Triangle Calculator

Last updated: April 2026

Triangle solver

Choose the information you know. The calculator applies the law of sines, law of cosines, or Heron's formula as needed.

Side a

units

Side b

units

Side c

units

Solved using SSS: three known sides.

Side a3 units

Side b4 units

Side c5 units

Angle A36.87°

Angle B53.13°

Angle C90°

Perimeter12 units

Area6 units²

Typescalene, right

Step-by-step working

cos(C) = (a^2 + b^2 - c^2) / 2ab = (3^2 + 4^2 - 5^2) / (2 x 3 x 4) = 0
C = cos^-1(0) = 90 degrees
sin(A) / a = sin(C) / c, so A = sin^-1((3 x sin(90)) / 5) = 36.87 degrees
B = 180 - 36.87 - 90 = 53.13 degrees
s = (3 + 4 + 5) / 2 = 6
Area = sqrt(s(s-a)(s-b)(s-c)) = 6 square units

Law of Sines explained

The law of sines says that each side divided by the sine of its opposite angle is the same ratio: a/sin(A) = b/sin(B) = c/sin(C). It is most useful when you know two angles and one side, or in the SSA ambiguous case.

Law of Cosines explained

The law of cosines extends Pythagoras to any triangle: c² = a² + b² - 2ab cos(C). It is the standard route for SSS and SAS problems.

The ambiguous SSA case

SSA can produce zero, one, or two valid triangles because the known angle is not between the two known sides. When two solutions exist, this calculator shows both possible triangles separately.

About this calculator

The Triangle Calculator solves common triangle measurements such as area, perimeter, missing sides, missing angles, and height where enough information is provided. It is useful for geometry homework, design checks, construction layouts, and trigonometry practice. Use this expanded guide when you need more than a quick result. It explains the assumptions behind the Triangle Calculator, the records to gather, and the decisions the estimate can support. It is especially useful for students, teachers, designers, and DIY users solving triangle area, perimeter, missing sides, missing angles, or right-triangle checks. The strongest use of the page is scenario comparison: change one input at a time, compare the output, and keep a note of which assumption changed.

Triangle calculation method

The calculator applies the appropriate triangle formula based on the known inputs, such as base and height, three sides, or side-angle combinations. The calculator result depends on the quality of the inputs and on the rule set or formula selected in the calculator above. For practical use, treat the output as a structured estimate: start with the core inputs, review the main outputs, then test the decision points that matter most to your situation. Key decisions include which triangle formula fits the known information, whether the side lengths can form a valid triangle, whether degrees or radians are being used.

area = base x height / 2
a^2 + b^2 = c^2 for right triangles
Heron area = sqrt(s x (s-a) x (s-b) x (s-c))
better estimate = accurate inputs + correct rule set + realistic assumptions
scenario difference = revised result - original result

How to use the triangle calculator

Choose the triangle problem type.
Enter known sides, angles, base, or height.
Use consistent units for side lengths.
Review calculated area, perimeter, missing side, or angle.
Check whether the inputs can form a valid triangle.
Gather the main inputs first: side lengths, angles, base.
Check supporting records such as diagram and measurements before relying on a final number.
Enter one realistic scenario first, using conservative assumptions where the future is uncertain.
Review the main outputs: area, perimeter, missing side.
Run at least one alternative scenario so you can see which input changes the answer most.
Compare the result with standard Euclidean geometry formulas or the relevant contract, bill, statement, or professional document.
Keep the calculation date and assumptions with your notes so you can revisit the estimate when rates, rules, or circumstances change.

Worked example

Area from base and height

Input: Base 10cm and height 6cm

Calculation: 10 x 6 / 2 = 30

Result: Triangle area is 30 square centimetres.

Roof pitch scenario

Input: A right triangle is formed by rise and run measurements.

Calculation: Pythagorean theorem and angle relationships are applied.

Result: The calculator estimates the missing side or angle for planning.

Three-side scenario

Input: All three side lengths are known but height is not.

Calculation: Heron formula uses the semi-perimeter to calculate area.

Result: Area can be found without directly measuring height.

Valid triangle checks

Not every set of side lengths can form a triangle. The sum of any two sides must be greater than the third side. Angle inputs should also make geometric sense, with interior angles totalling 180 degrees.

What to check before relying on the result

A useful Triangle Calculator result starts with the same evidence you would use if you were checking the answer manually. The calculator can organise the arithmetic, but it cannot know whether a payslip is final, a bill is estimated, a quote excludes fees, or a personal circumstance has changed since the last statement.

Before making a decision, compare the calculator result with the source document that controls the real outcome. For this topic, that usually means checking standard Euclidean geometry formulas. If there is a difference between the calculator and an official statement, contract, assessment, or professional advice, treat the official document as the stronger source.

diagram: Use this as supporting evidence for the calculation. If it is out of date, estimated, or based on a different period, the calculator output may look precise while still being wrong for the decision.
measurements: Use this as supporting evidence for the calculation. If it is out of date, estimated, or based on a different period, the calculator output may look precise while still being wrong for the decision.
angle units: Use this as supporting evidence for the calculation. If it is out of date, estimated, or based on a different period, the calculator output may look precise while still being wrong for the decision.
working notes: Use this as supporting evidence for the calculation. If it is out of date, estimated, or based on a different period, the calculator output may look precise while still being wrong for the decision.

