Chapter 5 of 8

All 8 chapters
  1. 1Straight lines and parallel lines, readunderstood
  2. 2Polygons, readunderstood
  3. 3Right angles and turns, readunderstood
  4. 4Measuring angles in degrees, readunderstood
  5. 5Triangles and four sided shapes, readunderstood
  6. 6Lines of symmetry, readunderstood
  7. 7Solid shapes and nets, readunderstood
  8. 8Grids and directions, readunderstood

M2.5.5 Chapter 5

Triangles and four sided shapes

Is a square also a rectangle?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Name a square, rectangle, rhombus, parallelogram and trapezium
  • Sort four sided shapes by their sides and angles
  • Sort triangles by how many sides are equal

You know shapes with 4 straight sides. Some of them have special names.

The name comes from the sides and the corners. Look for three things: equal sides, right angles and parallel sides.

Draw it first

Five shapes with four sides

  1. Five four sided shapes built up in four steps, each with its name under it: square, rectangle, rhombus, parallelogram and trapezium. Ticks mark equal sides, small squares mark right angles and arrow heads mark parallel sides.square

    1 A square has 4 equal sides and 4 right angles. Ticks and small squares show this.

  2. Five four sided shapes built up in four steps, each with its name under it: square, rectangle, rhombus, parallelogram and trapezium. Ticks mark equal sides, small squares mark right angles and arrow heads mark parallel sides.squarerectangle

    2 A rectangle has 4 right angles. Its opposite sides are equal.

  3. Five four sided shapes built up in four steps, each with its name under it: square, rectangle, rhombus, parallelogram and trapezium. Ticks mark equal sides, small squares mark right angles and arrow heads mark parallel sides.squarerectanglerhombus

    3 Push the square sideways. You get a rhombus. It has 4 equal sides.

  4. Five four sided shapes built up in four steps, each with its name under it: square, rectangle, rhombus, parallelogram and trapezium. Ticks mark equal sides, small squares mark right angles and arrow heads mark parallel sides.squarerectanglerhombusparallelogramtrapezium

    4 A parallelogram has two pairs of parallel sides. A trapezium has one pair.

A key to the marks. Equal sides: two lines, each with one tick. Right angle: two lines meeting at a small square. Parallel sides: two lines, each with an arrow head.equal sidesright angleparallel sides
Use these marks to read any shape in this chapter.

In a rhombus, the corners opposite each other are equal. In a parallelogram, the opposite sides are equal.

In some countries a trapezium is called a trapezoid. There, the word trapezium can mean a shape with no parallel sides. Ask which one your book uses.

Now test a square. It has 4 equal sides, so it is a rhombus. It has 4 right angles, so it is a rectangle.

Triangles have 3 sides. Count the sides that are equal.

Three triangles. The first is equilateral, with 3 equal sides. The second is isosceles, with 2 equal sides. The third is scalene, with no equal sides. Ticks mark the sides. Sides with the same number of ticks are equal, and the scalene sides have 1, 2 and 3 ticks.equilateral3 equal sidesisosceles2 equal sidesscaleneno equal sides
Matching ticks mark equal sides. Each corner of the equilateral triangle is 60°.

Look at the ticks. Equal sides have the same number of ticks.

Equilateral means 3 equal sides. Isosceles means 2 equal sides. Scalene means no equal sides.

See it three ways

With things

Lay 4 equal sticks end to end in a loop. Make a square. Then push it sideways.

As a picture

A square and a rhombus side by side, each with four ticks showing four equal sides. The square has small squares at its corners. The labels are square and rhombus.squarerhombus

In symbols

Square: 4 equal sides and 4 right angles. This rhombus: 4 equal sides and no right angles.

Try it

Push a square into a rhombus

You need: 4 sticks of the same length, such as pencils or straws, your right angle tester from chapter 3.

  1. Lay the sticks in a loop so the ends touch. Make a square.
  2. Check each corner with your tester. All four fit.
  3. Push the loop sideways. Keep every end touching.
  4. Check the corners again. Two are smaller than your tester. Two are bigger.

What you will see: The sticks did not change, so the 4 sides stay equal. Only the corners changed. The square became a rhombus.

If you cannot try it, this is what happens.
A square and a rhombus. The tester fits the corner of the square. At the corner of the rhombus a dashed line shows a gap. The labels are tester fits and gap shows.tester fitsgap shows

A wrong idea many people have

A square is not a rectangle.

Why it feels right: Each shape has one name. So people think it is only one thing.

The test that breaks it: Check the rules of a rectangle: 4 right angles, equal opposite sides. Test a square with your tester. All four corners fit. Its opposite sides match.

What is true: A square follows every rule of a rectangle. It is a special rectangle. It is also a rhombus.

Key words

rhombus
a shape with 4 equal sides.
parallelogram
a shape with two pairs of parallel sides.
trapezium
a shape with one pair of parallel sides.

Back to your guess

The answer is B: Yes. A square has everything a rectangle needs.

A rectangle has 4 right angles and equal opposite sides. A square has both, so a square is a special rectangle. Size does not matter.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Which shape has 4 equal sides and no right angles?

    Three shapes. Shape A is a square with 4 ticks and 4 small squares. Shape B is a rectangle with 1 tick on each long side and 2 ticks on each short side. Shape C is a rhombus with 4 ticks and no small squares. The labels are A, B and C.ABC
    Tap your answer.

