Toolarithm

Math Calculator

Pythagorean Theorem Calculator

Find a side. Check the right angle.

Pythagorean inputs

Find one side or check three given lengths.

Each method retains its own values. Only active fields are validated.

Use a finite decimal, scientific e notation or an integer fraction.

Use a finite decimal, scientific e notation or an integer fraction.

Units and display choices

The two known legs must meet at a right angle. A rectangle's width and height can be used here to find its diagonal.

Applies to every active side. Convert mixed measurements to one unit first. Changing this selector reinterprets the retained numbers.

Converts sides and perimeter. Squared-side rows use the square of this unit.

Area converts independently, including international acres and hectares.

Exact expressions and classification stay unchanged. Very small nonzero values retain scientific views.

Load a worked example

Finite decimals and fractions retain exact values. Source limits and thin-diagram omissions are explained beside the results. Checking three sides does not model measurement uncertainty.

One right triangle. The missing positive side, coverage and perimeter retain exact expressions.

Interpreted active measurements

a=3 cm,b=4 cma=3\,\mathrm{cm},\quad b=4\,\mathrm{cm}

Hypotenuse

c=5 cmc=5\,\mathrm{cm}

Rounded side view

c=5 cmc=5\,\mathrm{cm}

Area

K=6 cm2K=6\,\mathrm{cm}^{2}

Perimeter

P=12 cmP=12\,\mathrm{cm}

Exact means exact for the entered numbers, not a claim about measurement uncertainty. No tolerance silently turns a nearly right triangle into a right triangle.

Sources allow sixty significant decimal digits or sixty digits per fraction integer, up to one hundred eighty characters and scientific exponents from negative one hundred through positive one hundred. Exact intermediate numerators and denominators are limited to eighteen hundred digits.

Small nonzero results retain a scientific view instead of being displayed as zero. Radical extraction checks small square factors and a remaining perfect square; it does not promise arbitrary integer factorization.

Verified right triangle diagram

Solved right triangle with two perpendicular legsThe horizontal side is leg a, the vertical side is leg b, and the sloping side is hypotenuse c. Both directions use equal units. A corner square marks the right angle.
aa
bb
cc
Equal horizontal and vertical scale
The horizontal leg is aa, the vertical leg is bb and the sloping hypotenuse is cc. Both axes use one scale. The small corner marker denotes the verified right angle; labels are symbolic so converted values remain in the result table.

Calculation step by step

  1. Identify the active measurements

    a=3 cm,b=4 cma=3\,\mathrm{cm},\quad b=4\,\mathrm{cm}

    The hypotenuse is opposite the right angle. The two given lengths use the shared source unit; they are not independent answers to unrelated problems.

  2. Add the two leg squares

    c2=a2+b2=9 cm2+16 cm2=25 cm2c^2=a^2+b^2=9\,\mathrm{cm}^{2}+16\,\mathrm{cm}^{2}=25\,\mathrm{cm}^{2}

    All decimal and fraction sources are exact rational numbers. Squaring and subtraction occur before any decimal rounding, even when two given sides are almost equal.

  3. Take the positive length

    c=25 cm2=5 cmc=\sqrt{25\,\mathrm{cm}^{2}}=5\,\mathrm{cm}

    Lengths use the nonnegative square-root branch. A negative algebraic root is not a second physical side length. Radical extraction is verified but bounded; a large remaining radicand need not be fully factored.

  4. Calculate coverage and the boundary

    K=ab2,P=a+b+cK=\frac{ab}{2},\qquad P=a+b+c

    The area uses the perpendicular legs from this same triangle. Perimeter adds all three exact side lengths.

  5. Convert dimensions before rounding

    Lout=Lsource(1),Kout=Ksource(1)L_{\mathrm{out}}=L_{\mathrm{source}}\left(1\right),\qquad K_{\mathrm{out}}=K_{\mathrm{source}}\left(1\right)

    A length factor and an area factor have different dimensions. Each output is rounded independently from its exact expression. Precision changes the decimal view, not the source or the classification.

All side and area results

Squared quantities use the square of the output length unit. Area has its own output unit. Rounded views never replace exact values.

