Area of a rhombus
Cut a rhombus along its two diagonals and the four triangles make a rectangle.
Diamond and rectangle
A rhombus has two diagonals, one 8 long (across) and one 6 long (up and down). The dashed rectangle is 8 long, the length of one diagonal, and 3 high, half of the other diagonal.
Which has the greater area?
What this lesson covers
The idea
Area of a rhombus = 1/2 × product of its diagonals, because its diagonals are perpendicular bisectors of each other and it dissects into a rectangle with sides one diagonal and half the other.
Diamond and rectangle
A rhombus has two diagonals, one 8 long (across) and one 6 long (up and down). The dashed rectangle is 8 long, the length of one diagonal, and 3 high, half of the other diagonal.
Which has the greater area?
- The rhombus
- The dashed rectangle
- Both have the same area
Four triangles
The two diagonals cut the rhombus into four right triangles. Pick the orange triangle and the green triangle in turn and move each one up to fill the dashed rectangle.
Half the product of the diagonals
Area of a rhombus = ½ × product of its diagonals, because its diagonals are perpendicular bisectors of each other and it dissects into a rectangle with sides one diagonal and half the other.
The diagonals cross at right angles and cut each other in half, so the rhombus falls into four right triangles that are copies of each other. Move the two lower triangles up beside the top one and they fill the two corners of a rectangle.
The rectangle is as long as one diagonal, 8, and as high as half of the other, 6 ÷ 2 = 3. So the area is 8 × 3 = 24, and that is ½ × 8 × 6 = 24.
A rhombus is also a parallelogram, so base × height works too. Its side is 5, so 5 × height = 24 and the height is 4.8.
Notes
Area of a rhombus = ½ × product of its diagonals, because its diagonals are perpendicular bisectors of each other and it dissects into a rectangle with sides one diagonal and half the other.
Check yourself
The diagonals of a rhombus are 14 cm and 9 cm. What is its area, in cm²?
Answer: 63
Area = ½ × 14 × 9 = 63 cm².
A rhombus has an area of 48 cm² and one diagonal of 12 cm. How long is the other diagonal, in cm?
Answer: 8
48 = ½ × 12 × d = 6 × d, so d = 48 ÷ 6 = 8 cm.
Which fact about the diagonals of a rhombus makes the rectangle dissection work?
A rhombus has sides of 13 cm and one diagonal of 10 cm. What is its area, in cm²?
Answer: 120
Half of the diagonal is 5, and 5² + 12² = 13², so the other half-diagonal is 12 and the other diagonal is 24 cm. Area = ½ × 10 × 24 = 120 cm².
- They have the same length. That is true for a square, not for every rhombus. The diagonals of our rhombus are 8 and 6.
- They cross at right angles and cut each other in half — correct. Yes! That gives four right triangles that are copies of each other, which fit into a rectangle of one diagonal by half the other.
- They are parallel. The diagonals cross each other, so they cannot be parallel.