Matemáticas
Why Is a Square Also a Rectangle?
A rectangle is a quadrilateral with four right angles, and a square has four right angles. The full quadrilateral hierarchy, and why the reverse is not true.
Respuesta breve
Is a square also a rectangle?
Yes. A rectangle is a quadrilateral with four right angles, and a square has four right angles, so every square is a rectangle. A rectangle is a square only when all four sides are equal.
La respuesta corta
Every square is a rectangle, because a rectangle is defined as a quadrilateral with four right angles and a square has four right angles. The square carries one extra property on top - four equal sides - which makes it a special kind of rectangle, in the same way a rectangle is a special kind of parallelogram.
Qué cambia la respuesta
- Some primary resources define a rectangle as having two long sides and two short ones, which excludes squares; that is the older word oblong wearing the wrong name.
- When a question asks you to name a shape, give the most specific name that fits - square rather than rectangle - because the specific name carries more information.
- In everyday speech rectangle usually means the non-square kind, and no harm comes of that until a geometry question is being answered.
Run the two definitions against each other
A rectangle is a quadrilateral with four right angles. That is the whole definition, and it says nothing whatever about the sides having to be different lengths. Everything else usually attached to rectangles - opposite sides equal, opposite sides parallel, diagonals equal in length and bisecting each other - follows from the four right angles rather than forming part of the definition.
A square is a quadrilateral with four right angles and four equal sides. Read that slowly and the first half is the definition of a rectangle, word for word. A square is a rectangle with a condition added, and adding a condition never removes a shape from a category - it places it in a smaller category inside it.
Which is why the relationship runs one way only. Every square passes the rectangle test. Most rectangles fail the square test, because their adjacent sides are not equal. Yes to the first question, no to the reverse, and both answers come from the same two definitions.
Where each quadrilateral sits
The indentation below is the point of the diagram: each level inherits everything above it.
Read downwards and a square is a rhombus, a rectangle, a parallelogram, a trapezium and a quadrilateral, all at once and all correctly. Read upwards and nothing is guaranteed - a parallelogram need not be a rectangle, and a rectangle need not be a square.
The inheritance is the useful part rather than the naming. Any fact proved about parallelograms - opposite angles equal, diagonals bisecting each other - is automatically true of rectangles, rhombuses and squares with nothing further to prove. Students who keep squares and rectangles in separate mental boxes end up proving the same result twice.
- Quadrilateral - any four-sided shape
- Trapezium - a pair of parallel sides
- Parallelogram - both pairs of opposite sides parallel
- Rectangle - a parallelogram with a right angle, which forces all four
- Rhombus - a parallelogram with four equal sides
- Square - both at once: four right angles and four equal sides
- Kite - two pairs of adjacent sides equal; every rhombus is also a kite
The one definition that is genuinely not settled
There is a real disagreement inside that hierarchy, and it is better known about than stumbled into. A trapezium is defined by some books as a quadrilateral with at least one pair of parallel sides, and by others as one with exactly one pair. Under the first, every parallelogram is a trapezium. Under the second, none is.
Textbooks differ and both versions are in use in UK classrooms. Nothing on an exam paper turns on it, because questions ask for the area of a trapezium or for a shape to be identified, and neither requires you to take a position on parallelograms.
The general lesson is worth more than the particular case. In geometry the definition is the authority. If a question or a textbook has told you what a word means, use that meaning for the rest of the question rather than the one you walked in with.
What to write when a question asks for a name
Give the most specific correct name. If a shape has four right angles and four equal sides, write square. Rectangle is not false, but it withholds the information the question was asking for, and a name that fits many shapes is a weaker answer than one that fits this one.
When the question runs the other way, answer from the definition and name the deciding property. Yes, a square is a rectangle, because it has four right angles. No, a rectangle need not be a square, because its adjacent sides may be different lengths. Both answers are one sentence long and both sentences contain a reason, which is what the mark is for.
The national curriculum for England expects Year 5 pupils to use the properties of rectangles to deduce related facts and find missing lengths and angles. That is the point in school where a definition has to take over from a picture, because a fact given about rectangles is now also a fact about squares, and a student still sorting shapes by how they look will not use it.
Preguntas frecuentes
Is a rectangle a square?
Only when all four of its sides are equal. Most rectangles are not squares. The inclusion runs in one direction: every square meets the rectangle definition, but the rectangle definition says nothing about side lengths, so it does not force a rectangle to meet the square definition.
Is a square a rhombus?
Yes. A rhombus is a quadrilateral with four equal sides, and a square has four equal sides. A square is precisely the shape that is both a rectangle and a rhombus - the four right angles from one, the four equal sides from the other.
Is a square a parallelogram?
Yes. Its opposite sides are parallel, which is the definition of a parallelogram, and a square qualifies. It follows that every property of parallelograms applies to squares as well: opposite angles equal, diagonals bisecting one another, and area equal to the base times the perpendicular height.
What is an oblong?
The older word for a rectangle that is not a square. It exists precisely because rectangle includes squares, so a separate word was needed for the elongated kind. Some primary schemes still teach it, and it is a useful word for exactly the situation people are trying to describe when they say a rectangle is not a square.
Why are children taught that squares and rectangles are different shapes?
Early work sorts shapes by appearance before it sorts them by properties, and at that stage the sorting is not wrong - it is just not yet a definition. What matters is that it gets revisited when definitions arrive, rather than being left as a fact the student has to unlearn silently.
Fuentes
- National curriculum in England: mathematics programmes of study — Department for Education
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