Math Calculator

Right Triangle Calculator

Connect the sides to the angles.

Right triangle inputs

Choose two known measurements to solve one right triangle.

Solve using

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

Leg

a

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

Leg

b

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

Units and display choices

The right angle is at the third vertex. Each side is opposite its matching uppercase angle. At least one length is needed to determine scale.

Output angle convention

Converts angle views without reinterpreting the source problem.

Source length unit

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

Output length unit

Converts sides, perimeter, altitude, projections and circumradius.

Output area unit

Area converts independently, including international acres and hectares.

Rounded decimal places

Exact side-based expressions stay unchanged. Numerical trigonometry uses at most ten places; nonzero small values retain scientific views.

Load a worked example

Show right triangle diagramShow all unit conversions

Side pairs retain exact squared magnitudes. Special degree and pi-radian angles retain exact geometry. General trigonometry is approximate, with explicit boundary limits. The source and output angle selectors are independent.

One right triangle. Exact side-based or special-angle geometry; both acute-angle ratio sets are included.

Interpreted active measurements

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

Solved right triangle

c=5\,\mathrm{cm}

Rounded hypotenuse view

c=5\,\mathrm{cm}

Area

K=6\,\mathrm{cm}^{2}

Perimeter

P=12\,\mathrm{cm}

Acute angle

A\approx\;{36.869898}^{\circ}

Complementary acute angle

B\approx\;{53.130102}^{\circ}

Sides, coverage and acute-angle ratios retain exact rational or verified radical expressions. Angle views are numerical unless a recognized exact angular relationship is available.

Exact means exact for the typed numbers, without measurement uncertainty. Sources support sixty significant decimal digits or sixty digits per fraction integer, one hundred eighty characters and exponents from negative one hundred through positive one hundred. Exact intermediate magnitudes are limited to eighteen hundred digits.

Root extraction is verified but bounded, not a promise of complete integer factorization. Small nonzero values retain scientific views. Exact displays support twelve decimal places; numerical trigonometry uses at most ten.

Opposite sides and acute angles

The right angle is at

C

. Side

a

is opposite

A

, side

b

is opposite

B

, and hypotenuse

c

joins the two acute vertices. The angle arcs belong to those actual vertices. Horizontal and vertical directions use equal units.

Calculation step by step

Identify opposite and adjacent sides
$a=3\,\mathrm{cm},\quad b=4\,\mathrm{cm},\quad C=90^{\circ}$
The third angle is right. Each lowercase side is opposite its matching uppercase angle. A hypotenuse never serves as the adjacent leg in a tangent ratio.
Find the hypotenuse from perpendicular legs
$c^2=a^2+b^2=9\,\mathrm{cm}^{2}+16\,\mathrm{cm}^{2}=25\,\mathrm{cm}^{2}$
Exact squared source lengths are added before the positive root is evaluated. The source unit applies to both legs.
Evaluate both acute-angle ratios
$\sin A=\frac ac,\quad\cos A=\frac bc,\quad\tan A=\frac ab,\qquad\sin B=\frac bc,\quad\cos B=\frac ac,\quad\tan B=\frac ba$
Cosecant, secant and cotangent are the reciprocals of sine, cosine and tangent. Positive nondegenerate legs keep all six acute-angle ratios finite.
Recover coverage and derived distances
$K=\frac{ab}{2},\quad P=a+b+c,\quad h=\frac{ab}{c},\quad R=\frac c2,\quad p_a=\frac{a^2}{c},\quad p_b=\frac{b^2}{c}$
Coverage uses perpendicular legs. The altitude and projections refer to the hypotenuse line, not to an unrelated base-height example. Projections sum to the hypotenuse.
Convert dimensions independently
$L_{\mathrm{out}}=L_{\mathrm{source}}\left(1\right),\quad K_{\mathrm{out}}=K_{\mathrm{source}}\left(1\right),\quad A+B+C=180^{\circ}$
Output selectors convert numeric measurements of the same triangle. Changing a source-unit or input-angle convention instead reinterprets the retained input numbers. Length factors are squared before conversion to an independent area unit.

All triangle measurements

The first value retains an exact expression where available. Numerical trigonometry is marked as approximate. The rounded column never becomes a new source.

