Golden Rectangle Calculator
Calculate golden rectangle dimensions for geometry, design, architecture, art, and photography. Enter square side a or strip width b; the tool applies φ and shows total width a + b and area instantly.
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Golden Rectangle Calculator
Matches the classic subdivision: square a × a on the left, strip of width b on the right, full height a. Enter a or b; φ links them.
Side of the a × a square; also the height of the full golden rectangle.
Width of the smaller rectangle beside the square (height is still a).
a + b
Total width of the golden rectangle.
Area
Height × width = a × (a + b).
Using this calculator
- Enter a (square side) or b (width of the right-hand piece).
- If you type a, the tool sets b = a ÷ φ. If you type b, it sets a = b × φ (since a ÷ b = φ).
- Read a + b for total width and Area for a × (a + b).
- Press Reset to clear all fields.
Example calculations
- a = 5 units: b ≈ 3.09, a + b ≈ 8.09, area ≈ 40.45 square units
- a = 10 cm: b ≈ 6.18 cm, a + b ≈ 16.18 cm, area ≈ 161.8 cm²
- b = 4 in (strip width): a ≈ 6.47 in, a + b ≈ 10.47 in, area ≈ 67.7 in²
What Is a Golden Rectangle?
A golden rectangle is a rectangle whose sides follow the golden ratio φ in the square-plus-strip layout used on this site.
An a × a square sits on the left. A companion piece of height a and width b sits on the right. Together they form one rectangle of height a and total width a + b.
The shape is golden when (a + b) ÷ a = φ and a ÷ b = φ, with φ ≈ 1.618. Designers, architects, and photographers use these proportions for balanced frames.
Read the full guide: What Is a Golden Rectangle?.
- Definition: (a + b) ÷ a = a ÷ b = φ
- a: square side and full height
- b: width of the strip beside the square
- Applications: layout grids, facades, photo frames, logo boxes
Golden Rectangle Formula
The core relationship links total width, square side, and strip width to φ.
(a + b) ÷ a = a ÷ b = φ, where φ = (1 + √5) ÷ 2 ≈ 1.6180339887.
From square side a: b = a ÷ φ, total width = a + b, area = a × (a + b). From strip b: a = b × φ, then use the same sum and area formulas.
Formula article: Golden Rectangle Formula.
Golden ratio form
Total width divided by height equals φ. Height divided by strip width equals φ. Both checks must agree.
Side calculations
b = a ÷ φ and a = b × φ. Equivalently, a + b = a × φ.
Area
Area = a × (a + b). Do not use a × b for the full golden rectangle.
How to Calculate a Golden Rectangle
You need one known dimension and φ. The calculator above applies these steps automatically.
Step 1: Identify whether you know a (square side) or b (strip width).
Step 2: If you know a, set b = a ÷ φ. If you know b, set a = b × φ.
Step 3: Compute total width a + b and area a × (a + b).
Step 4: Verify (a + b) ÷ a and a ÷ b both equal φ within your rounding.
Walkthrough: How to Calculate a Golden Rectangle.
Calculate width from length
When length means square side a, strip width is b = a ÷ φ and total width is a + b.
Calculate length from width
When you know strip width b, square side is a = b × φ. Add for total width.
Golden Rectangle Examples
Sample dimensions using φ ≈ 1.618. Enter the same values in the calculator to confirm.
Geometry: a = 1 gives b ≈ 0.618, a + b ≈ 1.618, area ≈ 1.618 square units.
Design: a = 960 px gives b ≈ 594 px and total width ≈ 1554 px for a wide layout block.
Photography: true golden width from height a is a + b = a × φ, not exactly 3:2 unless you approximate.
More cases: Golden Rectangle Examples.
Golden Ratio Calculator
φ is the number behind golden rectangles. This page’s tool applies φ to sides a and b and shows total width and area.
φ ≈ 1.6180339887. Exact value: (1 + √5) ÷ 2.
Ratio verification: compute (a + b) ÷ a and a ÷ b. Both should match φ.
