Math CalculatorsExtend the pattern geometrically

Third Proportional Calculator

2026-09-29 8 min

Third proportional calculator — continue a:b as b:c with c = b²/a. Free online solver with geometric-sequence check, visuals and practice.

Square the bridge term and divide by the start: c = b²/a. The triple then climbs by the same ratio twice.

Grade level: 7–10; Curriculum band
Difficulty: Intermediate; For this topic
Time to master: 1–2 weeks; With daily practice
Tools on page: 10 on page; Calculator + practice

Two Terms — enter a and b

First term (a)

Second term (b)

Try:

c = 9 · k = 1.5

Results update live as you type

Status:4:6 = 6:9

Third Proportional Summary

Given pair a, b4, 6

Third proportional c9

Common ratio k1.5

Breakdown

Property	Value
Computation b² ÷ a	36 ÷ 4
Cross a × c	36
Cross b²	36
Triple a, b, c	4, 6, 9

Results

Optimal

Third proportional (c)

b² ÷ a — continues the proportion

Continued proportion

4:6 = 6:9

Middle term bridges both sides

Common ratio

1.5

b/a = c/b — geometric step

Cross check

36 = 36

a×c vs b² — must match

Results follow continued-proportion rules (a:b = b:c) for the inputs given. The first term must be nonzero.

Statistics

Live readouts from your current inputs — the staircase, ratios and cards all move together. Change a or b to watch c and k update.

Third proportional

b² ÷ a

Common ratio

1.5

×k twice

Cross a×c

must equal b²

b²

middle squared

Visualizations

Geometric staircase a → b → c

Growth per step (common ratio)

b ÷ a1.5×

Each step multiplies by the same ratio

Formula

The third proportional extends two terms into a three-term geometric progression: each step multiplies by the same ratio k, so the middle term bridges both sides.

a:b = b:c ⟺ c = b²/a (a ≠ 0)

Alternative Form

Common ratio k = b/a = c/b · triple a, ak, ak²

Variables

Symbol	Name	Unit	Description
a	First term	number	Start of the triple (≠ 0)
b	Mean	number	Geometric mean of a and c
c	Third proportional	number	b²/a — the answer
k	Common ratio	number	b/a = c/b

Explanation

If 4:6 continues proportionally, the step ×1.5 repeats: 6×1.5 = 9. Squaring the bridge (b²) and dividing by the start (a) jumps two steps at once.

Derivation

From a:b = b:c, cross: a·c = b². Divide by a ≠ 0: c = b²/a. Check: c/b = (b²/a)/b = b/a = k. Triple: a, ak, ak² with k = b/a.

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Write the continued proportion with c unknown.

a:b = b:c

a:4

b:6

Intermediate: 4:6 = 6:c

Result: Framed

Step 2

Cross-multiply: outer × outer = inner × inner.

a×c = b×b

Intermediate: 4×c = 6²

Result: = 36

Step 3

Isolate c by dividing both sides by a.

c = b²/a

Intermediate: 36 ÷ 4

Result: c = 9

Step 4

Evaluate the division.

b²/a

Intermediate: 36/4

Result: 9

Step 5

Verify the common ratio b/a equals c/b.

b/a = c/b?

Intermediate: 1.5 vs 1.5

Result: Match ✓

Step 6

State the completed triple.

a, b, c geometric

Intermediate: 4, 6, 9

Result: Done ✓

Final Answer

Worked Examples

See how the calculator works with different scenarios.

Scenario

Continue 4:6 proportionally.

Inputs

a:4

b:6

Calculation

c = 6²/4 = 36/4 = 9

Result

c = 9 (4:6 = 6:9)

Interpretation

Common ratio 1.5 twice: 4 → 6 → 9.

Understanding Your Result

Your Result: Third proportional (c): 9

Typical range: Ratios 1.1–3 per step; textbook triples like 4, 6, 9 and 2, 4, 8.

What This Means

c extends the multiplicative pattern one more step — the triple a, b, c grows by the same factor twice.

