Triangle Calculator

A triangle calculator solves a whole triangle from three measurements. From three sides it uses Heron's formula for the area, area = √(s(s−a)(s−b)(s−c)), where s is half the perimeter: a classic 3-4-5 triangle gives an area of 6 square units, a perimeter of 12, and angles of 36.87°, 53.13° and 90°, the example the calculator opens with. It also solves from two sides and an angle (SAS or SSA) or two angles and a side (ASA or AAS), and returns every side, angle and height instantly as you type.

How to find the area of a triangle

There are three common situations, and this tool handles all of them. When you know the three side lengths, the calculator uses Heron's formula:

Here s is the semi-perimeter (half the perimeter), and a, b and c are the three sides. When you instead know a base and the perpendicular height, the area is simpler:

And when you know two sides and the angle between them, the area is ½ × b × c × sin A, the formula high-school geometry derives by dropping a perpendicular from one vertex. All three formulas give the same answer for the same triangle; they just start from different information. The key requirement for the base-and-height formula is that the height must be measured perpendicular to the base, not along a slanted side.

How the angles are calculated

When all three sides are known, each angle is found with the law of cosines. The angle opposite side a is:

The same pattern gives angles B and C. When one or two angles are given, the law of sines fills in the sides:

The calculator reports each angle in degrees (or radians if you switch), so you read familiar values like 60° or 90°. As a built-in check, the three angles always sum to exactly 180° for any flat (Euclidean) triangle, and the largest angle always sits opposite the longest side.

How to use this triangle calculator

You only need three numbers (two for area from base and height). Follow these steps:

1. Pick a mode: 3 sides (SSS), 2 sides + angle between (SAS), 2 angles + side between (ASA), 2 angles + another side (AAS), 2 sides + angle not between (SSA), or Base & height for area only.
2. Choose degrees or radians for angles. Switching converts the angles you already typed, so the triangle stays the same.
3. Enter your values. Side a is opposite angle A, side b opposite B and side c opposite C. Any length unit works as long as you keep it consistent.
4. Read the results: the large number is the area in square units; below it you get the perimeter, the three angles, the triangle type, the altitude to each side, and the inscribed and circumscribed circle radii.
5. Watch for messages: if the values cannot form a triangle, the calculator says why; in the ambiguous SSA case it shows both possible triangles.

Try the preset buttons (Right, Equilateral, Isosceles, Obtuse) in SSS mode to see how different shapes change the angles and area at a glance.

Who this calculator is for

• Students checking geometry or trigonometry homework on area, perimeter, the law of sines and the law of cosines.
• Teachers generating quick worked examples with consistent, correct numbers.
• DIY and trades measuring a triangular patch of floor, roof, garden or fabric to estimate material.
• Designers and makers who need the angles of a triangular piece before cutting.
• Anyone who remembers ½ × base × height but wants area and angles straight from measured sides or angles.

Key terms explained

• Perimeter: the total distance around the triangle, a + b + c.
• Semi-perimeter (s): half the perimeter, used inside Heron's formula.
• Altitude (height): the perpendicular distance from a side to the opposite vertex. Each side has its own altitude.
• Law of cosines: a generalization of the Pythagorean theorem that works for any triangle, used here to find a side from two sides and the included angle and to recover angles from sides.
• Law of sines: the ratio of each side to the sine of its opposite angle is the same for all three sides; used when angles are known.
• SSS, SAS, ASA, AAS, SSA: shorthand for which measurements you know, read in order around the triangle (S = side, A = angle).
• Triangle inequality: the rule that any two sides must together be longer than the third.
• Scalene / isosceles / equilateral: classifications by how many sides are equal (none, two, or all three).
• Acute / right / obtuse: classifications by the largest angle (under 90°, exactly 90°, or over 90°).
• Inradius and circumradius: the radius of the circle that fits inside the triangle touching all three sides (area ÷ s), and of the circle through all three corners (abc ÷ 4 × area).

