Shapes and Polygons

By Stage 9 you do not just recall the polygon angle rules — you reason with them, writing a clear justification for every step.

The two key facts:

• The interior angle sum of an $n$-sided polygon is $(n - 2) \times 180\degree$.
• The exterior angles of any polygon always add up to $360\degree$.

For a regular polygon every angle is equal, so both rules become quick:

• Each exterior angle $= 360\degree \div n$.
• Each interior angle $= 180\degree - \text{exterior}$, because the two sit on a straight line.

A regular pentagon ($n = 5$) has each exterior angle $360\degree \div 5 = 72\degree$, and each interior angle $180\degree - 72\degree = 108\degree$. You can work backwards too: a regular polygon with interior angles of $150\degree$ has exterior angles of $30\degree$, so $n = 360\degree \div 30\degree = 12$ sides. Whatever you find, write the reason beside it.

Two shapes are congruent when they are identical — same shape and same size. Cut one out and it fits perfectly on the other, even after a turn, flip or slide.

Two shapes are similar when they are the same shape but a different size. All matching angles are equal, and every length has been multiplied by the same number — the scale factor.

To find the scale factor, divide a length on the larger shape by the matching length on the smaller one. Above, $6 \div 3 = 2$ and $8 \div 4 = 2$ — a scale factor of $2$, so the larger triangle is an enlargement of the smaller.

The key distinction: congruent shapes have a scale factor of exactly 1 (no change in size). Similar shapes have any other positive scale factor. All congruent shapes are similar, but most similar shapes are not congruent.

A scale factor tells you how much bigger or smaller a shape becomes. To create a similar shape, multiply every length by the scale factor.

• A scale factor greater than 1 makes the shape larger (an enlargement).
• A scale factor between 0 and 1 makes the shape smaller (a reduction).
• A scale factor of exactly 1 leaves the shape unchanged — the shapes are congruent.

A $3\text{ cm}$ by $2\text{ cm}$ rectangle enlarged by scale factor $3$ becomes $9\text{ cm}$ by $6\text{ cm}$ — every length tripled.

To go the other way, divide by the scale factor. A $9\text{ cm}$ by $6\text{ cm}$ rectangle reduced by scale factor $3$ returns to $3\text{ cm}$ by $2\text{ cm}$. The crucial point: a scale factor changes lengths, while every angle stays exactly the same. That is precisely why the enlarged shape is similar to the original.

A construction is a precise drawing made with proper instruments — a sharp pencil, a ruler and a pair of compasses — rather than by guessing or measuring with a protractor.

A useful construction is bisecting a line: finding its exact midpoint with a perpendicular line through it.

To bisect a line:

1. Open the compasses to more than half the line's length.
2. Draw an arc from each end of the line — above and below.
3. The arcs cross at two points. Join those points with a straight line.

That join passes exactly through the midpoint and meets the line at a perfect right angle — a perpendicular bisector. The same compass-and-arc idea constructs triangles from three given sides, and bisects angles. Always leave your construction arcs showing — they are the evidence your drawing is accurate, not guessed.

A scale drawing is an accurate, smaller (or larger) picture of a real object — a map, a floor plan, a model design. It uses a fixed scale that links a length on the drawing to a length in real life.

A scale is written as a ratio or a statement, such as 1 : 100 or "1 cm represents 1 m". Every length on the drawing has been divided by the scale factor.

To work with a scale drawing:

• Drawing to real: multiply the measured length by the scale. With "1 cm to 2 m", a $5\text{ cm}$ wall is $5 \times 2 = 10\text{ m}$ in real life.
• Real to drawing: divide the real length by the scale. A real $6\text{ m}$ wall is drawn $6 \div 2 = 3\text{ cm}$ long.

Always keep the units consistent and state the scale clearly. A scale drawing is only useful if a reader knows exactly what each centimetre represents.

The shape and polygon skills you have sharpened set you up for much more geometry:

• Transformations — enlargements use scale factors directly, and reflections, rotations and translations all produce congruent copies.
• Similar triangles — comparing matching sides with a scale factor leads into trigonometry, where similar right-angled triangles do the work.
• Maps and plans — scale drawings appear in geography, design and architecture.
• In everyday life, this thinking shapes model-making, navigation, engineering drawings and computer graphics.

If a later geometry topic feels tricky, a shape property, a scale factor or a construction is usually underneath it. Keep this guide close — careful, reasoned geometry gets easier every time you practise.