Right triangle measurements
Quantity	Exact or numerical value	Rounded view	Interpretation
Leg $a$	$3\,\mathrm{cm}$	$3\,\mathrm{cm}$	Exact positive length from the selected geometry.
Leg $b$	$4\,\mathrm{cm}$	$4\,\mathrm{cm}$	Exact positive length from the selected geometry.
Hypotenuse $c$	$5\,\mathrm{cm}$	$5\,\mathrm{cm}$	Exact positive length from the selected geometry.
Area $K$	$6\,\mathrm{cm}^{2}$	$6\,\mathrm{cm}^{2}$	Half the product of the perpendicular legs of this same triangle.
Perimeter $P$	$12\,\mathrm{cm}$	$12\,\mathrm{cm}$	Sum of the full side expressions; displayed rounded sides are not reused.
Semiperimeter $s$	$6\,\mathrm{cm}$	$6\,\mathrm{cm}$	Half the perimeter, evaluated before rounding.
Altitude to hypotenuse $h$	$\frac{12}{5}\,\mathrm{cm}$	$2.4\,\mathrm{cm}$	Perpendicular distance from the right vertex to the hypotenuse line.
Circumradius $R$	$\frac{5}{2}\,\mathrm{cm}$	$2.5\,\mathrm{cm}$	Half the hypotenuse; its midpoint is the circumcenter.
Projection $p_a$	$\frac{9}{5}\,\mathrm{cm}$	$1.8\,\mathrm{cm}$	Projection of the first leg onto the hypotenuse.
Projection $p_b$	$\frac{16}{5}\,\mathrm{cm}$	$3.2\,\mathrm{cm}$	Projection of the second leg onto the hypotenuse.
Acute angle $A$	$\approx\;{36.869898}^{\circ}$	$\approx\;{36.869898}^{\circ}$	Numerical inverse tangent using both perpendicular legs.
Acute angle $B$	$\approx\;{53.130102}^{\circ}$	$\approx\;{53.130102}^{\circ}$	Numerical complement of the first acute angle.
Right angle $C$	${90}^{\circ}$	${90}^{\circ}$	The defining right angle; the two acute angles are complementary.

Six trig ratios for each acute angle

The first value retains an exact expression where available. Numerical trigonometry is marked as approximate. The rounded column never becomes a new source.

Acute angle trigonometric ratios
Quantity	Exact or numerical value	Rounded view	Interpretation
Sine of first angle $\sin A$	$\frac{3}{5}$	$0.6$	Dimensionless ratio; changing length units does not change its value.
Cosine of first angle $\cos A$	$\frac{4}{5}$	$0.8$	Dimensionless ratio; changing length units does not change its value.
Tangent of first angle $\tan A$	$\frac{3}{4}$	$0.75$	Dimensionless ratio; changing length units does not change its value.
Cosecant of first angle $\csc A$	$\frac{5}{3}$	$\approx\;1.666667$	Dimensionless ratio; changing length units does not change its value.
Secant of first angle $\sec A$	$\frac{5}{4}$	$1.25$	Dimensionless ratio; changing length units does not change its value.
Cotangent of first angle $\cot A$	$\frac{4}{3}$	$\approx\;1.333333$	Dimensionless ratio; changing length units does not change its value.
Sine of second angle $\sin B$	$\frac{4}{5}$	$0.8$	Dimensionless ratio; changing length units does not change its value.
Cosine of second angle $\cos B$	$\frac{3}{5}$	$0.6$	Dimensionless ratio; changing length units does not change its value.
Tangent of second angle $\tan B$	$\frac{4}{3}$	$\approx\;1.333333$	Dimensionless ratio; changing length units does not change its value.
Cosecant of second angle $\csc B$	$\frac{5}{4}$	$1.25$	Dimensionless ratio; changing length units does not change its value.
Secant of second angle $\sec B$	$\frac{5}{3}$	$\approx\;1.666667$	Dimensionless ratio; changing length units does not change its value.
Cotangent of second angle $\cot B$	$\frac{3}{4}$	$0.75$	Dimensionless ratio; changing length units does not change its value.