Worked example 1: the 3-4-5 right triangle

With sides a = 3, b = 4, c = 5, the semi-perimeter is s = (3 + 4 + 5) / 2 = 6. Heron's formula gives area = √(6 × 3 × 2 × 1) = √36 = 6 square units. The law of cosines puts the angle opposite the longest side (5) at exactly 90°, with the other two angles at roughly 36.87° and 53.13°. Because 3² + 4² = 9 + 16 = 25 = 5², it is the textbook right triangle. Its inscribed circle has radius 1 and its circumscribed circle radius 2.5 (half the hypotenuse, as for every right triangle).

Worked example 2: an equilateral triangle

For sides a = b = c = 6, the semi-perimeter is 9, and the area is √(9 × 3 × 3 × 3) = √243 ≈ 15.59 square units. Every angle is exactly 60°, and the triangle is classified as equilateral. This matches the shortcut for equilateral area, (√3 / 4) × side², which equals (1.732 / 4) × 36 ≈ 15.59.

Worked example 3: area from base and height

Suppose a triangular garden bed has a base of 12 feet and a perpendicular height of 5 feet. The area is ½ × 12 × 5 = 30 square feet. Notice you do not need the slanted sides for the area, only the base and the perpendicular height. To also find the angles, you would measure the three sides (or two sides and an angle) and use one of the other modes.

Worked example 4: two sides and the included angle (SAS)

Two sides of 8 and 5 meet at a 60° angle. The law of cosines gives the third side: a² = 8² + 5² − 2 × 8 × 5 × cos 60° = 64 + 25 − 40 = 49, so a = 7. The area is ½ × 8 × 5 × sin 60° ≈ 17.3205 square units, and the remaining angles are about 81.79° (opposite the side of 8) and 38.21° (opposite the side of 5). Choose "2 sides + angle between" and enter b = 8, A = 60, c = 5 to reproduce it.

Worked example 5: the ambiguous case (SSA)

Now suppose you know side a = 8, side b = 10 and angle A = 40°, the angle opposite a. The height from the corner between b and c is b × sin 40° ≈ 6.43. Side a (8) is longer than that height but shorter than b (10), so it can swing to meet the third side in two places. The law of sines gives sin B = 10 × sin 40° ÷ 8 ≈ 0.8035, so B is either 53.46° or its supplement 126.54°:

• Triangle 1: B ≈ 53.46°, C ≈ 86.54°, c ≈ 12.423, area ≈ 39.9269.
• Triangle 2: B ≈ 126.54°, C ≈ 13.46°, c ≈ 2.8978, area ≈ 9.3135.

Both are genuine triangles. That is why SSA, unlike SSS, SAS, ASA and AAS, is not a congruence rule: the same three measurements can describe two different shapes. The calculator shows both whenever this happens, and tells you when side a is too short to make any triangle (for example a = 4, b = 10, A = 30°).

Triangle area chart for common base and height

Because area = ½ × base × height, you can read the area for everyday base-and-height pairs straight off this chart. Values are in square units (square feet if you measure in feet), computed with the exact formula.

Quick reference table

Common triangles and their results, computed with the formulas above (lengths in the same unit, area in square units, rounded to two decimals):

Equilateral triangle area and height by side length

For an equilateral triangle (all three sides equal), the perimeter is 3 × side, the height is (√3 ÷ 2) × side, and the area is (√3 ÷ 4) × side². Each angle is always 60°. Values below are computed for common side lengths.

How to calculate triangle area step by step (Heron's formula)

Here is every step for a 5-12-13 triangle, the numbers you would key into SSS mode. Each row is one computation, so you can follow it by hand.

The 90° angle opposite the longest side (13) confirms 5-12-13 is a right triangle, matching 5² + 12² = 25 + 144 = 169 = 13².

Similar and congruent triangles

Two triangles are congruent when they have exactly the same size and shape: all three sides and all three angles match. Knowing SSS, SAS, ASA or AAS is enough to pin a triangle down completely, which is why those are the congruence rules and why each has its own mode here; SSA is not, because of the two-triangle case above. Triangles are similar when they have the same shape but a different size: the angles are identical and the sides are in the same proportion. Similarity is what lets you measure something you cannot reach. The classic example is finding the height of a tree by comparing the length of its shadow to the shadow of a stick of known height: the two right triangles formed by the sun's rays are similar, so the ratios are equal. Because similar triangles share their angles, the angles this calculator reports stay the same when you double or halve every side; only the area changes, and it changes by the square of the scale factor. Double the sides of a triangle and its area becomes four times larger, not twice as large.

