Grade 3 · 3.G.A.1

Shapes in more than one category, and quadrilaterals in none

A square has four equal sides and four square corners. So which group does it belong in, and where does a four-sided shape go when it fits none of the groups?

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Rhombuses, rectangles and squaresGrade 3

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Parallelograms, rectangles, rhombuses and squares: what each one guaranteesGrade 5

Where you are 7 in Quadrilaterals

Geometry11

Two-dimensional shapes15

Quadrilaterals7

▴ 2 above Rectangles, rhombuses and squares Rhombuses, rectangles and squares

Shapes in more than one category, and quadrilaterals in none

The quadrilateral family tree Parallelograms, rectangles, rhombuses and squares: what each one guarantees

Why this exists

The lesson Rhombuses, rectangles and squares gave three names, each with a test:

a rhombus passes the ruler test: all four sides are the same length;
a rectangle passes the card test: the corner of an index card fits all four corners;
a square passes both.

That last line has a surprise in it. A square passes the ruler test, so it is a rhombus. It passes the card test, so it is a rectangle. One shape, three names, and every one of them is right. People are like this too: one person can be a sister, a soccer player and a drummer all at once. Each name says one true thing about her.

The other surprise goes the opposite way. Plenty of four-sided shapes pass neither test. They are still quadrilaterals, but none of the three names fits them.

Sorting shapes on a mat makes both surprises easy to see. This lesson builds that mat, reads it, and then hunts for the shapes that land outside everything.

Definition

A category is a group of shapes that all have some attribute: something the shape must have, checked with a tool or by counting. Size, color and which way the shape is turned are not attributes here, because none of them changes what the ruler and the card say.

The categories in this lesson:

Category	Attribute every shape in it has
quadrilateral	four straight sides (and so four corners), closed up
rhombus	a quadrilateral with four sides of the same length
rectangle	a quadrilateral with four square corners
square	a quadrilateral with four equal sides and four square corners

A shape is in a category when it has that category’s attribute, whatever else it has. A shape can be in more than one category at once: it only has to have each attribute. A quadrilateral can also be in none of the categories rhombus, rectangle and square: it only has to fail both tests.

A square corner is one the corner of an index card fits exactly, as in the card test. Later lessons call it a right angle. In the figures, tick marks show sides of equal length and a small square marks a square corner, as before.

Intuition

This lesson runs over three days: one shape with several names, then the sorting mat, then the quadrilaterals that fit none of the names.

Part 1: one shape, several names

Hold up a square and ask the two questions from the last lesson.

Are all four sides the same length? Yes, so it is a rhombus.
Are all four corners square corners? Yes, so it is a rectangle.

Neither answer cancels the other. The word “rhombus” only asks about sides, and the word “rectangle” only asks about corners, so a square can answer yes to both. Its most exact name is square, because that name tells both facts at once, but “rhombus” and “rectangle” are true too. Calling a square a rectangle is like calling a robin a bird: less exact, still true.

Now go the other way. Line up a rhombus that leans, a long rectangle and a square. What do all three have? Each has four straight sides and four corners. That shared attribute is exactly what makes a shape a quadrilateral. So the three smaller categories all sit inside one larger category: the category is defined by what its members share, and every rhombus, rectangle and square shares the four sides.

Part 2: the sorting mat

Lay two loops of string on a mat so that they overlap. Label one loop four equal sides and the other four square corners. The whole mat is for quadrilaterals: a triangle or a five-sided shape does not go on it at all.

Figure 1The sorting mat. Every quadrilateral has exactly one place on it.

The two loops cut the mat into four parts, and each part answers the two questions its own way.

Part	Equal sides?	Square corners?	Shapes there
left only	yes	no	rhombus, not square
overlap	yes	yes	square
right only	no	yes	rectangle, not square
outside	no	no	none of the three

Every quadrilateral answers each question yes or no, so every quadrilateral has exactly one place on the mat. Two questions, two answers each, four places.

The mat also shows how the categories fit together. The whole left loop is the rhombuses, and the overlap is part of it: every square is a rhombus. The whole right loop is the rectangles, and the overlap is part of that too: every square is a rectangle. But each loop has a part outside the overlap, with shapes in it, so not every rhombus is a square and not every rectangle is a square.

Part 3: quadrilaterals in neither loop

The part of the mat outside both loops is not empty. Any quadrilateral that fails the ruler test and fails the card test goes there, and there are many of them: a door pushed out of shape, a slice of a pizza box cut on a slant, a lopsided scrap of paper.