Pythagorean calculation details
QuantityExact valueRounded viewInterpretation
Leg aa3 cm3\,\mathrm{cm}3 cm3\,\mathrm{cm}Converted known source length.
Leg bb4 cm4\,\mathrm{cm}4 cm4\,\mathrm{cm}Converted known source length.
Hypotenuse cc5 cm5\,\mathrm{cm}5 cm5\,\mathrm{cm}Solved positive root.
Area KK6 cm26\,\mathrm{cm}^{2}6 cm26\,\mathrm{cm}^{2}Half the product of the perpendicular legs.
Perimeter PP12 cm12\,\mathrm{cm}12 cm12\,\mathrm{cm}Exact sum evaluated before rounding; no rounded side is reused.
Squared side a2a^29 cm29\,\mathrm{cm}^{2}9 cm29\,\mathrm{cm}^{2}Squared output length units, independent of the area-output selector.
Squared side b2b^216 cm216\,\mathrm{cm}^{2}16 cm216\,\mathrm{cm}^{2}Squared output length units, independent of the area-output selector.
Squared side c2c^225 cm225\,\mathrm{cm}^{2}25 cm225\,\mathrm{cm}^{2}Squared output length units, independent of the area-output selector.
Sum a2+b2a^2+b^225 cm225\,\mathrm{cm}^{2}25 cm225\,\mathrm{cm}^{2}Exact sum of the two shorter-side squares in check mode.
Residual a2+b2−c2a^2+b^2-c^20 cm20\,\mathrm{cm}^{2}0 cm20\,\mathrm{cm}^{2}Zero means right; positive means acute; negative means obtuse after the strict triangle check.

Side cc in all length units

Side c in all length units
UnitExact valueRounded view
Nanometers5×107 nm5\times10^{7}\,\mathrm{nm}50000000 nm50000000\,\mathrm{nm}
Micrometers50000 μm50000\,\mathrm{\mu m}50000 μm50000\,\mathrm{\mu m}
Millimeters50 mm50\,\mathrm{mm}50 mm50\,\mathrm{mm}
Centimeters5 cm5\,\mathrm{cm}5 cm5\,\mathrm{cm}
Decimeters12 dm\frac{1}{2}\,\mathrm{dm}0.5 dm0.5\,\mathrm{dm}
Meters120 m\frac{1}{20}\,\mathrm{m}0.05 m0.05\,\mathrm{m}
Decameters1200 dam\frac{1}{200}\,\mathrm{dam}0.005 dam0.005\,\mathrm{dam}
Hectometers12000 hm\frac{1}{2000}\,\mathrm{hm}0.0005 hm0.0005\,\mathrm{hm}
Kilometers120000 km\frac{1}{20000}\,\mathrm{km}0.00005 km0.00005\,\mathrm{km}
Mils (one thousandth inch)250000127 mil\frac{250000}{127}\,\mathrm{mil}≈  1968.503937 mil\approx\;1968.503937\,\mathrm{mil}
Inches250127 in\frac{250}{127}\,\mathrm{in}≈  1.968504 in\approx\;1.968504\,\mathrm{in}
International feet125762 ft\frac{125}{762}\,\mathrm{ft}≈  0.164042 ft\approx\;0.164042\,\mathrm{ft}
International yards1252286 yd\frac{125}{2286}\,\mathrm{yd}≈  0.054681 yd\approx\;0.054681\,\mathrm{yd}
International miles25804672 mi\frac{25}{804672}\,\mathrm{mi}≈  0.000031 mi\approx\;0.000031\,\mathrm{mi}
International nautical miles137040 nmi\frac{1}{37040}\,\mathrm{nmi}≈  0.000027 nmi\approx\;0.000027\,\mathrm{nmi}
Typographic points (one seventy-second inch)18000127 pt\frac{18000}{127}\,\mathrm{pt}≈  141.732283 pt\approx\;141.732283\,\mathrm{pt}
Typographic picas (one sixth inch)1500127 pc\frac{1500}{127}\,\mathrm{pc}≈  11.811024 pc\approx\;11.811024\,\mathrm{pc}