Inputs that usually change the answer

The most important input is not always the largest number on the form. Sometimes a date, threshold, percentage, eligibility flag, or timing assumption changes the result more than the headline amount. This is why scenario testing is more useful than a single calculation.

Input	Why it matters	What to double-check
side lengths	It feeds directly into the estimate or changes which rule is applied.	Check the period, units, eligibility, and whether the figure is final or estimated.
angles	It feeds directly into the estimate or changes which rule is applied.	Check the period, units, eligibility, and whether the figure is final or estimated.
base	It feeds directly into the estimate or changes which rule is applied.	Check the period, units, eligibility, and whether the figure is final or estimated.
height	It feeds directly into the estimate or changes which rule is applied.	Check the period, units, eligibility, and whether the figure is final or estimated.
unit system	It feeds directly into the estimate or changes which rule is applied.	Check the period, units, eligibility, and whether the figure is final or estimated.

How to interpret the output

The output should be read as a decision aid, not just a number. For Triangle Calculator, the useful question is often what the result means for timing, affordability, eligibility, comparison, or next steps.

area: Use this output alongside the other results rather than in isolation. A monthly amount, percentage, date, or payback figure can look acceptable until fees, timing, evidence, or eligibility conditions are added.
perimeter: Use this output alongside the other results rather than in isolation. A monthly amount, percentage, date, or payback figure can look acceptable until fees, timing, evidence, or eligibility conditions are added.
missing side: Use this output alongside the other results rather than in isolation. A monthly amount, percentage, date, or payback figure can look acceptable until fees, timing, evidence, or eligibility conditions are added.
missing angle: Use this output alongside the other results rather than in isolation. A monthly amount, percentage, date, or payback figure can look acceptable until fees, timing, evidence, or eligibility conditions are added.
validity check: Use this output alongside the other results rather than in isolation. A monthly amount, percentage, date, or payback figure can look acceptable until fees, timing, evidence, or eligibility conditions are added.

Scenarios worth comparing

A single estimate is a snapshot. A better approach is to save a base case, then adjust one assumption at a time. This shows whether the result is stable or whether a small change in timing, rate, usage, income, or cost creates a very different answer.

Scenario	Change one assumption	What the comparison shows
Base case	Use the best current evidence.	Shows the result you would expect if nothing important changes.
Conservative case	Use lower income, higher cost, slower growth, or less favourable timing.	Shows whether the decision still works with less optimistic assumptions.
Improved case	Use the realistic upside, such as lower cost, better rate, higher usage, or stronger evidence.	Shows the potential benefit without treating it as guaranteed.

Common mistakes and edge cases

Most errors come from using the right formula with the wrong assumption. Dates can be counted differently, rates can change, official thresholds can move, and real bills or contracts often include conditions that a simple calculator cannot infer automatically.

The sum of two sides must exceed the third.: Check this point before using the estimate for a payment, claim, purchase, application, employment decision, or health-related decision.
Angles in a flat triangle total 180 degrees.: Check this point before using the estimate for a payment, claim, purchase, application, employment decision, or health-related decision.
Height must be perpendicular to the base.: Check this point before using the estimate for a payment, claim, purchase, application, employment decision, or health-related decision.
Rounding can affect trigonometry results.: Check this point before using the estimate for a payment, claim, purchase, application, employment decision, or health-related decision.

Next steps after calculating

Once you have a result, write down the key assumptions and compare them with standard Euclidean geometry formulas. If the number affects a deadline, tax return, benefit claim, employment issue, medical question, finance agreement, or major purchase, use the calculator as preparation for a more formal check.

For lower-stakes use, the next step may simply be comparing two or three scenarios. For higher-stakes use, the next step should be checking the official guidance, speaking to the relevant organisation, or getting qualified advice before acting.

Important edge cases

The sum of two sides must exceed the third.
Angles in a flat triangle total 180 degrees.
Height must be perpendicular to the base.
Rounding can affect trigonometry results.

Limitations

This calculator is for mathematical and educational use. This is educational geometry information, not engineering advice. The calculator is designed to support understanding and planning, but it cannot verify documents, predict future rule changes, or account for every exception. Use it as an estimate and check the official source before acting where the result matters.

It assumes flat Euclidean geometry.
Rounding can slightly change angle or side results.
Construction or engineering use may require tolerances and professional checks.
Check standard Euclidean geometry formulas for current rules, rates, definitions, and eligibility where relevant.
Do not rely on a single scenario where income, costs, dates, rates, usage, or health circumstances may change.
Keep records of the inputs used so that the estimate can be reviewed later.

Frequently asked questions

What is Heron formula?

Heron formula calculates triangle area from three side lengths using the semi-perimeter.

Do triangle angles always add to 180 degrees?

In flat Euclidean geometry, yes.

What makes a right triangle?

A right triangle has one 90-degree angle and follows the Pythagorean theorem.

What if my sides fail the triangle test?

The measurements cannot form a valid flat triangle and should be checked.

Can this solve any triangle?

Only when enough valid information is provided for the selected method.

Why does angle unit matter?

Trigonometric functions need the correct unit, usually degrees or radians, to return the expected result.

Related calculators

Area Calculator
Volume Calculator
Circle Calculator
Distance Calculator

What does this mean?

This calculator is designed to help you understand the likely number before you make a decision or start an application.

Your result should be checked against official UK guidance, especially if your circumstances include dependants, exemptions, prior leave, or a complex immigration history.

Treat the figure as a planning tool rather than legal advice. Where the answer affects an application deadline or major payment, speak to an authorised adviser.

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