A shape lands outside a loop when even one side or one corner breaks the rule. Three equal sides are not four equal sides. Two square corners are not four. So a good way to make a shape for this part of the mat is to start from a rectangle or a rhombus and move one corner.

Results

A shape can be in more than one category. A square is a quadrilateral, a rhombus, a rectangle and a square, all at once.

Every square is a rhombus, and every square is a rectangle. On the mat, the overlap sits inside each loop.

Not every rhombus is a square, and not every rectangle is a square. A rhombus that leans and a rectangle with two different side lengths are in a loop but not in the overlap.

Every rhombus, rectangle and square is a quadrilateral. They share the attribute “four straight sides”, which defines the larger category.

Some quadrilaterals are in none of the three smaller categories. They fail both tests and go on the mat outside both loops.

A quadrilateral with three square corners has a fourth. So no quadrilateral has exactly three square corners.

Why it is true

A shape is in every category whose attribute it has. That is what being in a category means (see the Definition). A square has four equal sides, so it is in the rhombus category. It has four square corners, so it is in the rectangle category. Having one attribute does not take the other away.

“Every” and “not every”. A sentence “every square is a rectangle” is true when no square can fail the card test, and the definition of a square promises all four corners pass. A sentence “every rectangle is a square” is false as soon as one rectangle is not a square: a rectangle 6 cm by 3 cm passes the card test and fails the ruler test, and that one shape is enough.

Three square corners force the fourth. Call the corners in order around the shape P, Q, R and S, and say P, Q and R are square. Lay the shape on grid paper with side PQ along a grid line, P on the left. The corners do not have to land where grid lines cross; only the directions matter, “straight across” like a row and “straight up” like a column. The corner at P is square, so side PS goes straight up. The corner at Q is square, so side QR goes straight up too. The corner at R is square, so side RS runs straight across. Now the last corner S is the end of PS, which goes straight up, and the end of RS, which goes straight across. A side going straight up and a side going straight across meet in a square corner, just as a column and a row do, so the corner at S is square as well.

Worked examples

Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, and Examples 5 to 7 with Part 3.

Example 1

All the names of one card

Hauwa trims an index card into a shape with four straight sides. With a ruler she finds the sides are 3 inches, 3 inches, 3 inches and 3 inches. She takes a second index card and does the card test: it fits all four corners exactly. Which of the names quadrilateral, rhombus, rectangle and square fit her shape?

Solution

Take the names one at a time and ask each one its own question.

Quadrilateral: the shape has four straight sides, so yes.
Rhombus: all four sides are 3 inches, so the ruler test passes. Yes.
Rectangle: the card fits all four corners, so the card test passes. Yes.
Square: it passes both tests. Yes.

All four names fit. The name that says the most is “square”, because it carries both facts at once. A friend told only “it is a rectangle” knows the corners are square but not that the sides match; that name is true, just less exact.

Example 2

What the cards have in common

Chukwu has five shape cards:

a rhombus with sides of 4 cm that leans, with no square corners;
a rectangle 6 cm long and 3 cm wide;
a square with sides of 3 cm;
a triangle;
a shape with five sides.

(a) Which name from the lesson fits the rhombus, the rectangle and the square, but not the last two cards? What attribute do those three share?

(b) Which name fits the rectangle and the square but not the rhombus?

Solution

(a) The rhombus, the rectangle and the square each have four straight sides and four corners. That shared attribute is what “quadrilateral” means, so all three are quadrilaterals. The triangle has three sides and the last card has five, so neither is a quadrilateral.

(b) The rectangle and the square both have four square corners, so both are rectangles. The rhombus leans and has no square corner, so it is not a rectangle.

(c) The rhombus and the square both have four equal sides, so both are rhombuses. The rectangle has sides of 6 cm and 3 cm, which are not equal, so it is not a rhombus.

The square showed up in both (b) and (c). That is the whole idea of this lesson: the square is the one card that is in both groups.

Example 3

Twelve cards on the sorting mat

Wangari sets out the sorting mat of Figure 1 and twelve quadrilateral cards. Every card comes with its four side lengths in centimeters, in order around the shape, and how many of its corners pass the card test.