Why It Matters

Geometric sequences, similar-figure chains, and mean-proportional constructions all hinge on third proportionals.

Professional Interpretation

Report c with the ratio proof: “c = 9 (b/a = c/b = 1.5).”

Recommendations

•Square the middle, divide by the first
•Verify b/a equals c/b always
•Reject a = 0 up front
•Read b as the geometric mean of a and c
•Extend further with ×k steps
•Keep sign rules for negative inputs

Learn — The Complete Guide

The third proportional c completes a:b as b:c — a three-term geometric progression with common ratio k = b/a.

The core concept

c = b²/a. The middle term b is the geometric mean of the outer pair: b = √(a×c).

Third proportional reuses b as the bridge
c = b²/a: square middle, divide by first
b/a = c/b = k proves geometric growth
b is the geometric mean √(a×c)

Derivations — why the formulas work

Formula — cross and isolate

1a:b = b:c
2Cross: a·c = b·b
3Divide by a ≠ 0
4c = b²/a ✓

Ratio — k twice

1k = b/a
2c/b = (b²/a)/b = b/a
3So c/b = k
4Triple a, ak, ak² ✓

Mean — solve for b

1a·c = b²
2Take square roots (positive)
3b = √(a·c)
4Middle = geometric mean ✓

Where you meet this in real life

▸Geometric sequence homework (find next term)

▸Mean-proportional geometry constructions

▸Similar-figure scaling chains

▸Compound-growth step continuation

▸Photo/print size progressions

▸Musical interval stacking (ratios)

▸Geometric sequence homework and tests

▸Photo/print size progressions

▸Compound-growth step forecasts

▸Musical interval stacking

▸Geometric-mean engineering estimates

Learn this first — prerequisites in order

1Ratios: write and cross a:b = c:d
2Powers: square two-digit numbers
3Roots: square-root basics for mean reading
4Sequences: arithmetic vs geometric sense

Careers that use this daily

•Teacher — sequences curriculum
•Photographer — print sizing chains
•Financial planner — growth-step models
•Luthier — fret-ratio layout
•Architect — proportional progressions

Why it helps

•c with cross + ratio double proof
•Staircase visual shows ×k twice
•Mean-proportional insight (b = √ac)
•Steps model cross-then-divide
•Gateway to geometric sequences

Best practices

•Write a:b = b:c before computing
•Cross to a×c = b² explicitly
•Verify the common ratio both ways
•Interpret b as √(a×c)
•Continue the chain with ×k for more terms

Limitations — what this tool does not do

•Requires a ≠ 0
•Two-term input only (a, b)
•Geometric (not arithmetic) continuation
•Display rounds to 12 significant digits

Curriculum map — where this topic lives in school

Level	Grades	Standards	Skills
Middle School	7–8	7.RP, 8.EE	Proportions · Powers
Algebra I	9–10	HSA, HSF-LE	Geometric sequences · Mean proportionals
Geometry	9–11	HSG-SRT	Similar figures · Geometric-mean constructions

Timeline — how this idea grew up

~300 BC
Euclid’s mean proportionals
Elements VI constructs mean proportionals geometrically — b between a and c in stone-and-compass form.
~250 AD
Diophantine ratios
Greek arithmetic treats continued proportions as the solvable core of ratio problems.
1202
Fibonacci exercises
Liber Abaci drills continued proportions for merchants splitting growth fairly.
Today
Sequences curriculum
Third proportionals open the geometric-sequences chapter in algebra worldwide.

A little history

• Euclid VI.13 constructs the mean proportional to two lines geometrically.
• “Continued proportion” is the classical name for a:b = b:c.
• Mean proportionals solved doubling-the-cube approximations in antiquity.

Exam-day checklist

• Name the pattern “a:b = b:c” explicitly.
• Show the cross a×c = b² for method marks.
• Verify k = b/a = c/b on the answer line.
• Flag arithmetic traps (equal differences) as wrong.

Myths vs facts

Myth: Third proportional adds the difference (6 + 2 = 8).

