Mathematics Year 6: Polygons: concave or convex, regular or irregular
Mathematics, Year 6 (Portugal): Tell concave from convex and regular from irregular polygons, and use the perimeter to solve problems with them.
Two questions about the shape of a polygon
Question one: is it convex? Pick any two points inside the polygon and join them with a segment. If that segment always stays inside, the polygon is convex; if some segment goes out of the figure, it is concave (it has a corner “pushed in”). Question two: is it regular? A polygon is regular when all its sides are congruent and all its angles are congruent; if either fails, it is irregular. A square is regular; a scalene triangle is irregular (and, like every triangle, convex).
Why two questions and not one
Because they check different things: one looks at whether the shape has “dents”, the other at whether it is the same all the way round. A square is convex and regular; a 6 by 3 rectangle is convex and irregular; an “L” or an arrow shape is concave and irregular. Describing a polygon well means saying both. And what you know helps you calculate: in a regular polygon with n sides of length ℓ, the perimeter is n × ℓ, with no need to add side by side. All the regular polygons you know are convex.
A case: the “L” and the square
On a 1 cm grid, the “L” polygon has vertices (0, 0), (4, 0), (4, 2), (2, 2), (2, 4), (0, 4). Convex? Take A = (3.5, 1) and B = (1, 3.5), both inside. The midpoint of [AB] is (2.25, 2.25), in the missing corner: the segment leaves the figure, so the “L” is concave (and irregular: sides of 4 cm and 2 cm). Perimeter: 4 + 2 + 2 + 2 + 2 + 4 = 16 cm, the same as a square of side 4 cm (4 × 4 = 16 cm). Areas: 4 × 4 − 2 × 2 = 12 cm² against 16 cm².
The trap: equal sides are not enough
“All sides equal” sounds like enough, but it is not. A rhombus with 5 cm sides has four congruent sides, but its angles can be, for example, 60° and 120°: it is irregular. A rectangle 6 cm by 3 cm has four right angles, but different sides: also irregular. Among quadrilaterals, only the square has both. Before you say “regular”, check the sides AND the angles, and do not trust the drawing alone: measure.
Where you see this outside school
A STOP sign is a regular octagon: eight equal sides and eight equal angles. The cells of a honeycomb look like regular hexagons, and a square tile is a regular polygon. The plan of an “L”-shaped house, the outline of a map or an arrow painted on the road are irregular polygons, and many are concave. Anyone who designs a game or a floor plan uses the same test: does a straight line between two points of the room leave the room?
Test what you learned
How can you tell that a polygon is concave?
- a) All its sides are congruent
- b) It has more than four sides
- c) Two points inside it can be joined by a segment that leaves the figure
- d) It has at least one right angle
Correct answer: c) Two points inside it can be joined by a segment that leaves the figure — Right: that is the test. If some segment between inner points goes outside, the polygon has a corner pushed in, so it is concave.
A regular pentagon with 12 cm sides and a regular hexagon have the same perimeter. How long is a side of the hexagon?
- a) 10 cm
- b) 12 cm
- c) 72 cm
- d) 5 cm
Correct answer: a) 10 cm — Yes: the pentagon has 5 × 12 = 60 cm, and the hexagon shares that among 6 equal sides: 60 ÷ 6 = 10 cm.
A rhombus has four sides of 5 cm. Is it a regular polygon?
- a) Yes, because all its sides are congruent
- b) No, a rhombus can never be regular
- c) Yes, as long as it is convex
- d) Only if its four angles are congruent too; then it is a square
Correct answer: d) Only if its four angles are congruent too; then it is a square — Right: regular needs equal sides AND equal angles. A rhombus that also has equal angles is a square.
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