A Triangle Calculator solves missing side lengths, angles, perimeter, and area when enough independent information about a triangle is known.

Depending on the values entered, it may use the Pythagorean theorem, trigonometric ratios, the law of sines, the law of cosines, Heron’s formula, or the triangle angle-sum relationship.

The calculator can also check basic validity, including whether:

A + B + C = 180°

and whether the side lengths satisfy the triangle inequality.

Match every side with its opposite angle, enter enough independent measurements, and keep extra precision until the entire triangle has been solved.

How triangle labels work

A triangle is commonly labeled with angles:

A, B, C

and opposite sides:

a, b, c

That means:

Side a is opposite angle A

Side b is opposite angle B

Side c is opposite angle C

Keeping these pairs aligned is essential when using the law of sines or law of cosines.

For example, if:

A = 40°

then the side labeled:

a

must be the side directly opposite that 40° angle.

How to use the Triangle Calculator

1. Identify the known sides and angles.
2. Match each side with its opposite angle.
3. Select the triangle case when the calculator provides that option.
4. Enter at least enough independent information to determine the triangle.
5. Use degree mode unless the problem explicitly uses radians.
6. Calculate the missing sides and angles.
7. Review the perimeter and area.
8. Check the angle sum and triangle inequality.
9. If the input is SSA, review whether a second valid triangle exists.

Usually, three independent measurements are needed, including at least one side.

The five common triangle cases

Triangle problems are often classified by the measurements given.

Case	Known information	Typical first method
SSS	Three sides	Law of cosines
SAS	Two sides and included angle	Law of cosines
ASA	Two angles and included side	Angle sum, then law of sines
AAS	Two angles and a nonincluded side	Angle sum, then law of sines
SSA	Two sides and a nonincluded angle	Law of sines, with ambiguity check

A right triangle may be solved more directly using trigonometric ratios and the Pythagorean theorem.

Three angles alone are not enough

If you know only:

A, B, and C

you know the triangle’s shape but not its size.

For example, all of these triangles can have angles:

30°, 60°, 90°

while having very different side lengths.

So:

AAA

does not uniquely determine a triangle’s scale.

At least one side length is needed.

The interior angles must total 180 degrees

For any ordinary plane triangle:

A + B + C = 180°

So if two angles are known:

C = 180° − A − B

For example:

A = 52°

B = 68°

Then:

C = 180° − 52° − 68°

= 60°

This is often the first step in ASA or AAS problems.

Angle inputs must describe a valid triangle

Three positive interior angles must add to:

180°

For example:

70° + 60° + 50° = 180°

so the set is valid.

But:

90° + 60° + 40° = 190°

cannot describe an ordinary plane triangle.

Likewise, a single interior angle cannot be:

0°

or:

180°

The triangle inequality checks side validity

For side lengths:

a, b, c

each side must be shorter than the sum of the other two.

So:

a < b + c

b < a + c

c < a + b

For example:

3, 4, 5

forms a valid triangle because:

3 + 4 > 5

3 + 5 > 4

and:

4 + 5 > 3

An invalid side example

Consider:

2, 3, 6

Check:

2 + 3 = 5

But:

5 < 6

So the triangle inequality fails.

These lengths cannot form a nondegenerate triangle.

A calculator should reject them rather than continuing with area or angle calculations.

Equality creates a degenerate triangle

Suppose:

2 + 3 = 5

The points can lie on one straight line, but they do not enclose ordinary triangular area.

So:

2, 3, 5

does not form a proper triangle.

For a valid nondegenerate triangle:

the sum of any two sides must be strictly greater than the third

not merely equal.

Right triangles use the Pythagorean theorem

For a right triangle:

a² + b² = c²

where:

c

is the hypotenuse.

The hypotenuse is the side opposite the:

90° angle

and is always the longest side.

For example:

a = 3

b = 4

Then:

c = √(3² + 4²)

= √(9 + 16)

= √25

= 5

So:

3, 4, 5

is a right triangle.

Finding a missing leg

Suppose:

c = 13

and:

a = 5

Then:

b² = c² − a²

= 13² − 5²

= 169 − 25

= 144

So:

b = 12

The triangle is:

5, 12, 13

Trigonometric ratios solve right-triangle angles

For an acute angle:

θ

in a right triangle:

sin(θ) = Opposite / Hypotenuse

cos(θ) = Adjacent / Hypotenuse

tan(θ) = Opposite / Adjacent

These are often remembered as:

SOH-CAH-TOA

Right-triangle angle example

Suppose the side opposite angle A is:

3

and the adjacent leg is:

4

Then:

tan(A) = 3/4

So:

A = arctan(3/4)

≈ 36.87°

The other acute angle is:

90° − 36.87° ≈ 53.13°

The three angles then sum to:

180°

Degree and radian mode must match the problem

Most geometry problems express triangle angles in:

degrees

If the calculator is accidentally in radian mode, trigonometric results can be completely different.