Card	Sides (cm)	Square corners
A	4, 4, 4, 4	4
B	6, 3, 6, 3	4
C	5, 5, 5, 5	0
D	2, 2, 2, 2	4
E	7, 2, 7, 2	4
F	3, 3, 3, 3	0
G	5, 3, 5, 3	0
H	4, 6, 3, 5	0
I	8, 8, 8, 8	4
J	6, 6, 4, 4	0
K	4, 3, 5, 6	2
L	6, 6, 6, 6	0

Place every card on the mat.

Solution

Ask each card the two questions, and let the two answers choose its place.

Four equal sides? Yes for A, C, D, F, I and L, whose four lengths match. No for the others.

Four square corners? Yes for A, B, D, E and I. No for the others. Card K has two square corners, but the loop asks for four, so K’s answer is no.

Now put the answers together. In the table, “equal sides” means all four sides equal and “square corners” means all four corners square.

	Square corners	Not
Equal sides	A, D, I	C, F, L
Not	B, E	G, H, J, K

Figure 2Wangari’s twelve cards, sorted.

A, D and I answer yes twice, so they go in the overlap. They are squares.
C, F and L go in the “four equal sides” loop only. They are rhombuses that are not squares.
B and E go in the “four square corners” loop only. They are rectangles that are not squares.
G, H, J and K go outside both loops. Card H is the plainest case: no two of its sides are equal and none of its corners is square.

As a check, 3 + 3 + 2 + 4 = 12, so every card has a place and none was placed twice.

Example 4

Reading the mat: true or false

Use the sorted mat from Example 3 to decide whether each sentence is true or false.

(a) Every square is a rectangle.

(b) Every rectangle is a square.

(d) Every quadrilateral is a rhombus.

Solution

A sentence that starts “every” makes a promise about all the shapes in a group. It is false as soon as one shape breaks it.

(a) True. The squares live in the overlap, and the overlap is part of the “four square corners” loop. A square must have four square corners, so no square can be outside that loop.

(b) False. Card B is in the “four square corners” loop, so it is a rectangle, but it sits outside the overlap: its sides are 6 cm and 3 cm. One such card is enough. Sentences (a) and (b) use the same two words in the opposite order, and only one of them is true: every square is a rectangle, but not every rectangle is a square.

(d) False. Card G is on the mat, so it is a quadrilateral, but it is outside the “four equal sides” loop. It is not a rhombus.

Example 5

Moving one corner

Chukwu draws a rectangle 5 cm long and 3 cm tall on centimeter grid paper. He wants a shape that goes outside both loops, so he keeps the bottom side and the left side and moves the top right corner one square to the right, as in Figure 3. Where does the new shape go on the mat?

Figure 3Chukwu moves one corner of a rectangle. The squares of the grid are 1 centimeter on each side.

Solution

First the sides. The bottom still runs along a grid line for 5 squares, so it is 5 cm. The left side is 3 cm. The top now runs 6 squares, so it is 6 cm. The right side slants, so it cannot be measured by counting squares; a ruler says it is a little more than 3 cm. Already 5 cm and 6 cm differ, so the four sides are not all equal. The shape is outside the “four equal sides” loop.

Now the corners. The two corners on the left were not touched, and they still sit where a row meets a column of the grid, so the card fits both. At the two right corners the side slants away from the grid lines: at the bottom right the card leaves a gap, and at the top right the card sticks out. Two square corners are not four, so the shape is outside the “four square corners” loop as well.

The shape goes on the mat outside both loops. Moving one corner broke both tests at once: it changed the length of two sides and the shape of two corners.

Example 6

Can a shape have exactly three square corners?

Hauwa wants a card for the mat that has exactly three square corners. Can she make one? Where would it go?

Solution

Try to draw it. Put the first square corner at the bottom left of some grid paper and draw the bottom side along a grid line, say 4 squares. Make the bottom right corner square too: the right side has to go straight up a grid line, say 2 squares. Now make the third corner, at the top right, square: the top side has to go straight left along a row.

Look at what is left. The left side starts at the bottom left corner and must go straight up, since that corner is square. The top side runs straight left along a row. The two meet where a column of the grid meets a row, and a column and a row always meet in a square corner. The fourth corner comes out square whether Hauwa wants it or not.

The same thing happens however long she makes the sides: this is the reason given in Why it is true. So Hauwa cannot make a quadrilateral with exactly three square corners. Any card she draws this way has four, and it goes in the “four square corners” loop. With the lengths above, its sides are 4, 2, 4 and 2 squares, so it is a rectangle that is not a square.

Three equal sides are a different story: a quadrilateral can have three equal sides and a fourth that is longer or shorter (Exercise 20 asks for one).

