Rhombus
Four equal sides.
Equal sides, parallel sides?
A square has four equal sides. But a square also has right angles. What if a quadrilateral has four equal sides and no right angles? Will its opposite sides still be parallel?
If all four sides of a quadrilateral are equal, are its opposite sides parallel?
What this lesson covers
The idea
A quadrilateral in which all the sides have the same length is a rhombus; every rhombus is a parallelogram, so all parallelogram properties hold for it.
Equal sides, parallel sides?
A square has four equal sides. But a square also has right angles. What if a quadrilateral has four equal sides and no right angles? Will its opposite sides still be parallel?
If all four sides of a quadrilateral are equal, are its opposite sides parallel?
- Yes, always
- Only sometimes
- Never
Two tests
Tap a shape. Run the equal sides test and the parallel sides test. Then decide: has it four equal sides?
Equal sides, parallel sides
A quadrilateral in which all the sides have the same length is a rhombus; every rhombus is a parallelogram, so all properties of a parallelogram hold for it.
Look back at the shapes that passed the equal sides test: every one of them also passed the parallel sides test.
But some shapes passed the parallel test without having four equal sides: a rhombus is a special parallelogram, not the other way round.
So everything we know about parallelograms is true for a rhombus: opposite angles are equal, neighbouring angles add up to 180°, the diagonals bisect each other.
Notes
A quadrilateral in which all the sides have the same length is a rhombus; every rhombus is a parallelogram, so all properties of a parallelogram hold for it.
Check yourself
Each side of a rhombus is 7 cm. What is its perimeter?
Answer: 28 cm
All four sides of a rhombus are equal, so the perimeter is 7 + 7 + 7 + 7 = 28 cm.
ABCD is a rhombus with ∠A = 70°. How large is ∠B?
Answer: 110 °
A rhombus is a parallelogram, so its neighbouring angles add up to 180°. ∠B = 180° − 70° = 110°.
The diagonals of rhombus ABCD meet at O. AC = 10 cm. How long is OA?
Answer: 5 cm
A rhombus is a parallelogram, so its diagonals bisect each other. O is the middle of AC, so OA = 10 ÷ 2 = 5 cm.
Which statement is true?
- Every rhombus is a parallelogram — correct. Yes! A rhombus has opposite sides parallel, so it is a parallelogram.
- Every parallelogram is a rhombus. A parallelogram with sides 6 cm and 4 cm is not a rhombus. Its sides are not all equal.
- A rhombus has only one pair of parallel sides. A rhombus is a parallelogram, so both pairs of its opposite sides are parallel.