Golden Rectangle Calculator
Enter the short or long side of a golden rectangle to find the other side, the area, perimeter and diagonal.
Updated
Golden Rectangle Calculator
Our Golden Rectangle Calculator finds the missing side of a golden rectangle from the side you know, using the golden ratio φ ≈ 1.618034. It also gives the leftover strip, the area, the perimeter, the diagonal and exact values with √5.
Quick answer
The long side of a golden rectangle is L = a × φ: 10 × 1.618034 ≈ 16.18034.
- Golden ratio: φ = (1 + √5) ÷ 2 ≈ 1.6180339887
- From the long side: a = L ÷ φ, so L = 10 gives a ≈ 6.18034
- Area: a × L = a² × φ, so a = 10 gives ≈ 161.803399
- Key property: cut off a square and the strip left over is another golden rectangle
What Is a Golden Rectangle?
A golden rectangle is a rectangle whose long side divided by its short side is the golden ratio, φ ≈ 1.618. So a golden rectangle 10 cm tall is about 16.18 cm wide.
It is the only rectangle with this property: cut a square off one end and the strip that is left, b by a, has exactly the same shape as the whole rectangle.
Golden Rectangle Formulas
L = a × φ
a = L ÷ φ
b = L − a = a ÷ φ
| Value | a = 10 example |
|---|---|
Long sidea × φ | ≈ 16.18034 |
Leftover stripa ÷ φ | ≈ 6.18034 |
Areaa² × φ | ≈ 161.803399 |
Perimeter2(a + L) | ≈ 52.36068 |
Diagonal√(a² + L²) | ≈ 19.02113 |
Exact values use √5: a = 10 gives L = 5 + 5√5 and b = 5√5 − 5. Dividing by φ is the same as multiplying by φ − 1 ≈ 0.618034.
Why the Ratio Is (1 + √5) ÷ 2
The strip left after cutting off the square must have the same shape as the whole rectangle, so L ÷ a = a ÷ b. Call that ratio φ and set a = 1:
- Sides: L = φ and b = L − a = φ − 1.
- Same shape: φ ÷ 1 = 1 ÷ (φ − 1), so φ² − φ = 1.
- Solve: φ² − φ − 1 = 0 has the positive root φ = (1 + √5) ÷ 2 ≈ 1.618034.
That is also why φ² = φ + 1 ≈ 2.618034 and 1 ÷ φ = φ − 1 ≈ 0.618034. For the ratio on its own, such as splitting a line, use the Golden Ratio Calculator.
How to Draw a Golden Rectangle
You only need a ruler and a compass:
- Draw a square with side a.
- Mark the midpoint M of the bottom side.
- Set the compass from M to a top corner on the far side. That length is a√5 ÷ 2.
- Swing it down to the extended bottom line, then finish the rectangle.
The Golden Spiral
Keep cutting squares off each new golden rectangle and the squares wind inward. A quarter circle in each square joins up into the golden spiral, which grows by a factor of φ every quarter turn.
Golden Rectangle Examples
| Short a | Long L | Area |
|---|---|---|
| 1 | ≈ 1.618 | ≈ 1.618 |
| 5 cm | ≈ 8.09 cm | ≈ 40.45 cm² |
| 8 in | ≈ 12.94 in | ≈ 103.55 in² |
| 10 cm | ≈ 16.18 cm | ≈ 161.8 cm² |
| 21 m | ≈ 33.98 m | ≈ 713.55 m² |
Example: a picture frame opening 30 cm tall would be 30 × φ ≈ 48.54 cm wide to be a golden rectangle.
Golden Rectangle vs Other Rectangles
| Shape | Long ÷ short |
|---|---|
| A4 paper (√2) | ≈ 1.414 |
| 3:2 photo | 1.5 |
| 16:10 screen, 8:5 | 1.6 |
| Golden rectangle | ≈ 1.618 |
| 13:8 | 1.625 |
| 16:9 screen | ≈ 1.778 |
Ratios of neighbouring Fibonacci numbers (5:3, 8:5, 13:8, 21:13) get closer and closer to φ, so Fibonacci-sized rectangles are good whole-number stand-ins.
Common Mistakes
- Multiplying when you should divide. From the long side, divide by φ: 10 ÷ φ ≈ 6.18, not 16.18.
- Rounding φ to 1.6. That is an 8:5 rectangle; on a 100 cm side it is off by about 1.8 cm.
- Mixing up b and a. b is the leftover strip L − a, not the short side.
- Trusting famous claims. Golden ratios in the Parthenon or the Mona Lisa are disputed; measure first.
Key Terms
- Golden ratio (φ)
- (1 + √5) ÷ 2 ≈ 1.618034.
- Short side (a)
- The height, and the side of the square.
- Long side (L)
- a × φ, the width.
- Strip (b)
- L − a, a smaller golden rectangle.
- Golden spiral
- Quarter circles through the nested squares.
How to Use the Golden Rectangle Calculator
- Choose the side you know. Pick Short side to find the long side, or Long side to find the short side.
- Enter the length. Type it, such as 10, and pick a unit if you want one (mm, cm, m, km, in, ft, yd or mi). Decimals and fractions such as 3/4 work.
- Read and copy the result. The missing side appears as you type, with a drawing, the working, the exact value with √5, the leftover strip, the area, the perimeter and the diagonal. Press Copy to copy it, or Clear or Esc to start again.
Example: a web page is 1200 px wide and you want a golden split. Choose Long side and enter 1200. The short side is about 741.640786, so a 742 px main column and a 458 px sidebar.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
What is a golden rectangle?
A rectangle whose long side divided by its short side is the golden ratio, φ ≈ 1.618. Cutting a square off one end leaves a smaller rectangle with the same shape.
How do you calculate a golden rectangle?
Multiply the short side by φ ≈ 1.618034 to get the long side, or divide the long side by φ to get the short side. A short side of 10 gives a long side of about 16.18.
What is the ratio of a golden rectangle?
φ : 1, where φ = (1 + √5) ÷ 2 ≈ 1.6180339887. It is close to 8:5 (1.6) and 13:8 (1.625).
How do you find the area of a golden rectangle?
Multiply the two sides, which is the same as a² × φ. A short side of 10 cm gives about 161.8 cm².
How do you find the short side from the long side?
Divide by φ, or multiply by 0.618034. A long side of 10 gives a short side of about 6.18.
Why is the golden ratio (1 + √5) ÷ 2?
The leftover strip must have the same shape as the whole, so φ = 1 ÷ (φ − 1). That gives φ² − φ − 1 = 0, whose positive root is (1 + √5) ÷ 2.
Is a 16:9 screen a golden rectangle?
No. 16:9 is about 1.778, wider than φ ≈ 1.618. A 16:10 screen (1.6) is much closer.
How is the golden spiral made from a golden rectangle?
Keep cutting a square off each new golden rectangle and draw a quarter circle in each square. The arcs join into a spiral that grows by φ every quarter turn.
Is the Golden Rectangle Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Try our Golden Rectangle Calculator to find the missing side, area, perimeter and diagonal from either side.