Example 7

Guess the card

Wangari hides one card and gives three clues.

It is inside the “four square corners” loop.
It is not in the overlap.
One side is 5 cm, and the side next to it is 2 cm.

What are all four side lengths? Which names fit the card? How far is it once around the edge?

Solution

Clue 1 says all four corners are square, so the card is a rectangle. Clue 2 says it is not in the overlap, so it does not have four equal sides: it is not a rhombus and not a square. That fits clue 3, where two sides already differ.

In a rectangle, sides across from each other are equal, as the last lesson showed. So the side across from the 5 cm side is 5 cm, and the side across from the 2 cm side is 2 cm. The sides in order are 5, 2, 5 and 2 cm.

The names that fit are quadrilateral and rectangle. Once around the edge is

5 + 2 + 5 + 2 = 14 \text{ cm}

That distance around is the card’s Perimeter.

Boundaries

The mat is for quadrilaterals only. The loops ask about four sides and four corners. A triangle or a five-sided shape is not placed outside the loops; it is not placed on the mat at all.

All four, not most. A loop asks about all four sides or all four corners. Three equal sides, or two square corners, put a shape outside the loop.

“Every” does not turn around. “Every square is a rectangle” is true and “every rectangle is a square” is false. Swapping the two words gives a different sentence that has to be checked on its own.

The mat records two answers, nothing more. Shapes in the same part of the mat can still be very different. Cards B and E in Example 3 sit side by side in the “four square corners” loop, though one is 6 cm by 3 cm and the other 7 cm by 2 cm. The mat says only that both are rectangles that are not squares.

Outside both loops is not nameless forever. Those shapes are still quadrilaterals, and some of them get names of their own later, such as parallelogram, trapezoid and kite, in Trapezoids and kites and The quadrilateral family tree.

Where this goes next

The two loops are the start of a family tree of shapes. Later, two new tools arrive: sides that run in the same direction without ever meeting, and corners measured in degrees. With them, more loops can be drawn, and the categories nest inside one another in more ways, as in Classifying shapes in a hierarchy and Parallelograms, rectangles, rhombuses and squares: what each one guarantees.

The “every” sentences carry over too. Anything true of every rectangle, such as sides across from each other being equal, is then true of every square without checking again, because every square is a rectangle.

The square corners that put a shape in the right loop are also what let a rectangle be covered by rows of unit squares, which leads to the area of a rectangle.

Exercises

Part 1: one shape, several names

A quadrilateral has sides of 9 cm, 9 cm, 9 cm and 9 cm, and the card fits all four corners. List every name from the lesson that fits it.
A quadrilateral has four sides of 6 cm, and the card fits none of its corners. Which names fit it?
A quadrilateral has sides of 8 inches, 5 inches, 8 inches and 5 inches, and four square corners. Which names fit it?
A triangle has three sides of 4 cm. Which of the names quadrilateral, rhombus, rectangle and square fit it?
What attribute does every rhombus share with every square, that a rectangle does not have to have?
What attribute does every rectangle share with every square, that a rhombus does not have to have?
A rhombus, a rectangle and a lopsided four-sided shape are on a table. Name one category all three belong to, and the attribute they share that puts them in it.

Part 2: the sorting mat

Exercise 8 uses this table of Wangari’s second set of cards.

Card	Sides (cm)	Square corners
M	9, 9, 9, 9	4
N	4, 4, 4, 4	0
P	8, 3, 8, 3	4
Q	2, 5, 3, 6	0
R	6, 4, 6, 4	0
S	7, 7, 7, 7	0
T	5, 5, 5, 5	4
U	3, 3, 5, 5	0

Place each card on the sorting mat of Figure 1.
Why does the part of the mat for squares have to be inside both loops?
A rectangle is 7 inches long and 4 inches wide. Where does it go on the mat?
Wangari puts a card in the overlap. One of its sides is 6 cm. How long are the other three sides? How far is it once around the edge?
A card is in the “four square corners” loop but not in the overlap. One side is 9 cm and the side next to it is 2 cm. What are all four sides, and how far is it once around?

True or false? Explain each answer using the mat.

Every rhombus is a square.
Every square is a rhombus.
Every rectangle has four sides.
Every shape with four square corners is a square.

