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Free Online Triangle Calculator With Angles and Area

Solve any triangle from three sides, from two sides and the angle between them, or from two angles and the side between them. Every remaining side and angle is given, along with the area, the perimeter, the heights and both circle radii. Sets of lengths that cannot form a triangle are named rather than returning not a number.

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Free Online Triangle Calculator With Angles and Area

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🔺 Triangle Calculator Three ways in · every side, angle and area · impossible sets explained

The sum runs in your browser. Nothing you type is sent anywhere.

What you know

🔺 The measurements

Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. Angles are in degrees.

Side a

Side b

Side c

Try one of these

The last two cannot be triangles, and the sixth is where the usual area formula loses its digits.

The area

—

Sides—

Angles—

What kind—

Perimeter—

Heights—

Inscribed circle radius—

Circumscribed circle radius—

Each result with your own numbers in it, and the check that the three angles add to 180 degrees.

Step	With your numbers

Recent triangles, kept in this browser only.

Nothing here yet Press Log to history on the Calculate tab to keep a triangle here.

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How to Solve a Triangle

Three steps, and the third is where the detail is.

Choose which three facts you have Three sides, two sides with the angle between them, or two angles with the side between them. Each of those three fixes the triangle completely, and only the boxes that option needs are shown.

Read every remaining measurement All three sides, all three angles, the area, the perimeter, the three heights and the radii of the inscribed and circumscribed circles, all at once.

Open the working The law of cosines or the law of sines with your own numbers in it, and the check that the three angles add to exactly 180 degrees.

What to Know About Triangles

Which three facts are enough, and which are not.

The area formula matters for a thin triangle. Heron’s formula computes the area from the three sides, and on a long thin triangle it collapses: the half perimeter and one of the sides are nearly equal, and subtracting them destroys the digits that carry the answer. Kahan’s rearrangement computes exactly the same quantity in an order that does not cancel. It only works if the sides are sorted largest first, which is what is done here. A triangle with sides 100, 100 and a thousandth is where the difference shows.

Three lengths do not always make a triangle. The two shorter sides have to add up to more than the longest, or they cannot reach. Sides of 1, 2 and 5 fail badly. Sides of 1, 2 and 3 fail by exactly nothing: the shape collapses into a straight line with no height and no area. Both cases are named here, with how far short they fall, rather than returning not a number.

Two sides and a non included angle is deliberately absent. Knowing two sides and an angle that is not between them can describe two different triangles, one triangle or none at all, depending on the numbers. It is called the ambiguous case for good reason. Showing one answer would be a lie and showing both would confuse the page, so the three options here are the ones that always have a single answer.

The angles are checked against 180 degrees. Every triangle has angles that add to exactly 180. They are computed independently from the sides, so adding them is a real check on the arithmetic rather than a repeat of it, and the total is shown in the working.

Both circle radii come with the triangle. The inscribed circle touches all three sides from inside and its radius is the area divided by the half perimeter. The circumscribed circle passes through all three corners and its radius is the three sides multiplied together over four times the area. Both are asked for often enough to be worth giving unprompted.

It runs in your browser with nothing uploaded. The arithmetic is in the page, so it works offline once loaded and nothing you type is sent anywhere.

Key Features & Capabilities

What it does, and what it deliberately does not.

Three ways in Three sides, two sides and the angle between, or two angles and the side between.

An area that survives thin triangles Kahan’s rearrangement of Heron, with the sides sorted, which keeps its digits where Heron loses them.

Everything at once Sides, angles, area, perimeter, the three heights and both circle radii.

Impossible sets explained Lengths that cannot close, and lengths that close exactly flat, each named with the shortfall.

The angle sum checked Computed from the sides and then added, which catches an error the working alone would hide.

Nothing is uploaded It runs in this tab and keeps working offline.

About Solving Triangles

Solving a triangle means finding every side and angle from the few you were given. Three facts are enough provided at least one of them is a side, because angles alone fix the shape but not the size. The three combinations that always give a single answer are three sides, two sides with the angle between them, and two angles with the side between them, and those are the three options here.

Three sides is handled by the law of cosines, which gives each angle from the three lengths. Two sides and the included angle uses the same law in the other direction to find the third side, and then proceeds as if three sides had been given. Two angles and the included side uses the law of sines, after finding the third angle by subtraction from 180 degrees.

The case that is missing is two sides and an angle that is not between them. That combination is genuinely ambiguous: for some numbers there are two different triangles that fit, for others exactly one, and for others none. A calculator that prints a single answer there is hiding a real mathematical fact from you, so rather than do that this page offers the three combinations that are never ambiguous.

The accuracy question in triangles is the area. Heron’s formula, built from the half perimeter, is the one everybody learns, and it is unstable for thin triangles: when one side is nearly as long as the other two combined, the half perimeter and that side are nearly equal, and the subtraction between them loses most of its significant digits. Kahan published a rearrangement that computes the same value without that cancellation, on condition that the sides are sorted from longest to shortest first. That sorting step is easy to omit and it is what makes the method work, so it is done here.

Frequently Asked Questions

Sides, angles, impossible sets and what comes out.

Three, and at least one of them must be a side. Three angles fix the shape but not the size, so there are infinitely many triangles with the same three angles.

Because that case can describe two different triangles, one, or none, depending on the numbers. It is known as the ambiguous case. Showing a single answer would be misleading, so the three options offered are the ones with a unique solution.

That the two shorter ones do not add up to more than the longest, so they cannot meet. The page says how far short they fall. If they add up to exactly the longest, the triangle is flat: a straight line with no area.

Side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C. It is the standard labelling and it is what the law of sines and the law of cosines assume.

Each is the perpendicular distance from one corner to the opposite side. They are what you need to draw the triangle accurately or to work with it in construction, and each one times its side gives twice the area.

The inscribed circle sits inside and touches all three sides. The circumscribed circle passes through all three corners. Both are determined by the triangle, and both are given.

No. The arithmetic runs in your browser, so nothing you type leaves the page and it works offline.