Where triangle math is used

Triangles are everywhere once you look. In construction and roofing, the pitch of a roof is a right triangle, and the rafter length is its hypotenuse. In navigation and surveying, triangulation fixes an unknown position by measuring the angles to it from the two ends of a known baseline: the ASA case, solved with the law of sines. In art and design, triangular composition guides the eye. In engineering, the triangle is the rigid polygon, which is why bridges, cranes and roof trusses are built from triangular frames that cannot deform without changing a side length. Even a simple home project (laying out a square deck, cutting a triangular shelf, or estimating turf for a corner lot) comes down to the area, perimeter and angle calculations on this page.

Tips for accurate results

• Keep one unit: mixing feet and inches gives a wrong area. Convert first, then enter.
• Measure the perpendicular height: in base-and-height mode, do not use a slanted side length as the height.
• Use more decimals for small triangles: rounding sides too early can noticeably shift the area.
• Sanity-check with the triangle inequality: if two sides barely exceed the third, the triangle is very "thin" and its area will be small.
• Check your angle mode: an angle of 60 entered while the calculator is set to radians is a very different triangle.

Common pitfalls

Most mistakes come from the wrong height, impossible sides or mislabeled measurements. Using a slanted edge instead of the perpendicular height overstates the area, and entering three lengths that violate the triangle inequality describes no real triangle at all. Mixing up which angle is opposite which side changes the answer in the SAS, ASA, AAS and SSA modes. The calculator guards against impossible inputs directly; the measuring and labeling are up to you. Another subtle point: a base and height alone fix only the area, never the angles, because the apex can slide along a parallel line without changing either the base or the height.

How a triangle relates to other shapes

A triangle is the simplest polygon, and many other shapes break down into triangles. A rectangle is two right triangles joined along the diagonal, which is why length × width is just two copies of ½ × base × height. Any convex polygon can be split into triangles from a single vertex, so its area is the sum of those triangle areas, the principle behind how mapping software measures irregular plots of land. A triangle also defines a plane in three dimensions: three points that are not in a straight line always lie on exactly one flat surface, which is why 3D models are built from many tiny triangles. Understanding the area, perimeter and angles of one triangle is therefore the foundation for measuring far more complex figures.

Right triangles and the Pythagorean theorem

The most useful triangle in practice is the right triangle, the one with a 90° angle. Its two short sides (the legs) and the long side opposite the right angle (the hypotenuse) obey the Pythagorean theorem, a² + b² = c². This is simply the law of cosines with the cosine of 90° equal to zero, so the cross-term vanishes. Builders use it constantly: measuring 3 feet along one wall and 4 feet along another, then checking that the diagonal is exactly 5 feet, proves the corner is square. The same 3-4-5 ratio scales up to 6-8-10, 9-12-15 and beyond, which is why those numbers appear so often. For right triangles specifically, the Pythagorean Theorem Calculator solves a missing side in one step, and the Square Root Calculator and Exponent Calculator help when you work the squares and roots by hand.

Related concepts and calculators

Triangles connect to a lot of everyday math. If you are computing the slanted side of a right triangle, the Pythagorean theorem (a² + b² = c²) is the special case of the law of cosines when one angle is 90°. For general arithmetic and trigonometric functions, the Scientific Calculator covers sine, cosine and tangent directly. When you need to scale a triangle up or down while keeping its shape, the Ratio Calculator helps with proportional resizing, and the Fraction Calculator keeps exact values when your side lengths are fractions rather than decimals.

For single-rule practice, the Law of Sines Calculator and Law of Cosines Calculator show each formula on its own, and the Trigonometry Calculator evaluates sine, cosine and tangent of any angle.

Sources

• Common Core State Standards: HSG-SRT.D.9 to D.11 (the area formula ½ab sin C and the laws of sines and cosines for solving right and non-right triangles) and 6.G.A.1 (area of triangles).
• Euclid, Elements, Book I, Proposition 20 (the triangle inequality: in any triangle the sum of any two sides is greater than the remaining one), D. E. Joyce edition, Clark University.