. The symbol for congruent is $\cong$.

To decide if two shapes are congruent, check all their measurements match:

• For triangles, all three sides and all three angles correspond.
• The shapes can be in different positions or orientations — that does not matter.

Be careful: shapes that are the same shape but a different size are not congruent. A small square and a large square have the same shape, but because the sizes differ they are only similar, not congruent.

Congruence is the precise way of saying "these two are truly identical".

Constructing a shape means drawing it precisely, using proper instruments rather than guessing. Your toolkit is a sharp pencil, a ruler, a pair of compasses and a protractor.

Suppose you must construct a triangle given all three sides — say $6\text{ cm}$, $5\text{ cm}$ and $4\text{ cm}$.

The steps:

• Draw the longest side, $6\text{ cm}$, with the ruler — this is the base.
• Open the compasses to $5\text{ cm}$ and draw an arc from one end of the base.
• Open the compasses to $4\text{ cm}$ and draw an arc from the other end.
• Where the two arcs cross is the third corner. Join it to both ends.

If you are given two sides and the angle between them, use the protractor to mark the angle, then the ruler for the sides. Leave your construction arcs showing — they are the evidence your drawing is accurate, not guessed.

Symmetry is a property you can use to describe and sort shapes precisely.

A line of symmetry is a fold line — fold the shape along it and the two halves match exactly. A square has $4$ lines of symmetry; a rectangle has $2$; an equilateral triangle has $3$.

Rotational symmetry is about turning. The order of rotational symmetry counts how many times a shape looks the same as you spin it once all the way round. A square has order $4$; a rectangle order $2$; a scalene triangle order $1$.

There is a neat rule for regular polygons: a regular $n$-sided polygon always has exactly $n$ lines of symmetry and rotational symmetry of order $n$. So a regular pentagon has $5$ of each, and a regular octagon has $8$.

Symmetry gives you a reliable extra check. If a shape is described as "regular with $6$ lines of symmetry", you know straight away it is a regular hexagon.

The shape and polygon skills you have sharpened this year open the door to lots more geometry:

• Transformations — reflections, rotations and translations all produce shapes congruent to the original.
• 3D solids and nets — every face of a prism or pyramid is a polygon you can already classify.
• Angle problems — finding missing angles in complex diagrams builds straight on the polygon angle rules.
• In everyday life — design, architecture, art, engineering and even computer graphics rely on knowing how shapes behave.

If a later geometry topic feels tricky, a shape property is often underneath it. Keep this guide close — careful, reasoned classification gets easier every time.