Area in all area units

Area in all area units
UnitExact valueRounded view
Square nanometers6×1014 nm26\times10^{14}\,\mathrm{nm}^{2}600000000000000 nm2600000000000000\,\mathrm{nm}^{2}
Square micrometers6×108 μm26\times10^{8}\,\mathrm{\mu m}^{2}600000000 μm2600000000\,\mathrm{\mu m}^{2}
Square millimeters600 mm2600\,\mathrm{mm}^{2}600 mm2600\,\mathrm{mm}^{2}
Square centimeters6 cm26\,\mathrm{cm}^{2}6 cm26\,\mathrm{cm}^{2}
Square decimeters350 dm2\frac{3}{50}\,\mathrm{dm}^{2}0.06 dm20.06\,\mathrm{dm}^{2}
Square meters35000 m2\frac{3}{5000}\,\mathrm{m}^{2}0.0006 m20.0006\,\mathrm{m}^{2}
Square decameters3500000 dam2\frac{3}{500000}\,\mathrm{dam}^{2}0.000006 dam20.000006\,\mathrm{dam}^{2}
Square hectometers35×107 hm2\frac{3}{5\times10^{7}}\,\mathrm{hm}^{2}6×10−8 hm26\times10^{-8}\,\mathrm{hm}^{2}
Square kilometers35×109 km2\frac{3}{5\times10^{9}}\,\mathrm{km}^{2}6×10−10 km26\times10^{-10}\,\mathrm{km}^{2}
Square mils (one thousandth inch)15×10916129 mil2\frac{15\times10^{9}}{16129}\,\mathrm{mil}^{2}≈  930001.860004 mil2\approx\;930001.860004\,\mathrm{mil}^{2}
Square inches1500016129 in2\frac{15000}{16129}\,\mathrm{in}^{2}≈  0.930002 in2\approx\;0.930002\,\mathrm{in}^{2}
Square international feet62596774 ft2\frac{625}{96774}\,\mathrm{ft}^{2}≈  0.006458 ft2\approx\;0.006458\,\mathrm{ft}^{2}
Square international yards625870966 yd2\frac{625}{870966}\,\mathrm{yd}^{2}≈  0.000718 yd2\approx\;0.000718\,\mathrm{yd}^{2}
Square international miles25107916171264 mi2\frac{25}{107916171264}\,\mathrm{mi}^{2}≈  2.316613×10−10 mi2\approx\;2.316613\times10^{-10}\,\mathrm{mi}^{2}
Square international nautical miles317149520000 nmi2\frac{3}{17149520000}\,\mathrm{nmi}^{2}≈  1.74932×10−10 nmi2\approx\;1.74932\times10^{-10}\,\mathrm{nmi}^{2}
Square typographic points (one seventy-second inch)7776000016129 pt2\frac{77760000}{16129}\,\mathrm{pt}^{2}≈  4821.129642 pt2\approx\;4821.129642\,\mathrm{pt}^{2}
Square typographic picas (one sixth inch)54000016129 pc2\frac{540000}{16129}\,\mathrm{pc}^{2}≈  33.480067 pc2\approx\;33.480067\,\mathrm{pc}^{2}
International acres125843095088 acre\frac{125}{843095088}\,\mathrm{acre}≈  1.482632×10−7 acre\approx\;1.482632\times10^{-7}\,\mathrm{acre}
Hectares35×107 ha\frac{3}{5\times10^{7}}\,\mathrm{ha}6×10−8 ha6\times10^{-8}\,\mathrm{ha}

Inputs saved with this calculation

Method
Find hypotenuse
Active source strings
Side a: 3; Side b: 4
Source unit
Centimeters
Output length unit
Centimeters
Output area unit
Square centimeters
Rounded decimal places
6
What Can You Create?

Make a right-triangle relationship useful

Recover the one side you are missing

Start from the two measurements you actually know. Perpendicular legs determine their hypotenuse, while a hypotenuse and a shorter leg determine the remaining leg. The selected method exposes only required fields and retains its own editor values. Exact squares, a positive root and a coherent result table replace the confusion of two unrelated sample triangles presented as one calculation.

Test a set of three supplied lengths

Use the converse as a separate check rather than forcing every input into a right-triangle model. The checker accepts any order, sorts lengths explicitly and validates that they form a positive-area triangle. Comparing exact squares distinguishes right, acute and obtuse cases. Non-right results keep their own valid area calculation and never acquire an invented perpendicular-leg diagram.

Interpret diagonals, coverage and boundaries

A flat rectangle diagonal is the hypotenuse formed by its perpendicular width and height. The calculator reports the associated triangle's coverage and boundary length with independent dimensional outputs. Remember that this triangular area is half the rectangle's area. Conversion tables translate a coherent idealized model into useful units while source strings, assumptions and exact expressions remain inspectable.

