Back to Exercise: Polygons: Properties, Construction, and Angle Sums

Exercises: Polygons, Construction, and Angle Sums

Use the interior angle sum formula $(n-2) \times 180\degree$ and remember: dividing by $n$ to get a single angle only works for regular polygons.

Grade 7·16 problems·~25 min·Uganda NCDC Primary 7 Mathematics·lesson·construction-03

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Recall / Warm-Up

Review what makes a polygon regular before working with angle sums.

$A square with equal sides and equal angles beside a rectangle with equal angles but unequal side lengths$

A rectangle has all four angles equal ( $90\degree$ each) but its sides are not all equal (two long, two short). Is a rectangle a regular polygon?

Yes — having all angles equal is enough on its own to make a polygon regular, regardless of what the side lengths happen to be, since the angle condition is treated as the defining one

No — a regular polygon needs both all sides equal AND all angles equal; the rectangle fails the equal-sides condition even though its angles are equal

No — a regular polygon needs all sides equal, but the angles do not need to match at all, since side length is treated as the only condition that determines regularity

What is the sum of the angles at a single point (all the way around, a full turn)?

$180\degree$

$90\degree$

$360\degree$

Fluency Practice

$A near-hexagon formed by stepping points around a circle, with a small gap where the sixth mark misses the starting point$

You are constructing a regular hexagon by stepping a circle's radius six times around its circumference, but midway through you accidentally widen the compass slightly. What happens?

Nothing — the hexagon still closes exactly, since only the very last of the six steps actually needs to use the correct, unchanged radius

The sixth mark misses the starting point, because the stepping radius no longer matches the circle's radius for every step

The hexagon still closes, but the shape it forms becomes a regular pentagon instead of a hexagon, since one step effectively merges with another

An equilateral triangle has all three sides of length $6$ cm. What can you say about its three angles?

They are all $60\degree$ , since equal sides in a triangle force equal angles, and the three angles of any triangle sum to $180\degree$

They could be any three values, since side lengths and angle sizes are treated as completely unrelated properties in a triangle

Only two of them are equal to each other; the third can be a different size even though all three sides are $6$ cm long

$A pentagon split into three triangles by diagonals drawn from one vertex$

A pentagon is split into triangles by drawing diagonals from one vertex. How many triangles result, and what is the pentagon's interior angle sum?

A regular pentagon has an interior angle sum of $540\degree$ . What is the size of EACH interior angle?

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