Rhombuses, rectangles and squares
Every four-sided shape is a quadrilateral. Which ones earn a name of their own, and how can a ruler and the corner of a card tell them apart?
Where you are 7 in Quadrilaterals
Why this exists
A quadrilateral is any closed shape with four straight sides. Shapes are named by what they must have, whatever their size, as in Triangles, quadrilaterals, pentagons and hexagons from their sides and angles. But “quadrilateral” covers a great deal: a door, a kite in the sky, a slice of a pizza box, a lopsided scrap of paper. The word alone says very little.
Two questions about a quadrilateral pick out the shapes people use most:
- Are all four sides the same length?
- Are all four corners square corners?
Each question has a simple tool. A ruler answers the first. The corner of an index card answers the second. The two answers give three names: rhombus, rectangle and square.
A name is a promise. If a carpenter is told a window is a rectangle, the carpenter knows every corner of the frame is square without seeing it. If a quilter is told a patch is a rhombus, the quilter knows all four edges can be cut to one length.
Definition
A square corner is a corner that the corner of an index card fits exactly: one edge of the card lies along each side, with no gap and nothing sticking out. Later lessons call a square corner a right angle.
- A rhombus is a quadrilateral whose four sides are all the same length.
- A rectangle is a quadrilateral whose four corners are all square corners.
- A square is a quadrilateral whose four sides are all the same length and whose four corners are all square corners.
Figures in this lesson use two marks. Sides with the same number of tick marks are the same length. A small square drawn in a corner means it is a square corner. A mark is a claim that the side or corner was checked; a shape with no marks claims nothing.
Nothing else is in the definitions. Size, color, and which way the shape is turned do not matter. In everyday talk a rhombus standing on its point is often called a “diamond”, but a rhombus does not have to stand on its point, and “diamond” is not a math name.
A square passes both tests, so a square is also a rhombus and also a rectangle. Sorting shapes that fit more than one name, and quadrilaterals that fit none, is the next lesson, Shapes in more than one category, and quadrilaterals in none.
Intuition
This lesson runs over three days: the side test and the rhombus, then the corner test with rectangles and squares, then drawing and sorting.
Part 1: the ruler test and the rhombus
Measure each side with a ruler, starting from the 0 mark, and write down the four lengths. If all four match, the shape is a rhombus. If even one side is different, it is not, however close it looks.
On a geoboard, or on dot or grid paper, a slanted side cannot be measured by counting spaces. It can still be compared with another slanted side. A side that goes 2 spaces across and 1 space up runs corner to corner across a box 2 spaces by 1 space. Every side that crosses a box of that same size is the same length, because one box can be turned and slid to sit exactly on top of the other, slanted side on slanted side. Turning a box 2 across and 1 up gives a box 1 across and 2 up, so those sides match too.
A rhombus can lean as far as you like. Join four equal sticks into a square and push on one corner: the shape leans over.
The four sides did not change, so the pushed shape is still a rhombus. The corners did change: none of them is square anymore. Equal sides do not decide the corners.
Part 2: the card test, rectangles and squares
Push the corner of an index card into a corner of the shape, with one edge of the card along one side. Three things can happen.
If the card fits exactly, the corner is a square corner. If a gap shows between the card and the second side, the corner is wider than square. If the card sticks out past the second side, the corner is narrower than square.
A rectangle needs all four corners to pass. Checking one corner is not enough: some quadrilaterals have one or two square corners and still are not rectangles (piece E in Example 6 has two).
A rectangle that is not a square has two long sides and two short sides, and the two sides across from each other always match. If all four sides of the shape also match, it passes the ruler test as well, and it is a square.
Part 3: drawing on dot and grid paper
The lines of grid paper meet in square corners, and so do the rows and columns of dot paper. So a shape drawn along the lines gets square corners at every turn for free, and its sides can be measured by counting spaces. That makes rectangles and squares easy to draw: go along the lines, and choose the lengths. Two different lengths give a rectangle that is not a square; one length all the way round gives a square.
A rhombus that is not a square has to lean, so some of its sides must slant. That is where the box idea from Part 1 earns its keep: slanted sides that cross boxes of the same size are the same length.
Results
The three names.
| Name | All four sides equal | All four corners square |
|---|---|---|
| rhombus | yes | not needed |
| rectangle | not needed | yes |
| square | yes | yes |
In a rectangle, sides across from each other are equal. A rectangle with one side 7 cm and the side next to it 3 cm has sides of 7 cm, 3 cm, 7 cm and 3 cm.