Hypotenuse in all length units

Hypotenuse in all length units
Unit	Exact or numerical value	Rounded view
Nanometers	$5\times10^{7}\,\mathrm{nm}$	$50000000\,\mathrm{nm}$
Micrometers	$50000\,\mathrm{\mu m}$	$50000\,\mathrm{\mu m}$
Millimeters	$50\,\mathrm{mm}$	$50\,\mathrm{mm}$
Centimeters	$5\,\mathrm{cm}$	$5\,\mathrm{cm}$
Decimeters	$\frac{1}{2}\,\mathrm{dm}$	$0.5\,\mathrm{dm}$
Meters	$\frac{1}{20}\,\mathrm{m}$	$0.05\,\mathrm{m}$
Decameters	$\frac{1}{200}\,\mathrm{dam}$	$0.005\,\mathrm{dam}$
Hectometers	$\frac{1}{2000}\,\mathrm{hm}$	$0.0005\,\mathrm{hm}$
Kilometers	$\frac{1}{20000}\,\mathrm{km}$	$0.00005\,\mathrm{km}$
Mils (one thousandth inch)	$\frac{250000}{127}\,\mathrm{mil}$	$\approx\;1968.503937\,\mathrm{mil}$
Inches	$\frac{250}{127}\,\mathrm{in}$	$\approx\;1.968504\,\mathrm{in}$
International feet	$\frac{125}{762}\,\mathrm{ft}$	$\approx\;0.164042\,\mathrm{ft}$
International yards	$\frac{125}{2286}\,\mathrm{yd}$	$\approx\;0.054681\,\mathrm{yd}$
International miles	$\frac{25}{804672}\,\mathrm{mi}$	$\approx\;0.000031\,\mathrm{mi}$
International nautical miles	$\frac{1}{37040}\,\mathrm{nmi}$	$\approx\;0.000027\,\mathrm{nmi}$
Typographic points (one seventy-second inch)	$\frac{18000}{127}\,\mathrm{pt}$	$\approx\;141.732283\,\mathrm{pt}$
Typographic picas (one sixth inch)	$\frac{1500}{127}\,\mathrm{pc}$	$\approx\;11.811024\,\mathrm{pc}$

Area in all area units

Area in all area units
Unit	Exact or numerical value	Rounded view
Square nanometers	$6\times10^{14}\,\mathrm{nm}^{2}$	$600000000000000\,\mathrm{nm}^{2}$
Square micrometers	$6\times10^{8}\,\mathrm{\mu m}^{2}$	$600000000\,\mathrm{\mu m}^{2}$
Square millimeters	$600\,\mathrm{mm}^{2}$	$600\,\mathrm{mm}^{2}$
Square centimeters	$6\,\mathrm{cm}^{2}$	$6\,\mathrm{cm}^{2}$
Square decimeters	$\frac{3}{50}\,\mathrm{dm}^{2}$	$0.06\,\mathrm{dm}^{2}$
Square meters	$\frac{3}{5000}\,\mathrm{m}^{2}$	$0.0006\,\mathrm{m}^{2}$
Square decameters	$\frac{3}{500000}\,\mathrm{dam}^{2}$	$0.000006\,\mathrm{dam}^{2}$
Square hectometers	$\frac{3}{5\times10^{7}}\,\mathrm{hm}^{2}$	$6\times10^{-8}\,\mathrm{hm}^{2}$
Square kilometers	$\frac{3}{5\times10^{9}}\,\mathrm{km}^{2}$	$6\times10^{-10}\,\mathrm{km}^{2}$
Square mils (one thousandth inch)	$\frac{15\times10^{9}}{16129}\,\mathrm{mil}^{2}$	$\approx\;930001.860004\,\mathrm{mil}^{2}$
Square inches	$\frac{15000}{16129}\,\mathrm{in}^{2}$	$\approx\;0.930002\,\mathrm{in}^{2}$
Square international feet	$\frac{625}{96774}\,\mathrm{ft}^{2}$	$\approx\;0.006458\,\mathrm{ft}^{2}$
Square international yards	$\frac{625}{870966}\,\mathrm{yd}^{2}$	$\approx\;0.000718\,\mathrm{yd}^{2}$
Square international miles	$\frac{25}{107916171264}\,\mathrm{mi}^{2}$	$\approx\;2.316613\times10^{-10}\,\mathrm{mi}^{2}$
Square international nautical miles	$\frac{3}{17149520000}\,\mathrm{nmi}^{2}$	$\approx\;1.74932\times10^{-10}\,\mathrm{nmi}^{2}$
Square typographic points (one seventy-second inch)	$\frac{77760000}{16129}\,\mathrm{pt}^{2}$	$\approx\;4821.129642\,\mathrm{pt}^{2}$
Square typographic picas (one sixth inch)	$\frac{540000}{16129}\,\mathrm{pc}^{2}$	$\approx\;33.480067\,\mathrm{pc}^{2}$
International acres	$\frac{125}{843095088}\,\mathrm{acre}$	$\approx\;1.482632\times10^{-7}\,\mathrm{acre}$
Hectares	$\frac{3}{5\times10^{7}}\,\mathrm{ha}$	$6\times10^{-8}\,\mathrm{ha}$

