Triangles, quadrilaterals, pentagons and hexagons from their sides and angles

When you reach for it

A child who has met defining attributes knows that a triangle is a closed shape with 3 straight sides, and that its size, color and the way it is turned do not matter. This lesson takes that habit to four families of shapes and runs it in both directions.

Naming starts from a shape and asks what it must be called. Count the straight sides, check that the shape is closed, and the count gives the name: 3 sides a triangle, 4 a quadrilateral, 5 a pentagon, 6 a hexagon.

Drawing starts from a rule and asks for a shape that fits it. “Draw a pentagon.” “Draw a shape with four sides and exactly one square corner.” A rule like that almost never has just one answer. Two children can draw quite different shapes and both be right, because the rule says only what the shape must have and leaves everything else free.

The skill matters because every later lesson on shapes talks this way. A rectangle is defined by its square corners, a rhombus by its equal sides, and a child who can read a rule and test a shape against it, or build a shape to fit it, can follow all of them.

The lesson runs over four class days, one part each.

Exercises 23 to 25 ask for reasons and fit the end of the third or fourth part. Have felt shapes and a felt board, drinking straws with small balls of clay (or tape) to join them, dot paper, plain paper with a square corner to test with, and, on a dry day, sidewalk chalk.

The procedure

Part 1: Sides and corners name the shape

The four families in this lesson are named by one thing only, how many straight sides the shape has. Every shape here is closed, so its edges join up all the way round with no gap, and every edge is a straight side.

The names carry their numbers: “tri” means three, “quadri” means four (and “lateral” means sides), “penta” means five and “hexa” means six.

A corner is a point where two sides meet and the edge turns. Many books also call a corner an angle, so “a triangle has 3 angles” means the same as “a triangle has 3 corners”. In this lesson an angle is only something to count, never something to measure.

Sides and corners always match. In the table the two counts are the same in every row, and that is no accident. Put a finger on one corner of a closed shape and walk it once around the edge. Each side the finger travels ends at a corner, and the next side starts there. When the finger is back where it began, it has passed exactly one corner at the end of each side. So a closed shape with straight sides has as many corners as sides. This gives a child two ways to name a shape and a way to check: count the sides, count the corners, and the two numbers must agree.

Counting all the way round. The hard part of naming a pentagon or a hexagon is counting its sides without skipping one or counting one twice. The fix is the same finger walk. Put a mark, a dot of chalk or a small sticker, on one corner. Start there and count each side as the finger runs along it. Stop when the finger comes back to the mark. Every side has been counted once, because the walk went round exactly once.

What does not change the name. A pentagon can be huge or tiny, red or blue, made of felt or of straws, lying flat on a side or balanced on a corner, with all its sides the same length or every side a different length. None of that is on the list. Only the count of straight sides, and the shape being closed, decides it. That is why a square and a rectangle are quadrilaterals too: each is closed with 4 straight sides. They have their own names as well, because they have square corners; one shape carrying two names is taken up later in shapes in more than one category.

Part 2: Drawing a shape from its number of sides

To draw a shape with a given number of sides, the easiest start is the corners. A pentagon has 5 corners, so put down 5 dots. Then join them with a ruler or a straight edge, going round from one dot to the next and back to the first. Each join is one straight side, and coming back to the first dot closes the shape. Five dots joined round give 5 sides.

The dots need a little care. Join them in order going round, so that no two lines cross: lines that cross make something that is not one of these shapes (see Boundaries). And no three dots in a row should sit on one straight line, or two of the sides join into one long side and the shape has one side fewer than planned (Example 8 shows this happening with straws).

Because the rule says only “5 straight sides, closed”, the dots can go anywhere that keeps those two promises. Spread them wide and the pentagon is big. Bunch them up and it is small. Put them round a circle and the sides come out nearly equal; scatter them and the pentagon is lopsided. Every one is a pentagon. This is the part of the lesson children enjoy most: a rule has many right answers, and checking an answer means checking the rule, not comparing pictures.

The straws show the same thing with materials. A straw is a side and a ball of clay is a corner. A closed shape made of 4 straws needs 4 balls of clay, one at each join, and it is a quadrilateral however the straws are bent at the joins. Changing the shape at a join, wider or narrower, never changes the name; adding or taking away a straw always does.

Part 3: Drawing to a rule with square corners

A rule can say more than the number of sides. The most useful extra in this lesson is the square corner, a corner shaped like the corner of a sheet of paper. To test a corner, lay the corner of a sheet of paper into it. If the two sides of the shape run exactly along the two edges of the paper, the corner is square. If the paper sticks out past a side, or leaves a gap, it is not. In drawings a square corner is often marked with a small square drawn inside it.

To draw a shape with a square corner in it, draw that corner first, by tracing round the corner of a sheet of paper or by following the lines on squared paper. Then finish the rest of the shape to fit the other parts of the rule. When the rule says exactly one square corner, check every other corner with the paper to make sure none of them is square by accident.

Rules like these still have many answers. “Four sides, exactly one square corner” fits a shape with a long side and a short side at the square corner and a slanted fourth side, and it also fits a shape with two equal sides at the square corner and a point sticking out opposite. What the rule decides is the count of sides and which corners are square; the lengths of the sides and how the other corners lean are left to the person drawing.