Turning or sliding a shape never changes its name. A rhombus standing on its point, lying on a side or tipped at any slant is the same rhombus.
Equal sides do not decide the corners. Four equal sides can make a square or a rhombus that leans by any amount.
Why it is true
The three names are definitions: they are what the words mean, so they need no proof. The other three facts do need a reason.
Turning does not change the name. Turning a shape moves it without stretching it. Each side measures the same with the ruler before and after, and the card fits the same corners before and after. Both tests give the same answers, so the name is the same.
Sides across from each other in a rectangle are equal. Lay a rectangle on grid paper with its bottom side along a grid line. The two bottom corners are square, so both the left and right sides go straight up, the same way as the up-and-down lines of the grid. The top corners are square too, so the top side runs straight across, level like the rows of the grid. Now the top side starts directly above where the bottom side starts and ends directly above where it ends, so the top and the bottom are the same length. The left and the right sides both go straight up from the bottom side to the level top side, so they are the same length too.
Equal sides do not decide the corners. Figure 2 shows it: four sides that make a square can be pushed into a leaning rhombus, and every side keeps its length. One example like that is enough to show that four equal sides do not force square corners.
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.
A rhombus on the geoboard that is not a square
Winona stretches a rubber band around four pegs of a geoboard, as in Figure 4. Is her shape a rhombus? Is it a rectangle? Is it a square?
The band is slanted on every side, so counting spaces along a side will not measure it. Look at the boxes instead. Starting at the bottom peg, the first side goes 2 spaces right and 1 space up. The next goes 2 spaces left and 1 space up. The next goes 2 left and 1 down, and the last goes 2 right and 1 down. Each side runs corner to corner across a box 2 spaces across and 1 space tall, and the four dashed boxes are all that size. One box can be turned or flipped to sit exactly on another, so the four sides are the same length. A ruler agrees: all four sides measure the same.
Four equal sides make the shape a rhombus.
Now the corners. At the left and right pegs the shape comes to a narrow point: the index card sticks out past a side. At the top and bottom pegs the corner is wide: the card leaves a gap. Not one corner is square, so the shape is not a rectangle, and it is not a square either, because a square needs square corners.
Winona’s shape is a rhombus that is not a square.
Two patches that look alike
Narantsetseg cuts two diamond-shaped patches for a quilt. They look almost the same. She measures them:
- Patch P has sides of 3 inches, 3 inches, 3 inches and 3 inches. The card sticks out at two corners and leaves a gap at the other two.
- Patch Q has sides of 3 inches, 3 inches, 2 inches and 2 inches, with the two 3-inch sides meeting at the top. The card does not fit any corner.
Which patch is a rhombus? Then Narantsetseg turns patch P a quarter turn so that it stands on a different point. Is it still a rhombus?
SolutionPatch P has four sides of the same length, so it is a rhombus. Its corners are not square, so it is not a rectangle or a square.
Patch Q has two sides of 3 inches and two of 2 inches. The lengths are not all the same, so Q is not a rhombus, however much it looks like P. No corner is square, so Q is not a rectangle either. It is a quadrilateral without one of these three names.
Turning patch P does not stretch it. Its sides still measure 3 inches each, so it is still a rhombus. The name belongs to the shape, not to the way it is facing.
A quilt block of one square and four rectangles
Narantsetseg sews a quilt block from four rectangles, each 4 inches long and 2 inches wide, wrapped around a center piece like a pinwheel, as in Figure 5. Is the center piece a square? Is the whole block a square? How long is each side?
Start with the edges of the whole block. Along the top edge there is the long side of one rectangle and the short side of the next one, so the top edge is
The pinwheel is the same all the way round, so every edge of the block is a long side plus a short side: all four edges are 6 inches. Each corner of the block is a corner of one of the rectangles, and every corner of a rectangle is square. Four equal sides and four square corners: the block is a square, 6 inches on each side.
Now the center piece. Its top edge lies along the long side of the top rectangle, but the left rectangle covers 2 inches of that long side. What is left for the center is
The same happens on every side, so the center piece has four sides of 2 inches. Each of its corners is a spot where the side of one rectangle comes straight off the long side of another. At the top left, for example, the right side of the left rectangle comes down from the long side of the top rectangle in a square corner, and the center piece sits just beside that corner, on the other side of the same line. An index card laid along the long side fits on either side of that line, so the center’s corner is square too, and the same is true at its other three corners. The center piece is a square, 2 inches on each side.