Inputs saved with this calculation

Method: Two perpendicular legs
Known side in angle mode: c
Known angle label: A
Input angle convention: degrees
Output angle convention: degrees
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?

Recover a complete right triangle from what you know

Solve perpendicular-side or hypotenuse cases

Choose the method that matches the known lengths instead of entering an unknown into an unrelated field. A pair of perpendicular legs determines the hypotenuse. A hypotenuse with either named leg determines the remaining leg through a difference of squares. Exact source arithmetic preserves small positive differences and makes the derived coverage, perimeter and projections belong to the same solved triangle.

Turn one acute angle into a side relationship

A side and a named acute angle establish both scale and shape. Identify whether the known side is opposite, adjacent or the hypotenuse relative to that angle. The solver handles both acute labels without relabelling the geometry. Independent input and output angle selectors prevent a unit conversion from silently changing the problem, while special-angle cases retain their useful exact radical forms.

Compare all six ratios at both vertices

Read sine, cosine and tangent beside their reciprocal functions, then compare the second acute vertex. The opposite and adjacent legs exchange roles when the reference angle changes. Neither a rotated drawing nor a physical unit conversion changes those dimensionless relationships. Keeping both complete ratio sets in one record helps distinguish complementary-angle identities from arithmetic coincidences in a single example.

Right-triangle formulas with connected examples

Solve a labelled side pair

With perpendicular legs $a=3\,\mathrm{cm}$ and $b=4\,\mathrm{cm}$ , the hypotenuse is $5\,\mathrm{cm}$ . The same triangle has coverage $6\,\mathrm{cm}^2$ and boundary length $12\,\mathrm{cm}$ . The inverse tangent uses the ratio of the first leg to the second for angle $A$ ; reversing those roles gives the complementary angle rather than the same vertex.

c=\sqrt{(3\,\mathrm{cm})^2+(4\,\mathrm{cm})^2}=5\,\mathrm{cm},\qquad A=\arctan\frac34\approx36.869898^{\circ},\qquad B\approx53.130102^{\circ}

Keep the reference vertex attached to each ratio

At the first acute vertex, the opposite leg is $a$ and the adjacent leg is $b$ . At the second vertex, they exchange roles. Their reciprocal functions describe the same triangle and remain finite because both perpendicular legs are positive. Converting every side to another common unit leaves the ratios below unchanged.

\sin A=\frac35,\quad\cos A=\frac45,\quad\tan A=\frac34,\quad\csc A=\frac53,\quad\sec A=\frac54,\quad\cot A=\frac43

\sin B=\cos A,\quad\cos B=\sin A,\quad\tan B=\cot A,\quad A+B=90^{\circ}

Recover scale from one side and an acute angle

For $c=10\,\mathrm{cm}$ and $A=30^{\circ}$ , sine recovers the opposite leg and cosine recovers the adjacent leg. The hypotenuse remains the supplied measurement. Selecting angle $B$ for the same thirty-degree source exchanges the two perpendicular leg lengths. This is why a numerical angle without its vertex label does not define the same labelled problem.