The theorem and its converse, step by step

Add the two leg squares

For perpendicular legs 3 cm3\,\mathrm{cm} and 4 cm4\,\mathrm{cm}, the squares sum to 25 cm225\,\mathrm{cm}^2. The positive root is the hypotenuse 5 cm5\,\mathrm{cm}. Coverage and perimeter describe this same triangle, rather than a separate example with unrelated measurements.

c=(3 cm)2+(4 cm)2=5 cm,K=(3 cm)(4 cm)2=6 cm2,P=12 cmc=\sqrt{(3\,\mathrm{cm})^2+(4\,\mathrm{cm})^2}=5\,\mathrm{cm},\qquad K=\frac{(3\,\mathrm{cm})(4\,\mathrm{cm})}{2}=6\,\mathrm{cm}^{2},\qquad P=12\,\mathrm{cm}

Subtract when the hypotenuse is known

With hypotenuse 1313 and known leg 55 in a common unit, the missing leg is 1212. A square root of a difference is not the difference of the square roots. An equal hypotenuse and leg would yield a collapsed figure, which the positive-side solver rejects.

a=c2−b2=132−52=144=12,c2−b2≠c−ba=\sqrt{c^2-b^2}=\sqrt{13^2-5^2}=\sqrt{144}=12,\qquad\sqrt{c^2-b^2}\ne c-b

Preserve the radical in the boundary length

Two unit legs give hypotenuse 2\sqrt{2} and perimeter 2+22+\sqrt{2}. A six-place hypotenuse view is approximately 1.4142141.414214, but it never replaces the radical. Each output is rounded from its exact quantity. The associated triangle covers half a square of the source length unit.

c=2,P=2+2≈3.414214,K=12c=\sqrt2,\qquad P=2+\sqrt2\approx3.414214,\qquad K=\frac12

Test the longest side and distinguish non-right cases

After sorting a≤b≤ca\le b\le c, first require a+b>ca+b>c. Sides 3,4,53,4,5 have zero residual. Sides 5,5,65,5,6 are acute, while sides 2,3,42,3,4 are obtuse. The obtuse example has Heron area 315/43\sqrt{15}/4, rather than the product of its two shorter sides divided by two.

R=a2+b2−c2,R=0⇒right,R>0⇒acute,R<0⇒obtuseR=a^2+b^2-c^2,\quad R=0\Rightarrow\text{right},\quad R>0\Rightarrow\text{acute},\quad R<0\Rightarrow\text{obtuse}

Convert each dimension with its own factor

The default hypotenuse becomes 0.05 m0.05\,\mathrm{m} and its area becomes 0.0006 m20.0006\,\mathrm{m}^2. Length uses a factor of 1/1001/100, while area uses 1/100001/10000. Changing output units preserves the figure; changing the source selector reinterprets the original numeric strings.

5 cm=0.05 m,12 cm=0.12 m,6 cm2=0.0006 m25\,\mathrm{cm}=0.05\,\mathrm{m},\quad12\,\mathrm{cm}=0.12\,\mathrm{m},\quad6\,\mathrm{cm}^{2}=0.0006\,\mathrm{m}^{2}
Why Users Love This Tool

Why exact sources and visible assumptions help

Classification comes before rounding

  • Strict positivity and triangle inequalities reject collapsed or impossible checked figures.
  • Exact rational squares preserve tiny differences between a leg and hypotenuse.
  • A nearly zero residual remains nonzero; no hidden tolerance certifies a right angle.
  • The corner marker appears only on a verified right triangle at an honest common scale.

The complete mathematical record stays available

  • Fractions and verified radicals remain beside independent rounded decimal views.
  • Perimeter uses an exact additive expression, never a sum of rounded displayed sides.
  • Seventeen physical length definitions and nineteen area definitions convert independently.
  • Sharing, plain copying, CSV and browser PDF retain the active problem and working.
Perfect For

Right-triangle reasoning for lessons and measured layouts

Students choosing addition or subtraction

Match the unknown side to its role before choosing an operation. Adding leg squares finds the longest side; subtracting a leg square from the hypotenuse square finds the other leg. Ordered working connects the rearranged identity to actual substitutions. Fractional and radical examples show why keeping an exact answer differs from simply choosing more decimal places.