The four other pieces are rectangles that are not squares, since 4 inches and 2 inches are not the same length.
A square that stands on its corner
Winona makes a new shape on a geoboard, as in Figure 6. Every side runs corner to corner across a box 2 spaces long and 1 space wide, and the card fits every corner exactly. What is the shape? How do its sides compare with the sides of her rhombus in Example 1?
Every side crosses a box 2 spaces by 1 space, so the four sides are the same length. The card fits all four corners, so all four corners are square. Four equal sides and four square corners: the shape is a square. Its sides do not run along the rows of pegs, but nothing in the definition asks them to.
The sides of the rhombus in Example 1 also crossed boxes 2 spaces by 1 space, so the two shapes have sides of exactly the same length. The difference is all in the corners. This is Figure 2 again, on a geoboard: the same four sides can make a square or a rhombus that leans.
Drawing a rectangle and a rhombus on dot paper
Odina has dot paper with dots 1 centimeter apart. She wants to draw a rectangle that is not a square, and a rhombus that is not a square with its bottom side lying along a row of dots.
SolutionThe rectangle is the easy one. Odina draws along the rows and columns of dots, which meet in square corners, so every corner of her shape is square. She makes it 5 spaces long and 2 spaces tall, as on the left of Figure 7. All four corners are square, so it is a rectangle. Its sides are 5 cm and 2 cm, which are not equal, so it is not a square.
For the rhombus she starts with a bottom side 5 spaces long, so it is 5 cm. Every other side must be 5 cm too. If she goes straight up 5 spaces she gets square corners, and that would make a square. So she leans: she tries a side that goes 3 spaces across and 4 spaces up, and measures it with a ruler. It measures 5 cm. Counting dots would have been wrong here: the slant side passes no dots on the way, and walking 3 across and then 4 up is 7 spaces of walking, yet the side itself is 5 cm long.
She draws the top side 5 spaces along a row, then the last side back to the start. The last side also goes 3 across and 4 up, so it crosses the same size of box as the first slanted side, and it is 5 cm too. All four sides are 5 cm, so the shape is a rhombus. The card does not fit any of its corners, so it is not a square.
Sorting stained glass by sides and corners
Odina is making a stained glass window from the five pieces in Figure 8. She checks each piece with a ruler and an index card:
| Piece | Sides (inches) | Square corners |
|---|---|---|
| A | 3, 3, 3, 3 | 4 |
| B | 4, 2, 4, 2 | 4 |
| C | 5, 5, 5, 5 | 0 |
| D | 4, 5, 4, 5 | 0 |
| E | 2, 5, 5, 4 | 2 |
Sort the pieces by the two questions, and name each piece.
Ask each piece both questions. Are all four sides equal? Yes for A and C, no for B, D and E. Are all four corners square? Yes for A and B, no for C, D and E. Piece E has two square corners, but two is not four, so its answer is no.
| All four corners square | Not all four square | |
|---|---|---|
| All four sides equal | A | C |
| Sides not all equal | B | D, E |
Now read off the names. A passes both tests, so A is a square. B has square corners but sides of two lengths, so B is a rectangle that is not a square. C has four equal sides and no square corners, so C is a rhombus that is not a square. D and E fail both tests, so none of the three names fits them. (D has a name of its own, parallelogram, and E is a trapezoid; those names come with later lessons.)
Four craft sticks
Narantsetseg joins craft sticks end to end with brass fasteners, so the corners can swing.
(a) She uses four sticks each 6 inches long. Which of the three shapes can she make?
(b) She uses two 6-inch sticks and two 4-inch sticks, joined in the order 6, 4, 6, 4. Which can she make?
(c) She uses the same four sticks in the order 6, 6, 4, 4. Which can she make?
Solution(a) The four sides are always 6 inches, so whatever she does to the corners, the shape is a rhombus. If she swings the corners until the card fits all four, it is a square. She cannot make a rectangle that is not a square, because a rectangle with four equal sides is a square.
(b) The sides are 6, 4, 6 and 4, so the shape is never a rhombus and never a square. If she swings the corners until the card fits all four, it is a rectangle 6 inches by 4 inches. Pushed over, it is none of the three.
(c) The sides are still not all equal, so it is never a rhombus. It is never a rectangle either, however she swings it. In a rectangle the sides across from each other are equal, but here the side across from a 6-inch stick is a 4-inch stick. So this order of sticks can never make any of the three shapes. The same sticks gave a rectangle in (b): the order of the sides matters as well as their lengths.