Truth: That is arithmetic. Proportional continuation MULTIPLIES: 6×1.5 = 9. Differences add; ratios multiply.

Myth: Third and fourth proportional are the same.

Truth: Third reuses b as the bridge (two inputs); fourth takes three distinct inputs a,b,c. Different problems, different formulas.

Myth: Any three numbers form a continued proportion.

Truth: Only geometric triples qualify: 4,6,9 ✓ (×1.5 twice) but 4,6,8 ✗ (1.5 then 1.33).

Myth: a = 0 just gives c = 0.

Truth: a = 0 makes c = b²/0 undefined — rejection, not zero.

Practical Tips

•Square middle, divide by first
•Cross line first: a×c = b²
•Verify k both ways
•Read b as √(a×c)
•Zero a = stop sign
•Multiply by k to extend
•Arithmetic ≠ geometric!

Common Mistakes

Adding differences instead of multiplying ratios

Continuation is geometric: multiply by k = b/a.

Using (a+b)/2 style averages

b is the GEOMETRIC mean √(ac), not arithmetic.

Confusing with fourth proportional

Two inputs + repeated middle = third; three inputs = fourth.

Dividing by zero a

Validate a ≠ 0 before the formula.

Skipping the ratio verification

Check b/a = c/b on every answer.

Assumptions

This calculator makes the following assumptions to provide estimates:

Nonzero a: The first term differs from zero.

Geometric Intent: Continuation is multiplicative, not additive.

Finite Range: Inputs stay within finite floating-point range.

Limitations

Be aware of these limitations when using this calculator:

a ≠ 0: Division by zero is rejected.

Two Inputs: Only the a, b → c case is solved.

Display Rounding: Results round to 12 significant digits.

Frequently Asked Questions

Answers to common questions about this calculator.

What is the third proportional?

The value c continuing a:b as b:c — i.e. a, b, c in geometric progression. Formula: c = b²/a. Example: a=4, b=6 → c = 36/4 = 9, giving 4:6 = 6:9.

How do you derive c = b²/a?

From a:b = b:c, cross-multiply: a×c = b×b, so c = b²/a (a ≠ 0). The middle term squared, divided by the first — one cross, one division.

What is the difference between third and fourth proportional?

Third continues TWO numbers (a:b = b:c, middle repeated — geometric sequence). Fourth completes THREE numbers (a:b = c:d). Third needs a,b; fourth needs a,b,c.

Must a, b, c be in geometric progression?

Yes — that is exactly what “third proportional” means: b/a = c/b = common ratio. 4, 6, 9 has ratio 1.5 throughout. Verify with that quotient.

Can a be zero or negative?

a = 0 is rejected (division by zero). Negatives work algebraically (a=−4, b=6 → c=−9) but most textbook problems use positive lengths and amounts.

How do I use this third proportional calculator?

Enter a and b (defaults 4 and 6). The page computes c = b²/a = 9, shows the continued proportion a:b = b:c, verifies the common ratio — plus steps, practice, and worksheet.

Related Calculators

Explore more calculators related to this topic.

Related Guides

Deepen your understanding with these educational resources.

Proportions Handbook

Cross, isolate, verify.

Geometric Sequences Guide

From triples to infinite series.

Mean Proportionals

Geometric means in geometry.

Glossary

Technical terms used in mortgage calculations explained in plain language.

Third proportional: c with a:b = b:c; equals b²/a.
Continued proportion: Three-term geometric equality.
Geometric mean: Middle b = √(a×c).
Common ratio: k = b/a = c/b step multiplier.
Geometric progression: Sequence multiplying by fixed k.

Authoritative References

Trusted sources and standards referenced by this calculator.

Khan Academy — Geometric sequences Math Is Fun — Geometric Sequences Wikipedia — Geometric progression Purplemath — Proportions

Applicable Standards

Common Core 7.RP & HSF-LE: proportional relationships and geometric sequences

Diagrams & Illustrations

Visual intuition for every core idea. Select a card to enlarge it.

Did You Know?

Short intuition builders — great for class warm-ups.