a=(10\,\mathrm{cm})\sin30^{\circ}=5\,\mathrm{cm},\qquad b=(10\,\mathrm{cm})\cos30^{\circ}=5\sqrt3\,\mathrm{cm},\qquad K=\frac{25\sqrt3}{2}\,\mathrm{cm}^{2}

Check derived distances on the hypotenuse

The default triangle's hypotenuse altitude is $12/5\,\mathrm{cm}$ . The leg projections are $9/5\,\mathrm{cm}$ and $16/5\,\mathrm{cm}$ , which sum to the original hypotenuse. Their product equals the altitude square. These distances are constructed from the same triangle, not from an unrelated pair of base-height measurements.

h=\frac{ab}{c}=\frac{12}{5}\,\mathrm{cm},\quad p_a=\frac{a^2}{c}=\frac95\,\mathrm{cm},\quad p_b=\frac{b^2}{c}=\frac{16}{5}\,\mathrm{cm},\quad p_a+p_b=c,\quad h^2=p_ap_b

Why Users Love This Tool

Why side labels and approximation boundaries matter

The right-angle model stays explicit

The right vertex and opposite-side convention remain visible in the source and drawing.
Positive-leg and strict hypotenuse checks reject collapsed or impossible inputs.
Known-side and known-angle selectors preserve their distinct geometric meanings.
Vertex-centered arcs and equal axis scales describe the solved triangle rather than an illustrative guess.

Exact and numerical answers stay distinguishable

Side pairs and recognized special angles retain verified rational and radical expressions.
General trigonometric outputs are explicitly approximate, with argument reduction near angular boundaries.
Perimeter is rounded from the full sum, without feeding rounded side views into another calculation.
Independent angle, length and coverage outputs retain coherent dimensions and complete downloadable working.

Perfect For

A bridge between plane geometry and trigonometry

Students identifying opposite and adjacent

Use the vertex labels before memorizing a ratio. The adjacent side in a tangent ratio is a perpendicular leg, never the hypotenuse. Comparing both acute angles makes the complementary relationships visible. Exact fractions, equal-leg triangles and numerical angle examples show how an answer can be algebraically exact in one case and a controlled approximation in another without changing the underlying geometric definitions.

Teachers preparing connected explanations

Demonstrate why a right triangle needs scale as well as shape, and why two angles alone cannot determine actual side lengths. Follow a side-pair example into its acute angles, then reconstruct the geometry from a side-angle source. Both ratio tables, ordered equations, labelled vector geometry and preserved source conventions support a coherent explanation instead of isolated calculator outputs without a shared model.

People estimating idealized slopes and rises

A horizontal run, vertical rise and straight sloping span can form a useful plane right-triangle model when the two legs are perpendicular. Identify which measured quantity is the hypotenuse before rearranging a ratio. Real layouts may need allowances, uncertainty estimates, material constraints or safety rules beyond this idealized geometry. The calculator supplies mathematical relationships for the entered dimensions rather than certifying a physical installation.

How It Works

How it works in three quick steps.

Choose two independent known measurements

Select two perpendicular legs, a hypotenuse with either named leg, or one side with an acute angle. In the angle workflow identify the known side and the named acute vertex before entering numbers. Choose the source angle convention explicitly and use a shared length unit for all supplied lengths.

Read the solved geometry and opposite-side ratios

Inspect the interpreted source and the right-angle assumption. The solver reports positive side lengths, coverage, boundary distances and complementary acute angles. Compare both acute-angle ratio tables. Exact side-based and recognized special-angle expressions remain separate from numerical trigonometry. The diagram keeps opposite-side labels and angle arcs attached to the actual solved vertices.

Convert the same triangle and save the working

Choose output length, coverage and angle conventions independently. Output changes convert a result, while source selectors reinterpret retained numbers. Inspect exact expressions and their separately rounded views. Save all quantities and hidden conversions as CSV, copy readable working, share the active problem or use browser printing to retain a mathematical PDF.

Download & Print

Preserve the angle convention with every result

Download every quantity and ratio

The CSV record includes dimensions, all six ratios for each acute vertex and full hypotenuse and coverage conversions. Exact squared magnitudes or root-sum certificates accompany side-based values, while numerical rows identify their approximation status. Active source strings and both angle conventions remain part of the record. Hidden conversion tables are included so a later reader can reproduce the calculation without relying on a screen view.