Boundaries
Only quadrilaterals. All three names are for closed shapes with exactly four straight sides. A shape with five equal sides is not a rhombus, and a shape with a curved edge is none of these.
Equal means measured equal. A side that looks the same length, or a corner that looks square, counts only once it has been checked with a ruler or a card, or once a mark on the figure says so. A card can look square and turn out to be 15 cm by 16 cm.
All four, not some. One square corner does not make a rectangle, and three equal sides do not make a rhombus. Each test asks about all four sides or all four corners.
A tool is only as exact as its use. A ruler reads to its smallest marks, and a card held loosely can seem to fit a corner that is a little off. Two sides that differ by less than the ruler can show will measure the same.
The names say nothing else. A rhombus has no promise about its corners, and a rectangle has no promise that its sides are equal. Anything more must be checked.
Where this goes next
The next lesson, Shapes in more than one category, and quadrilaterals in none, takes the two-question sort further: a square sits in every group at once, and plenty of quadrilaterals, like pieces D and E, sit in none.
The square corners of rectangles are what make them easy to measure. Covering a rectangle with unit squares, in rows and columns that fit because the corners are square, leads to the area of a rectangle, and walking all the way round its sides leads to Perimeter.
Later, two more ideas come in: sides that run in the same direction forever without meeting, and measuring corners in degrees. With them, the family grows to parallelograms, trapezoids and kites, and the rectangle, rhombus and square take their places in it, as in Parallelograms, rectangles, rhombuses and squares: what each one guarantees and The quadrilateral family tree.
Exercises
Part 1: sides, corners and rhombuses
- A quadrilateral has sides of 7 cm, 7 cm, 7 cm and 7 cm. Is it a rhombus?
- A quadrilateral has sides of 4 inches, 4 inches, 4 inches and 5 inches. Is it a rhombus?
- A stained glass piece is a rhombus with one side 8 cm long. How long is each of the other three sides? How much lead strip goes once all the way around the piece?
- Winona stretches a band around four pegs of a geoboard. Every side goes 1 space across and 2 spaces up or down. Is the shape a rhombus? Explain how you know.
- Odina pushes the corner of an index card into a corner of a glass piece. What does she see if the corner is square? If it is wider than square? If it is narrower than square?
Part 2: rectangles and squares
- A rectangle has one side 9 cm long, and the side next to it is 4 cm long. How long are the other two sides?
- A quadrilateral has four square corners and sides of 5 cm, 5 cm, 5 cm and 5 cm. Name it.
- A quadrilateral has four square corners and sides of 7 inches, 3 inches, 7 inches and 3 inches. Is it a rectangle? Is it a square?
- A quadrilateral has four sides of 6 cm. The card leaves a gap at two corners and sticks out at the other two. Is it a rhombus? A rectangle? A square?
- Narantsetseg has a square quilt block 8 inches on each side. She cuts straight across it, 3 inches below the top edge, and the cut makes square corners with both sides. What are the two pieces? Give the side lengths of each.
- A quilt strip is a rectangle 10 inches long and 5 inches wide. Narantsetseg cuts it across the middle of its long sides, making square corners. What are the two pieces?
- Narantsetseg makes another pinwheel block like the one in Example 3, but with four rectangles each 5 inches long and 2 inches wide. How long is each edge of the whole block? How long is each side of the center square?
- On a geoboard, two sides of Winona’s band each go 1 space across and 1 space up or down, and the other two each go 3 spaces across and 3 spaces up or down. The card fits all four corners. Is the shape a rectangle? Is it a square?
Part 3: drawing and sorting
- On grid paper, draw a rectangle that is not a square, with one side 6 units long. Describe your drawing.
- On dot paper, draw a square whose sides do not run along the rows or columns of dots. Describe your drawing.
- On dot paper, draw a rhombus that is not a square and is different from the ones in Examples 1 and 5. Describe it and explain why it is a rhombus. Exercise 17 uses this table of the pieces for Odina’s second window.
| Piece | Sides (inches) | Square corners |
|---|---|---|
| F | 2, 2, 2, 2 | 4 |
| G | 5, 3, 5, 3 | 0 |
| H | 4, 4, 4, 4 | 0 |
| J | 6, 2, 6, 2 | 4 |
| K | 3, 5, 6, 4 | 2 |
- Sort the pieces in the table by the two questions, and name each piece.
