Rhombus area
FoundationArea = product of diagonals ÷ 2.
After this lesson, you can:
- Rhombus area = diagonal × diagonal ÷ 2
- A rhombus has four equal sides and perpendicular diagonals
- One diagonal splits it into two congruent triangles
1Summary 重點摘要
Area = product of diagonals ÷ 2.
2Vocabulary cards Vocabulary
R
Rhombus菱形
A quadrilateral with four equal sides, like a tilted diamond.
D
Diagonal對角線
A line joining opposite vertices; a rhombus has two.
P
Perpendicular diagonals對角線互相垂直
The two diagonals of a rhombus cross at 90°.
A
Area面積
The flat space covered, in cm².
F
Formula公式
Rhombus area = diagonal × diagonal ÷ 2.
H
Halve折半
Divide by 2 — always part of the rhombus formula.
T
Triangle split分成三角形
One diagonal splits the rhombus into two congruent triangles.
S
Square (special rhombus)正方形（特別的菱形）
A rhombus whose diagonals are equal is a square.
3Step by step Steps
- Step 1: Know the rhombusA rhombus has four equal sides, and its two diagonals cross at right angles, one across and one up, like a diamond. Kites and floor tiles often show this shape. A diagonal joins opposite corners, and the two diagonals meet at the centre forming four right angles.
- Step 2: The formulaArea = diagonal × diagonal ÷ 2, written A = d₁ × d₂ ÷ 2. One line gives the area. The formula uses the diagonals, not the sides, which is the most common mix-up.
- Step 3: Why divide by 2Diagonals 6 and 4 split the rhombus into two triangles, each with base 6 and height 4 ÷ 2 = 2, so each has area 6 × 2 ÷ 2 = 6; together 12, exactly 6 × 4 ÷ 2. The product of the diagonals makes a rectangle, and the rhombus is half of it.
- Step 4: SubstituteWith d₁ = 6 and d₂ = 4: first 6 × 4 = 24, then 24 ÷ 2 = 12. Follow the order multiply first, divide second, and write down the middle result so checking is easy.
- Step 5: Use square unitsWrite the answer as 12 cm²: area always takes square units. The rhombus is a flat shape, measured in square centimetres or square metres, while plain centimetres measure the diagonals and sides. Units must match what they measure.
- Step 6: Find a missing diagonalWith the area and one diagonal, work backwards: area 24 and one diagonal 8 give the other as 24 × 2 ÷ 8 = 48 ÷ 8 = 6. The formula rearranges to: the two diagonals multiply to twice the area.
4Diagrams Diagrams
5Worked example Example
Example: a kite-shaped rhombus has diagonals 10 cm and 8 cm. What is its area?
- Write the formula: area = d₁ × d₂ ÷ 2, using the two diagonals.
- Substitute: area = 10 × 8 ÷ 2.
- Multiply first: 10 × 8 = 80, the product of the diagonals.
- Then divide: 80 ÷ 2 = 40.
- Attach square units: the kite's area is 40 cm².
- Answer: The kite's area is 40 cm². We multiplied the diagonals 10 cm and 8 cm to get 80, then divided by 2 to get 40. The division by 2 appears because the rectangle formed by the two diagonals has twice the rhombus's area. The square unit cm² completes the answer.
6Common mistakes Mistakes
✗ Forgetting to halve: 6 × 4 = 24.
✓ The formula ends with ÷ 2: 6 × 4 ÷ 2 = 12 cm².
✗ Multiplying side by side.
✓ A rhombus is not a square — use the diagonals: d₁ × d₂ ÷ 2.
✗ Mixing diagonals with sides.
✓ Only the two diagonals are multiplied; sides are irrelevant here.
✗ Finding a missing diagonal as 24 ÷ 8 = 3.
✓ Undo the halving first: 24 × 2 ÷ 8 = 6.
7Quick check Quick check
Q1Rhombus with diagonals 6 cm and 4 cm. Area?
Answer: 6 × 4 ÷ 2 = 12 cm².
Did you get it right?
Q2Diagonals 10 cm and 8 cm. Area?
Answer: 10 × 8 ÷ 2 = 40 cm².
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Q3Area 24 cm², one diagonal 8 cm. How long is the other?
Answer: 24 × 2 ÷ 8 = 6 cm.
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Q4At what angle do a rhombus's diagonals cross?
Answer: 90° — they are perpendicular.
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Q5A rhombus with equal diagonals is really which shape?
Answer: A square.
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8Recap Recap
- Rhombus area = diagonal × diagonal ÷ 2
- A rhombus has four equal sides and perpendicular diagonals
- One diagonal splits it into two congruent triangles
- Area units are cm²
- To find a diagonal: multiply the area by 2, then divide by the known diagonal