Print the mathematical explanation

Browser printing focuses on the solved geometry and its working, removing input controls and supporting marketing sections. Visible tables retain their headings across pages, and the vector drawing keeps its equal scale and vertex labels. Use Save as PDF and inspect paper orientation before distributing a worksheet. Very long expressions use explicit notices directing readers to complete CSV values instead of overflowing a narrow printed column.

Share a clearly defined problem

A shareable fragment stores one active method, its required measurements and the relevant side, angle and unit choices. Inactive source buffers are excluded. Extra, duplicate and malformed imported fields are rejected rather than becoming hidden measurements. Reset clears the fragment and restores familiar defaults. Plain copying produces readable mathematical text for destinations without visual typesetting while keeping the source assumptions beside the result.

FAQ

Frequently Asked Questions

Which measurements determine a right triangle?

A positive right triangle is determined by two perpendicular legs, a hypotenuse and a shorter leg, or one side and one acute angle. The angle workflow lets you identify any known side and either acute vertex. Two angles alone determine shape but not scale. This calculator therefore requires at least one length instead of inventing a size from angular information alone or silently interpreting an unspecified side.

How are sides and angles labelled?

The right angle is

C=90^\circ

. Lowercase side

a

is opposite angle

A

, side

b

is opposite angle

B

, and hypotenuse

c

is opposite the right angle. In the drawing,

a

is horizontal and

b

is vertical. The same drawing could be rotated without changing any ratios. Always identify the opposite and adjacent legs relative to the chosen acute vertex rather than their apparent position on a screen.

What are the six trig ratios for each acute angle?

For angle

A

, sine is

a/c

, cosine is

b/c

, and tangent is

a/b

. Cosecant, secant and cotangent are the reciprocals of those three values. At angle

B

, the opposite and adjacent legs exchange roles, while the hypotenuse stays fixed. Both tables are dimensionless: converting a triangle from centimeters to meters scales all sides together and leaves every ratio unchanged. All denominators are positive for the supported nondegenerate triangle.

Can I solve a side from an angle and hypotenuse?

Yes. Choose the side-angle workflow, identify the known side as the hypotenuse, select the known acute angle label and enter its convention. For a known first angle,

a=c\sin A

and

b=c\cos A

. For the second angle, their roles exchange. A degree source of thirty and a pi-radian source of one sixth describe the same special angle; an ordinary radian source is interpreted directly as a numerical radian measurement.

Which results retain exact expressions?

Two-side sources use exact finite decimals or integer fractions, so derived lengths, coverage, projections and trig ratios retain rational or verified radical expressions. Recognized degree or pi-radian special angles of thirty, forty-five and sixty degrees also retain exact geometry. General side-angle geometry uses numerical trigonometry and is marked approximate. Known lengths stay exact, and degree or pi-radian source relationships can remain exact even when derived side lengths require a numerical view.

Why are input and output angle conventions separate?

Changing the input convention changes what a retained source number means. A value of one sixth could mean a small number of degrees, ordinary radians, or

\pi/6

radians depending on that selector. Changing only the output convention expresses the same solved triangle differently. Degree and radian outputs report numerical angle measures, while the multiples-of-pi view reports a coefficient multiplying

\pi\,\mathrm{rad}

. The source and output conventions are preserved in sharing and exports.

Why are zero legs or equal hypotenuse and leg rejected?

The tool solves positive nondegenerate right triangles, which have two positive perpendicular legs and two genuinely acute interior angles. A zero leg collapses coverage and makes some reciprocal ratios undefined. A hypotenuse equal to the known leg forces that same collapse; a shorter hypotenuse gives a negative missing-side square. Validation reports these conditions explicitly rather than drawing a positive-looking triangle or returning an invented zero-valued trigonometric ratio.

What numerical boundaries apply near a right angle?

Side-angle sources must remain at least one hundred-millionth of a degree from zero and a right angle. Numerical evaluation reduces to the smaller acute argument to avoid subtracting nearly equal large values. Side-pair modes retain exact dimensions and ratios even for extreme scales, but acute-angle decimal views are omitted when both complementary angles cannot be resolved reliably. The drawing is also omitted when one leg is too short for the available common scale.