Definitions and checks: Golden Ratio Calculator (article, not a separate tool).
- Phi (φ): positive solution to x² = x + 1
- Dimension math: use b = a ÷ φ or a = b × φ
- Practical check: compare your measured ratio to 1.618
Golden Rectangle Construction
Build a golden rectangle with compass and straightedge, or see how the square-plus-strip diagram matches classical steps.
Start with square ABCD of side a. Bisect a base side, use the half-diagonal to mark a point so the full width becomes a × φ.
In the strip view, the piece beside the square has width b = a ÷ φ, so total width is a + b.
Construction guide: Golden Rectangle Construction.
Golden Rectangle vs Golden Ratio
φ is a number. A golden rectangle is a shape whose labeled sides realize that number.
Golden ratio: one constant used in sequences, spirals, and proportion problems.
Golden rectangle: a geometric figure where (a + b) ÷ a = a ÷ b = φ with a as square side and b as strip width.
Comparison: Golden Rectangle vs Golden Ratio.
Key difference
Every golden rectangle implies φ, but using φ in a ratio problem does not always mean you drew a rectangle.
Common misconception
Calling any pleasant-looking rectangle golden without checking (a + b) ÷ a or a ÷ b against φ.
Golden Rectangle in Design
Use golden proportions when you want calm, balanced layout width relative to height.
Web and UI: set a module height to a and width to a + b (or scale both) for hero blocks and cards.
Logos: grid cells sized with a and a + b keep marks and wordmarks aligned.
Photography: golden frames support hierarchy; pair with rule-of-thirds guides for composition.
Design article: Golden Rectangle in Design.
Common Golden Rectangle Mistakes
Most errors come from mislabeling b or using the wrong area formula.
Treating b as full width: total width is a + b, not b alone.
Using area = a × b: full area is a × (a + b).
Setting b = a × φ: with this diagram, b = a ÷ φ because a ÷ b = φ.
Equating 3:2 or 16:9 with golden: measure (a + b) ÷ a; only values near φ are golden.
Rounding too early: keep extra digits, then round for display.
Golden Rectangle and Fibonacci Sequence
Fibonacci ratios approach φ; golden rectangles use φ exactly in the side formulas.
Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, … Each term is the sum of the two before it.
Ratios F(n+1) ÷ F(n) approach φ. Rectangles with consecutive Fibonacci sides are almost, but not exactly, golden.
For exact dimensions, use the calculator or φ in the formulas.
Full article: Golden Rectangle and Fibonacci Sequence.
FAQs About Golden Rectangles
- What is a golden rectangle?
- A golden rectangle satisfies (a + b) ÷ a = a ÷ b = φ, with a as the square side (height) and b as the strip width. Total width is a + b.
- What is the golden rectangle formula?
- (a + b) ÷ a = a ÷ b = φ. From a: b = a ÷ φ and area = a(a + b). From b: a = b × φ, then compute sum and area.
- How do I calculate width from length?
- If length means square side a, strip width is b = a ÷ φ and total width is a + b. Enter a in the calculator to fill the rest.
- How do I calculate length from width?
- If you know strip width b, square side is a = b × φ. Total width is a + b.
- What does a mean in the diagram?
- a is the side of the square on the left and the height of the entire golden rectangle.
- What does b mean?
- b is only the width of the smaller rectangle on the right, not the full width.
- How is area calculated?
- Area = a × (a + b), height times total width. It is not a × b.
- What is φ (phi)?
- Phi is the golden ratio, about 1.618, equal to (1 + √5) ÷ 2.
- How do I verify golden proportions?
- Compute (a + b) ÷ a and a ÷ b. Both should equal φ within your rounding.
- How is a golden rectangle different from the golden ratio?
- φ is a number. A golden rectangle is a shape whose sides satisfy the golden formulas.
- Do Fibonacci numbers give an exact golden rectangle?
- Consecutive Fibonacci ratios approach φ but are rational approximations. Use φ in the formulas for exact sides.
- Is my data stored?
- No. All math runs locally in your browser.