Some rules have no answer. A triangle can have one square corner, but never two (Example 6). A quadrilateral can have no square corners, one, two, or four, but never exactly three (Exercise 24). Finding that a rule cannot be met is a real answer, and the reason is part of it.

Part 4: What is not in these families

Three things put a shape outside all four families, however much it looks like one of them.

• A gap. Straight lines that do not join up all the way round make an open shape. It has no inside that is shut in, so it is not a triangle, quadrilateral, pentagon or hexagon, even if it has the right number of lines.
• A curved edge. A curved edge is not a straight side. A shape with one curved edge and four straight sides is not a pentagon; it is not in these families at all.
• The wrong count. A closed shape with only straight sides but 7 or 8 or more sides is not in the four families. Such shapes have names too, met in a later grade in naming shapes by their sides.

Two things look odd but do not take a shape out of its family.

• A corner that points in. A quadrilateral may have one corner pushed in, like an arrowhead. It is still closed, with 4 straight sides and 4 corners, so it is still a quadrilateral.
• A dot or join in the middle of a side. A corner is where the edge turns. If two straws lie end to end in one straight line, or a dot sits partway along a straight side, the edge goes straight on and there is no corner there. The two pieces make one side.

Worked examples

Boundaries

• Closed and straight, or no name here. The four names belong only to closed shapes whose edges are all straight. A gap or a single curved edge puts a shape outside every family, whatever its count.
• Sides meet only at corners. Each side touches the next one at a corner and touches no other side anywhere. Four straight lines drawn as a bow tie, crossing in the middle, are closed but are not a quadrilateral: the crossing point is not a corner, and the shape is two triangles meeting at a point. Joining the dots in order going round avoids this.
• A side is from one turn to the next. Pieces that line up straight make one side. Count turns, not straws, sticks or dots.
• Corners match sides only for closed shapes. An open shape of 4 straight lines has 3 places where lines meet, so in an open shape the two counts do not agree, and counting corners will not name it.
• “Square corner” is the only corner size in this lesson. A rule may say a corner is square or is not square. How wide a corner is beyond that, in numbers, is measuring angles, much later.
• Some rules cannot be met. A triangle with two square corners, or a quadrilateral with exactly three, does not exist. A rule describes what a shape must have; it does not promise that such a shape can be drawn.
• Flat shapes only. These names are for flat shapes. Solid shapes are described by their faces, edges and corners, in Faces, edges and vertices.

Where this goes next

Next in the same unit, splitting a rectangle into rows and columns of squares uses a quadrilateral with four square corners and cuts it into equal squares to count. Equal shares cuts circles and rectangles into halves, thirds and fourths, and composing and decomposing shapes builds new shapes out of these, each named by its own sides, not by its pieces.

Later, the rules grow. Rhombuses, rectangles and squares are quadrilaterals picked out by equal sides and square corners, and Shapes in more than one category, and quadrilaterals in none and the family tree of quadrilaterals sort the four-sided shapes into groups inside groups. Naming shapes by their sides carries the names past six and gives the whole family, closed with straight sides, its own name. Every one of those lessons reads a rule and tests a shape against it, the move practiced here.

Exercises

Part 1: name it

For Exercises 1 to 5, use the shapes below. Name each shape and say how many corners it has.

1. Shape A.
2. Shape B.
3. Shape C.
4. Shape D.
5. Shape E.
6. A closed shape has 6 straight sides. How many corners does it have, and what is it called?
7. Farhana has a felt hexagon and a felt triangle. How many corners do the two shapes have altogether?

Part 2: draw it from its sides

8. Draw a triangle with one side much longer than the other two.
9. On dot paper, draw two quadrilaterals that look different from each other, with every corner on a dot. What makes both of them quadrilaterals?
10. Siddiq uses all of his 6 straws to build one closed shape. What shape is it, if no two straws lie in a straight line? How many balls of clay does he need?
11. Busisiwe has 7 straws. Can she build a triangle and a quadrilateral and use every straw? How many balls of clay will the two shapes need?
12. Siddiq has 10 straws. He builds two shapes from the same family and uses every straw. What shapes are they?

Part 3: draw it to a rule

Draw a shape that fits each rule, and mark its square corners. If you think a rule cannot be met, say why.

13. 4 straight sides and 4 square corners. What other name could the shape have?
14. A triangle with a square corner, smaller than your hand.
15. A pentagon with exactly 3 square corners.
16. A quadrilateral with exactly 2 square corners.
17. A hexagon with exactly 2 square corners.
18. A quadrilateral with no square corners at all.

Part 4: does it belong?

For Exercises 19 to 22, use the shapes below. Does the shape belong to one of the four families? If it does, name it.

19. Shape A.
20. Shape B.
21. Shape C.
22. Shape D.

Reasons

23. Can a closed shape with only straight sides have 4 sides and 5 corners? Explain.
24. Can a quadrilateral have exactly 3 square corners? Explain.
25. Make one straight cut across a corner of a paper quadrilateral, so that the cut does not reach another corner. What shape is the big piece that is left? Explain.