Part 3: shapes in neither loop

On centimeter grid paper, draw a quadrilateral that goes outside both loops and has exactly one square corner. Describe it.
On centimeter grid paper, draw a quadrilateral with no square corners and no two sides the same length. Describe it and say where it goes on the mat.
Can a quadrilateral have three square corners and still belong outside the “four square corners” loop? Explain.
Draw a quadrilateral with exactly three equal sides. Where does it go on the mat?
Chukwu cuts one corner off a rectangle card with a single straight cut, from a point partway along one side to a point partway along the next side. Where does the new shape go on the mat?

Reasoning

Explain why a square goes in the overlap rather than in just one of the loops.
Explain why every quadrilateral has exactly one place on the sorting mat.
Can a shape in the overlap ever be something other than a square? Explain.
Wangari labels the whole mat “four sides”. Explain why both loops fit inside that label, and why a square is in three categories on this mat.

Answers

Quadrilateral, rhombus, rectangle and square. It has four sides; all four sides are 9 cm, so it passes the ruler test; all four corners pass the card test; and passing both makes it a square.
Quadrilateral and rhombus. Four equal sides pass the ruler test, but with no square corner it is not a rectangle, and so not a square.
Quadrilateral and rectangle. The corners all pass the card test, but 8 inches and 5 inches are not equal, so it is not a rhombus or a square.
None of them. All four names are for shapes with four sides, and a triangle has three.
Four sides of the same length.
Four square corners.
Quadrilateral. Each has four straight sides (and so four corners), closed up.
M and T (equal sides, four square corners) go in the overlap. N and S (equal sides, no square corners) go in the “four equal sides” loop only. P (four square corners, sides 8 and 3 cm) goes in the “four square corners” loop only. Q, R and U fail both tests and go outside both loops. That places 2 + 2 + 1 + 3 = 8 cards.
A square has four equal sides, which puts it in the left loop, and four square corners, which puts it in the right loop. The only part of the mat in both loops is the overlap.
In the “four square corners” loop only. Its corners are square, but 7 inches and 4 inches are not equal.
Each of the other sides is 6 cm, because a card in the overlap has four equal sides. Once around is 6 + 6 + 6 + 6 = 24 cm, the same as 4 \times 6 = 24.
The sides are 9, 2, 9 and 2 cm, because it is a rectangle and sides across from each other match. Once around is 9 + 2 + 9 + 2 = 22 cm.
False. Card C from Example 3 is in the “four equal sides” loop but outside the overlap: a rhombus with no square corners.
True. The overlap is part of the “four equal sides” loop, and every square has four equal sides.
True. Every rectangle is on the mat, and only quadrilaterals go on the mat.
False. Card B from Example 3 has four square corners but sides of 6 cm and 3 cm, so it is a rectangle that is not a square.
For example, start where two grid lines cross and go 5 cm right along a line, then to a point 2 cm back left and 3 cm up, then to a point 3 cm left and 1 cm down, then 2 cm straight down back to the start. The only square corner is the bottom left. The sides are 5 cm, a little over 3\tfrac{1}{2} cm, a little over 3 cm and 2 cm, not all equal, so it goes outside both loops.
For example, start where two grid lines cross and go 6 cm right along a line, then to a point 2 cm back left and 3 cm up, then to a point 3 cm further left and 1 cm up, then back to the start. The sides measure 6 cm, a little over 3\tfrac{1}{2} cm, a little over 3 cm and a little over 4 cm, all different, and the card fits no corner. It goes outside both loops.
No. A quadrilateral with three square corners always has a fourth (Example 6), so it has four square corners and belongs inside that loop.
For example, go 5 cm right, 5 cm up, then 4 cm left and 3 cm down, then back to the start. The slanted side that goes 4 across and 3 down measures 5 cm with a ruler, so three sides are 5 cm, and the last side is a little over 2 cm. Only the bottom right corner is square. It goes outside both loops.
Off the mat. One straight cut across a corner takes away one corner and adds two, so the new shape has five sides and is not a quadrilateral.
Its four equal sides put it in the “four equal sides” loop and its four square corners put it in the “four square corners” loop. A shape in both loops is in the part they share.
Each quadrilateral answers two questions, four equal sides or not and four square corners or not. Each pair of answers matches exactly one of the four parts of the mat, so there is one place and only one.
No. A shape in the overlap has four equal sides and four square corners, and that is exactly what a square is.
Every shape in either loop is a quadrilateral, so it has four sides; the loops only pick out some quadrilaterals. A square sits on the “four sides” mat, inside the “four equal sides” loop and inside the “four square corners” loop, so it is a quadrilateral, a rhombus and a rectangle at once.