Reasoning
- Is every rhombus a square? Give an example to show your answer.
- Is it always true that a quadrilateral with four square corners has four equal sides? Give an example.
- Explain why checking that a shape is a square needs both the ruler and the card.
- A shape has five sides, each 4 cm long. Is it a rhombus? Explain.
- A floor tile looks square. Two sides that meet measure 10 cm and 11 cm, and the card fits all four corners. What is the tile?
- Winona turns a rectangle so that it stands on one corner. Is it still a rectangle? Explain.
- Narantsetseg has two craft sticks 8 inches long and two 3 inches long, joined end to end so the corners can swing. Can she make a square? A rhombus? A rectangle? Explain.
- Can four craft sticks of the same length make a rectangle that is not a square? Explain.
Answers
- Yes. All four sides are the same length, and a rhombus asks nothing about the corners.
- No. One side is 5 inches and the others are 4 inches, so the four sides are not all equal.
- Each side is 8 cm, because all four sides of a rhombus are equal. Once around is 8 + 8 + 8 + 8 = 32 cm of lead strip, the same as 4 \times 8 = 32.
- Yes. Each side runs corner to corner across a box 1 space across and 2 spaces tall. All four boxes are the same size, and one can be turned to sit on another, so the four sides are the same length. For example: start at a peg, go 1 right and 2 up, then 1 left and 2 up, then 1 left and 2 down, then 1 right and 2 down, back to the start.
- Square: the card fits exactly, one edge along each side. Wider: a gap shows between the card and one side. Narrower: the card sticks out past one side.
- 9 cm and 4 cm. In a rectangle, sides across from each other are equal, so the sides go 9, 4, 9, 4.
- A square: four equal sides and four square corners.
- It is a rectangle, because all four corners are square. It is not a square, because 7 inches and 3 inches are not equal.
- A rhombus, because all four sides are 6 cm. Not a rectangle and not a square, because no corner is square.
- Two rectangles. The top piece is 8 inches by 3 inches. The bottom piece is 8 inches by 8 - 3 = 5 inches. Neither is a square, since 8 is not 3 and 8 is not 5.
- Two squares. Each piece is 10 \div 2 = 5 inches long and 5 inches wide, with four square corners.
- Each edge of the block is 5 + 2 = 7 inches. Each side of the center square is 5 - 2 = 3 inches.
- A rectangle, because the card fits all four corners. Not a square: the long sides cross a box 3 spaces by 3 spaces and the short sides a box 1 space by 1 space, and the bigger box gives the longer side, so the sides are not all equal.
- For example, 6 units along a grid line, then 2 units up, 6 back and 2 down. Any second length other than 6 works; a second length of 6 would make a square.
- For example, start at a dot and go 1 right and 1 down, then 1 right and 1 up, then 1 left and 1 up, then 1 left and 1 down. Every side crosses a box 1 space by 1 space, so the sides are equal, and the card fits every corner.
- For example, go 3 right and 1 down, then 3 right and 1 up, then 3 left and 1 up, then 3 left and 1 down. Every side crosses a box 3 spaces by 1 space, so all four sides are equal and it is a rhombus. The shape is long and flat, and the card does not fit any corner, so it is not a square.
- F (all sides equal, all corners square) is a square. J (corners square, sides not equal) is a rectangle that is not a square. H (sides equal, corners not square) is a rhombus that is not a square. G and K pass neither test, so none of the three names fits them; K’s two square corners are not four.
- No. Winona’s shape in Example 1 has four equal sides and no square corners: a rhombus that is not a square.
- No. A rectangle 7 inches by 3 inches has four square corners but two different side lengths.
- Each tool checks only one thing. A leaning rhombus passes the ruler test and fails the card test; a long rectangle passes the card test and fails the ruler test. Only a shape that passes both is a square.
- No. A rhombus is a quadrilateral, so it has exactly four sides.
- A rectangle that is not a square. Its corners are all square, but sides of 10 cm and 11 cm are not equal.
- Yes. Turning does not change the length of any side or make any corner fit the card differently, so all four corners are still square.
- No square and no rhombus, because the sides are not all equal. A rectangle, yes: join them in the order 8, 3, 8, 3 and swing the corners until the card fits all four. In the order 8, 8, 3, 3 it can never be a rectangle, since sides across from each other would not match.
- No. If all four corners are square, the four equal sides make it a square.