Area · Lesson 8 of 12
Area of a Rhombus
“Use both base-height and diagonal methods for rhombus area, derive the diagonal formula by dissection and triangles, and solve inverse problems.”
• Use the parallelogram formula to find the area of a rhombus from base and height. • Recall the perpendicular-bisector relationship of rhombus diagonals. • Derive A = ½d₁d₂ by an area-preserving rearrangement. • Derive the same formula by adding triangle areas. • Find missing diagonal, base, or height from area information.
A rhombus is a parallelogram with all four sides equal, so the parallelogram formula A = bh already works. But a rhombus has extra structure: its diagonals are perpendicular and bisect each other. Those properties give a second area formula that can be more convenient when diagonal lengths are known.
Rhombus as a Parallelogram
Choose any side of a rhombus as base and measure the perpendicular distance to the opposite parallel side. The area is base × height. Equal side lengths do not mean the height equals the side; a slanted rhombus usually has a smaller perpendicular height.
Problem
A rhombus has side 10 cm and perpendicular height 8 cm. Find its area.
- 1.Use the side as base: b = 10 cm.
- 2.Corresponding height h = 8 cm.
- 3.Area = 10 × 8 = 80 cm².
What the Diagonals Do
The diagonals of a rhombus bisect one another at right angles. If their lengths are d₁ and d₂, then each half-diagonal has lengths d₁/2 and d₂/2. The four small triangles around the intersection are right triangles.
Segments joining opposite vertices. In a rhombus they bisect each other perpendicularly.
Dissection Derivation
A rhombus can be split into pieces and rearranged into a rectangle. In the resulting rectangle, one dimension equals one complete diagonal, while the other equals half the second diagonal. Since dissection preserves area, multiplying those rectangle dimensions gives the rhombus area.
Problem
A rhombus has diagonals 20 cm and 15 cm. Find its area.
- 1.A = ½d₁d₂.
- 2.A = ½ × 20 × 15.
- 3.A = 150 cm².
A Second Derivation Using Triangles
Let the diagonals AC and BD meet at O. Triangles ABD and CBD share base BD. Their heights to BD are AO and CO. Because AC is bisected, AO + CO = AC. Adding the two triangle areas gives ½ × BD × (AO + CO) = ½ × BD × AC.
The two derivations are useful for different reasons. The dissection makes the formula visually memorable. The triangle derivation links it directly to the general triangle area formula and the diagonal properties of the rhombus.
Finding a Missing Diagonal
If area and one diagonal are known, rearrange A = ½d₁d₂. Multiply the area by 2 and divide by the known diagonal.
Problem
A rhombus has area 96 cm² and one diagonal 12 cm. Find the other diagonal.
- 1.Use 96 = ½ × 12 × d₂.
- 2.96 = 6d₂.
- 3.d₂ = 16 cm.
Checking with the Base-Height Formula
When both a side-height pair and the diagonals are known, the two formulas must agree. This makes an excellent error check. If bh and ½d₁d₂ differ, at least one measurement or calculation is wrong.
Problem
A rhombus has base 13 cm and height 12 cm. Its diagonals are 24 cm and 13 cm. Verify the area both ways.
- 1.Base-height area = 13 × 12 = 156 cm².
- 2.Diagonal area = ½ × 24 × 13 = 156 cm².
- 3.The two independent methods agree.
Equal-Area Transformations with a Rhombus
Because the diagonal formula comes from a rectangle of dimensions d₁ by d₂/2, it also gives a construction recipe. To create a rectangle equal in area to a rhombus, use one diagonal as the rectangle length and half the other diagonal as its width. The reverse idea can guide a rectangle-to-rhombus construction.
Multiplying d₁d₂ gives the area of a rectangle twice as large as the rhombus area represented by the diagonal dissection. The factor ½ is essential.
Quiz
Which formula always works for a rhombus when a base and perpendicular height are known?
What special angle do the diagonals of a rhombus form?
A rhombus has diagonals 18 cm and 10 cm. Its area is
If A = 120 cm² and d₁ = 15 cm, d₂ equals
Why do the base-height and diagonal formulas give the same result?
Practice Problems
- Find the area of a rhombus with diagonals 22 cm and 14 cm.
- A rhombus has area 180 cm² and one diagonal 24 cm. Find the other diagonal.
- A rhombus has side 17 cm and height 9 cm. Find its area.
- A rhombus has diagonals 30 cm and 16 cm. Give dimensions of an equal-area rectangle suggested by the dissection.
- Explain why the four small triangles made by the diagonals are right triangles.
- Create numerical data for a rhombus where bh and ½d₁d₂ both give 144 cm².
Key Takeaways
• A rhombus is a parallelogram, so A = bh always applies with a perpendicular height. • Its diagonals bisect each other at right angles. • Dissection gives an equal-area rectangle of dimensions d₁ and d₂/2. • Therefore rhombus area is ½d₁d₂. • The same formula also follows by adding triangle areas. • Area information can be used backward to find a missing diagonal, base, or height.