What are the hypotenuse altitude and projections?

The altitude

h=ab/c

reaches the hypotenuse perpendicularly from the right-angle vertex. The first leg projects a distance

p_a=a^2/c

along that line, while the second projects

p_b=b^2/c

. These projections sum to the hypotenuse, and

h^2=p_ap_b

provides a useful independent check. The circumradius is half the hypotenuse. Each of these quantities is a length; coverage converts with a squared dimensional factor instead.

What does the downloadable calculation preserve?

CSV includes every main measurement, both complete trig-ratio tables, exact certificates where available, rounded views, active source strings, unit conventions and ordered working. Hypotenuse and area conversion tables remain included even when hidden on screen. Shared fragments exclude inactive buffers and reject malformed or extra fields. Browser printing removes controls while retaining visible mathematical tables and the available diagram. Exceptionally long expressions point to the complete CSV so printed pages remain readable.

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About This Tool

About this right triangle solver and ratio reference

Toolarithm's Right Triangle Calculator connects two kinds of reasoning: recovering a perpendicular-side geometry and interpreting a side ratio relative to a named acute vertex. The right-angle assumption is explicit, and the opposite-side convention stays fixed throughout the page. That consistency matters because a sine or tangent expression cannot be interpreted correctly until the relevant angle and side roles are identified. The calculator supports each pair of known sides and a side with either acute angle, while requiring a genuine positive length to establish scale.

Finite decimal and integer-fraction side sources retain exact rational squared magnitudes. Positive roots recover the missing dimensions, and direct root-sum evaluation supplies perimeter without reusing a rounded display. Thirty-, forty-five- and sixty-degree relationships supplied in degrees or pi-radians retain exact special-angle geometry. Other angle sources require numerical trigonometry, which is labelled distinctly. Known lengths and exact degree or pi-radian relationships remain separately inspectable. The independent output-angle selector converts a solved result without silently reinterpreting the original angle field.

The complete output includes both acute-angle ratio sets, coverage, boundary distances, the altitude to the hypotenuse, its two leg projections and the circumradius. A vertex-based vector figure places acute arcs at the actual solved corners and uses equal physical scale in both directions. An explicit notice explains omitted thin outlines or unresolved extreme angle views. Source records, unit conversions, copying, sharing, CSV and native PDF support repeatable learning and comparison. The model concerns ideal plane geometry and the supplied numbers; practical measurement uncertainty and construction requirements need additional evidence beyond these mathematical relationships.

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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Calculators hub

Right Triangle Calculator

Right triangle inputs

Units and display choices

Interpreted active measurements

Solved right triangle

Opposite sides and acute angles

Calculation step by step

Identify opposite and adjacent sides

Find the hypotenuse from perpendicular legs

Evaluate both acute-angle ratios

Recover coverage and derived distances

Convert dimensions independently

All triangle measurements

Six trig ratios for each acute angle

Hypotenuse in all length units

Area in all area units

Inputs saved with this calculation

Recover a complete right triangle from what you know

Solve perpendicular-side or hypotenuse cases

Turn one acute angle into a side relationship

Compare all six ratios at both vertices

Right-triangle formulas with connected examples

Solve a labelled side pair

Keep the reference vertex attached to each ratio

Recover scale from one side and an acute angle

Check derived distances on the hypotenuse

Why side labels and approximation boundaries matter

The right-angle model stays explicit

Exact and numerical answers stay distinguishable

A bridge between plane geometry and trigonometry

Students identifying opposite and adjacent

Teachers preparing connected explanations

People estimating idealized slopes and rises

How it works in three quick steps.

Choose two independent known measurements

Read the solved geometry and opposite-side ratios

Convert the same triangle and save the working

Preserve the angle convention with every result

Download every quantity and ratio

Print the mathematical explanation

Share a clearly defined problem

Frequently Asked Questions

Related calculators

Pythagorean Theorem Calculator

Triangle Calculator

Degrees to Radians Converter

About this right triangle solver and ratio reference

Who maintains this page

Ownership and review

Methodology

Dates and sources

Explore more calculators