Teachers explaining the converse

Compare a solved perpendicular-leg problem with a three-side test. The first assumes a right angle; the second establishes whether typed lengths imply one. Acute, right and obtuse examples demonstrate the squared-length residual's sign. The vector illustration shares one physical scale in both directions, and printed working keeps the source interpretation beside the equations.

People estimating flat diagonals and clearance

Use compatible measurements that genuinely meet at a right angle, such as a rectangle's width and height or a vertical rise and horizontal run. A sloping edge is not automatically a perpendicular leg. The result is an ideal plane-geometry estimate; measurement uncertainty, safety factors, material allowances and installation constraints need additional evidence beyond the typed dimensions.

How It Works

How it works in three quick steps.

1

Choose the missing side or check three lengths

Choose hypotenuse for two known perpendicular legs, or the matching missing-leg mode for a known hypotenuse and leg. Check mode accepts three lengths in any order. Enter decimals or integer fractions and select a common source length unit. Read the interpreted measurements before using the answer.

2

Inspect the exact squared-length working

Adding leg squares finds a hypotenuse; subtracting a leg square from the hypotenuse square finds the other leg. Check mode sorts lengths, validates the strict triangle inequality and compares squares exactly. Follow the positive root and the classification before interpreting area or a diagram.

3

Convert and preserve the full result

Choose length and area outputs independently. Compare exact expressions with rounded views, inspect the equal-scale diagram where available, and retain the original source strings. Copy working, share the active problem, download every conversion as CSV or use browser printing to save a mathematical PDF.

Download & Print

Keep the original problem with the answer

Download complete exact working

CSV includes active input strings, the method, units and exact classification. Result rows retain exact expressions and separately rounded views. Rational certificates identify direct values, while additive certificates describe a perimeter containing a radical. Hidden conversions remain included. Formula-safe cells prevent source text from becoming a spreadsheet instruction when the record is opened for comparison.

Print a solution or save PDF

Browser printing removes editor controls and supporting promotional sections. The available equal-scale diagram, equations, result details and visible conversion tables remain printable. Use the browser's Save as PDF destination and inspect paper size before sharing. Exceptionally long expressions use an explicit notice directing readers to complete CSV values instead of clipping a large number.

Share one reproducible active problem

A shared fragment records the chosen method, required source strings and current unit and display choices. Other editor buffers are excluded, so a recipient loads one defined problem. Strict importing rejects hidden measurements, extra fields and duplicate fragment keys. Reset restores familiar defaults and clears the fragment. Plain copying preserves readable mathematics for destinations without formula typesetting.