Euclid built means

Elements Book VI constructs the mean proportional with straightedge and compass — the geometric version of c = b²/a.

Photography loves 1.5×

Print series 4, 6, 9 inches step ×1.5 twice — geometric triples keep aspect ratios identical across sizes.

Music stacks thirds

Frequency ratios like 4:6:9 stack perfect intervals — continued proportions tune harmony, not just homework.

Golden ratio iterates thirds

φ satisfies 1:φ = φ:φ² — the golden ratio is a third proportional fixed point, generating Fibonacci scaling.

Compare Side by Side

Same start, different “continue”: adding gives 8 (wrong here); multiplying by k gives 9 (proportional).

Continuation	Next term	Step	Type
Arithmetic (+2)	8	+2	Adds — NOT proportional
Geometric (×k) ✓	9	×1.5	Multiplies — proportional

Recent Calculations

Your last runs on this page — stored only in this browser, never uploaded.

third prop: 4, 6 → 9
Current inputs — live

Study Plan

A week-by-week path from prerequisites to mastery. Your checklist progress saves in this browser.

Prerequisites

▸Ratios and cross-multiplication
▸Squaring numbers
▸Geometric sequences intuition

Mastery progress — 0%

I can write a:b = b:c from any wording
I compute c = b²/a fluently
I verify k = b/a = c/b
I read b as the geometric mean
I never continue arithmetically by mistake

1
Week 1 · ~15 min/day
Cross to c = b²/a
Goals
- Write a:b = b:c from words
- Cross to a×c = b²
- Divide to c fluently
Practice
Solve 10 third proportionals (4,6→9 family) — verify each common ratio.
2
Week 2 · ~20 min/day
Geometric meaning
Goals
- Read b as √(a×c)
- Compute k = b/a = c/b
- Distinguish arithmetic vs geometric continuation
Practice
For 5 triples, prove geometric or expose arithmetic impostors with k checks.
3
Week 3 · ~20 min/day
Triples in the wild
Goals
- Continue growth and size sequences
- Extend chains with ×k
- Apply mean proportionals in geometry
Practice
Continue one growth series, one size series, and one interval stack; state k each time.

Exam tips

•Write “a:b = b:c” first — it names the pattern.
•Show the a×c = b² cross line for method marks.
•Verify with the k-equality on the answer line.

Practice Problems

0/6 solved

Try each problem first, peek at the hint only if stuck, then reveal the solution and mark it solved.

Basicseasy

Find the third proportional to 4 and 6.

Basicseasy

Find the third proportional to 3 and 6.

Verifymedium

Is 4, 6, 8 a continued proportion?

Meanmedium

If a = 4 and c = 9, what is the mean proportional b?

Fractionsmedium

Third proportional to 2 and 5?

Appliedhard

A colony grows 4 → 6 thousand. Continue geometrically one step.

Printable Worksheet

Generate a fresh problem set for class or homework. Print it, or copy the answer key separately.

Problems Name

Arithmetic practice — set #1Name: __________ · Date: __________

1.18 × 47 =
2.8 ÷ 11 =
3.38 × 26 =
4.35 ÷ 4 =
5.21 − 16 =
6.2 + 26 =
7.14 ÷ 12 =
8.23 ÷ 3 =
9.49 ÷ 7 =
10.22 − 17 =
11.41 + 18 =
12.47 − 35 =

Formula Cheat Sheet

Every formula on this page in one printable table.

Formula	Meaning	Example
a:b = b:c	Continued pattern	4:6 = 6:9
c = b²/a	Third formula	36/4 = 9
a×c = b²	Cross form	4×9 = 36 = 6²
k = b/a = c/b	Common ratio	1.5 = 1.5 ✓
b = √(a×c)	Mean reading	√36 = 6
a, ak, ak²	Triple form	4, 6, 9 (k=1.5)

Tip: every interactive block on this page — checklist, practice, worksheet, history — works offline and saves progress in your browser.

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Page Information

Last Updated:

2026-09-29

Version:

3.1.0-math

References:

License:

Free to use for educational and professional purposes.