    1. Not this one. A has 4 equal sides, but it also has 4 right angles. Look for the shape with no small squares. Try another.

    2. Not this one. B has right angles, and its sides are not all equal. Check the ticks. Try another.

    3. Yes, that is right. C is a rhombus. Its 4 sides are equal and it has no right angles.

    See the answer

    Shape C

    C is a rhombus. Its 4 sides are equal and it has no right angles.

    • If you chose A: A has 4 equal sides, but it also has 4 right angles. Look for the shape with no small squares.
    • If you chose B: B has right angles, and its sides are not all equal. Check the ticks.

  2. A triangle has sides of 5 cm, 5 cm and 8 cm. What is it called?

    A triangle with its sides labelled 5 cm, 5 cm and 8 cm. The two shorter sides look equal.5 cm5 cm8 cm
    Tap your answer.

    1. Not this one. Equilateral means all 3 sides are equal. The 8 cm side is longer. Try another.

    2. Yes, that is right. Two sides are equal, 5 cm and 5 cm. The third side is different. That makes it isosceles.

    3. Not this one. Scalene means no sides are equal. Two sides here are both 5 cm. Try another.

    See the answer

    isosceles

    Two sides are equal, 5 cm and 5 cm. The third side is different. That makes it isosceles.

    • If you chose A: Equilateral means all 3 sides are equal. The 8 cm side is longer.
    • If you chose C: Scalene means no sides are equal. Two sides here are both 5 cm.

  3. Each corner of a triangle with 3 equal sides is 60°. What do the three corners add up to?

    See the answer

    180°

    60 + 60 + 60 = 180. That is the same turn as a straight line.

  4. Which sentence is true?

    Tap your answer.

    1. Yes, that is right. A square has 4 right angles and equal opposite sides. That is all a rectangle needs.

    2. Not this one. A rectangle can have long sides and short sides. Only some rectangles have 4 equal sides. Try another.

    3. Not this one. A rhombus can lean over, and then it has no right angles. A square is the rhombus with 4 right angles. Try another.

    See the answer

    A square is a rectangle.

    A square has 4 right angles and equal opposite sides. That is all a rectangle needs.

    • If you chose B: A rectangle can have long sides and short sides. Only some rectangles have 4 equal sides.
    • If you chose C: A rhombus can lean over, and then it has no right angles. A square is the rhombus with 4 right angles.

  5. A shape has 4 sides. Only one pair of sides is parallel. What is it?

    Tap your answer.

    1. Not this one. A parallelogram has two pairs of parallel sides, not one. Try another.

    2. Yes, that is right. A trapezium has exactly one pair of parallel sides.

    3. Not this one. A rectangle has two pairs of parallel sides, not one. Try another.

    See the answer

    trapezium

    A trapezium has exactly one pair of parallel sides.

    • If you chose A: A parallelogram has two pairs of parallel sides, not one.
    • If you chose C: A rectangle has two pairs of parallel sides, not one.

  6. From an earlier chapter

    A corner is bigger than a right angle. It is smaller than a straight line. What is it called?

    Tap your answer.

    1. Not this one. Acute is smaller than a right angle. Test the corner with your tester. Go to M2.5.3. Try another.

    2. Not this one. A right angle matches your tester exactly. This corner is bigger. Go to M2.5.3. Try another.

    3. Yes, that is right. Obtuse is bigger than a right angle. It is smaller than a straight line.

    See the answer

    obtuse

    Obtuse is bigger than a right angle. It is smaller than a straight line.

    • If you chose A: Acute is smaller than a right angle. Test the corner with your tester. Go to M2.5.3.
    • If you chose B: A right angle matches your tester exactly. This corner is bigger. Go to M2.5.3.

  7. Which shape is both a rhombus and a rectangle?

    Four shapes. Shape A leans, and has two pairs of equal sides of different lengths, marked with one and two ticks. Shape B leans and has one tick on every side. Shape C has matching ticks on opposite sides and a small square at each corner. Shape D has one tick on every side and a small square at each corner. The labels are A, B, C and D.ABCD
    Tap your answer.

    1. Not this one. A has equal opposite sides, but not all four are equal. It has no small squares, so no right angles. Try another.

    2. Not this one. B has 4 equal sides, so it is a rhombus. It has no right angles, so it is not a rectangle. Try another.

    3. Not this one. C has 4 right angles, so it is a rectangle. Its sides are not all equal, so it is not a rhombus. Try another.

    4. Yes, that is right. D has 4 equal sides, so it is a rhombus. It has 4 right angles, so it is a rectangle. It is a square.

    See the answer

    Shape D

    D has 4 equal sides, so it is a rhombus. It has 4 right angles, so it is a rectangle. It is a square.

    • If you chose A: A has equal opposite sides, but not all four are equal. It has no small squares, so no right angles.
    • If you chose B: B has 4 equal sides, so it is a rhombus. It has no right angles, so it is not a rectangle.
    • If you chose C: C has 4 right angles, so it is a rectangle. Its sides are not all equal, so it is not a rhombus.

Keep it

Copy this into your exercise book.

Square: 4 equal sides, 4 right angles. Rectangle: 4 right angles. Rhombus: 4 equal sides. Parallelogram: two pairs of parallel sides. Trapezium: one pair.

Draw: A square, a rectangle and a rhombus. Mark equal sides with ticks and right angles with small squares.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (773 KB). It opens on any phone.