FAQ

Frequently Asked Questions

What is the Pythagorean theorem formula?
For a right triangle, a2+b2=c2a^2+b^2=c^2, where aa and bb are perpendicular legs and cc is the hypotenuse opposite the right angle. It relates the areas of the squares on those sides. A missing hypotenuse is c=a2+b2c=\sqrt{a^2+b^2}. First confirm the right-angle assumption; applying this identity to an arbitrary triangle does not generally recover its third side.
How do I find a missing leg?
Select the missing-leg mode that matches the unknown label. With hypotenuse cc and known leg bb, use a=c2−b2a=\sqrt{c^2-b^2}. With hypotenuse cc and known leg aa, use b=c2−a2b=\sqrt{c^2-a^2}. The hypotenuse must be strictly longer than the supplied leg. The calculator subtracts exact squares before taking the positive root, so close source values are not prematurely rounded into equality.
Can I check whether three sides form a right triangle?
Yes. Check mode accepts three positive lengths in any order, sorts them, and labels the longest output side cc. It first requires a+b>ca+b>c; equality is a collapsed figure and a smaller sum is impossible. It then evaluates a2+b2−c2a^2+b^2-c^2 exactly. Zero proves a right triangle, a positive residual identifies an acute triangle, and a negative residual identifies an obtuse triangle. No decimal tolerance changes this classification.
Why does an almost right triangle fail the exact check?
Each typed decimal is treated as its exact stated rational value. Sides 11, 11 and 1.4142141.414214 do not satisfy a2+b2=c2a^2+b^2=c^2 exactly, even though the last number approximates 2\sqrt{2}. Choose hypotenuse mode with two unit legs to retain the radical. Real measurements have uncertainty, so an exact numerical check cannot by itself certify whether a physical corner is square within a particular construction tolerance.
Can I enter fractions and scientific notation?
Fields accept finite decimals, scientific e notation and signed integer fractions. Entering 1/31/3 preserves an exact third rather than a repeating decimal approximation. Each source supports sixty significant decimal digits, or sixty digits per fraction integer, up to one hundred eighty characters. Scientific exponents run from negative one hundred through positive one hundred. A zero denominator, nonpositive side, empty active field or unsupported expression receives validation instead of becoming zero.
Are radicals and rounded values both exact?
The radical retains the exact result for the entered source numbers. Its decimal view is rounded independently to the chosen number of places and is marked as approximate when necessary. Perimeter is rounded from its full rational-plus-radical expression rather than already rounded sides. Square-factor extraction verifies the recovered expression but performs bounded work; a remaining large radicand is valid even when it has not been completely factored.
Can I use this tool for a rectangle diagonal?
Yes, provided the width and height meet at a right angle in a flat rectangle. Choose hypotenuse mode, enter width and height as the two legs, and read the hypotenuse as the diagonal. The accompanying area is the right triangle's area K=ab/2K=ab/2, which is half the rectangle's area rather than its full coverage. Material thickness, cutting allowances and curved surfaces require separate modelling.
Why do area and squared-side rows use different units?
Area has its own output selector, while squared-side rows use the square of the selected output length unit. A side output in centimeters therefore produces squared-side rows in square centimeters even if coverage is shown in square meters. The dimension matters: 1 m=100 cm1\,\mathrm{m}=100\,\mathrm{cm} implies 1 m2=10000 cm21\,\mathrm{m}^2=10000\,\mathrm{cm}^2. Convert mixed source lengths to one common unit first. Changing the source selector reinterprets retained numbers rather than converting them.
Why is there sometimes no triangle diagram?
A right-angle illustration appears only when the solved or checked triangle is right and both legs are visible at the available common scale. An extremely short leg would produce a misleading nearly flat drawing, so that outline is omitted with a notice while exact results remain available. Acute and obtuse checked triangles receive their correct classification and Heron area without a false right-angle marker or perpendicular-leg formula.
What is included when I download, print or share?
CSV retains active source strings, units, classification, all result rows, ordered working and every physical conversion, including tables hidden on screen. It also preserves exact rational or additive radical certificates. Sharing stores only the selected method and active measurements; malformed, duplicate and extra imported fields are rejected. Printing removes editing controls and preserves readable mathematics and the available diagram. Very long expressions point to the full CSV instead of overflowing the printed page.
About This Tool

About this exact Pythagorean theorem calculator

Toolarithm's Pythagorean Theorem Calculator separates two mathematical questions that often look similar on a worksheet. Finding a missing side begins with the assumption that the legs meet perpendicularly. Checking three lengths instead asks whether a valid triangle with those lengths would have a right angle. Keeping these workflows distinct makes the source interpretation as important as the final number. The tool validates the required measurements, identifies the operation and retains a traceable exact expression for each result.

Finite decimal and integer-fraction sources are interpreted as rational numbers. Their squares and differences are calculated before decimal rounding, especially useful when a known leg lies close to the hypotenuse. The positive square root represents the missing physical length. Radical expressions are verified against exact squared quantities, while rounded views are calculated independently. A perimeter containing an irrational side is evaluated as an exact rational-plus-radical expression, so a displayed approximation never becomes a hidden input to a later quantity.

The checker sorts lengths explicitly, rejects collapsed or impossible outlines, and classifies valid triangles from the exact squared-length residual. Non-right triangles use Heron area rather than an unjustified perpendicular-leg product. The equal-scale diagram is reserved for right triangles with visible leg proportions; a notice explains omitted outlines. Independent length and area outputs, preserved source records and complete CSV working support study and reproducible comparisons. The calculator models ideal plane geometry and exact typed numbers; it does not infer measurement uncertainty or certify a physical construction tolerance.

Editorial Transparency

Who maintains this page

Ownership and review

Written and maintained by the Toolarithm editorial team. No review date is shown without a maintained editorial record. No independent professional review is claimed.

Methodology

Equations are checked by substituting worked examples into the stated relationship and comparing the result with the interactive implementation.

Read the editorial methodology

Dates and sources

Review dates change only after a substantive method or content check.

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