For example:

sin(30°) = 0.5

but:

sin(30 radians)

is not 0.5.

Use degree mode unless the problem explicitly specifies radians.

The law of sines

For any triangle:

a/sin(A) = b/sin(B) = c/sin(C)

This is especially useful when you know:

ASA

AAS

or sometimes:

SSA

For example, if:

A = 40°

B = 65°

and:

a = 8

first find:

C = 180° − 40° − 65° = 75°

Then:

b / sin(65°) = 8 / sin(40°)

So:

b = 8 × sin(65°) / sin(40°)

The remaining side can be found the same way.

The law of cosines

For a triangle with sides:

a, b, c

and opposite angles:

A, B, C

the law of cosines includes:

c² = a² + b² − 2ab cos(C)

Similarly:

a² = b² + c² − 2bc cos(A)

and:

b² = a² + c² − 2ac cos(B)

This is particularly useful for:

SSS

and:

SAS

cases.

SAS example

Suppose:

a = 7

b = 10

and the included angle:

C = 60°

Then:

c² = 7² + 10² − 2(7)(10)cos(60°)

Since:

cos(60°) = 0.5

then:

c² = 49 + 100 − 70

= 79

So:

c = √79

≈ 8.89

Once all three sides are known, the remaining angles can be found.

SSS example

Suppose:

a = 5

b = 7

c = 8

To find angle C:

c² = a² + b² − 2ab cos(C)

Rearrange:

cos(C) = (a² + b² − c²)/(2ab)

So:

cos(C) = (25 + 49 − 64)/(2 × 5 × 7)

= 10/70

= 1/7

Then:

C = arccos(1/7)

The other angles can be found with the law of sines, law of cosines, or the 180° angle sum.

Choose the law that fits the known information

Using the right method reduces unnecessary intermediate work.

For:

SSS

start with the law of cosines.

For:

SAS

use the law of cosines to find the third side.

For:

ASA or AAS

find the third angle first, then use the law of sines.

For:

SSA

use the law of sines but check for the ambiguous case.

For a:

right triangle

use the Pythagorean theorem and trigonometric ratios when possible.

The SSA case can be ambiguous

Suppose you know:

A

a

and:

b

where side:

a

is opposite angle:

A

The law of sines gives:

sin(B) = b sin(A) / a

But knowing:

sin(B)

does not always determine a unique angle.

For an acute angle:

B₁

another angle:

B₂ = 180° − B₁

has the same sine.

That means SSA data can sometimes produce:

two valid triangles

Why sine creates two possible angles

For example:

sin(30°) = 0.5

but:

sin(150°) = 0.5

So if a calculation gives:

sin(B) = 0.5

then B might be:

30°

or:

150°

You must test both possibilities against:

A + B < 180°

and the remaining side data.

SSA can produce zero, one, or two triangles

Depending on the measurements, an SSA setup may produce:

no valid triangle

one valid triangle

or:

two valid triangles

The calculator should not assume that the inverse-sine result is automatically the only answer.

Both the principal angle and its supplement may need to be checked.

Example of the ambiguous SSA idea

Suppose:

A = 30°

and the law of sines produces:

sin(B) = 0.8

The calculator first finds:

B₁ = arcsin(0.8)

which is approximately:

53.13°

The second possible angle is:

B₂ = 180° − 53.13°

≈ 126.87°

Now check:

A + B₁

and:

A + B₂

If each sum is less than:

180°

then each may define a possible triangle, subject to the remaining data.

Some SSA cases have only one solution

If the second possible angle would make:

A + B ≥ 180°

then that second triangle is impossible.

Likewise, the side lengths may eliminate one solution.

This is why the full validity check matters after inverse sine is used.

Area from base and perpendicular height

The most direct triangle-area formula is:

Area = Base × Perpendicular height ÷ 2

Suppose:

Base = 12 m

Height = 5 m

Then:

Area = 12 × 5 ÷ 2

= 30 m²

The height must be perpendicular to the selected base.

A sloping side is not automatically the height.

Heron’s formula finds area from three sides

When:

a, b, c

are known, first calculate the semiperimeter:

s = (a + b + c)/2

Then:

Area = √(s(s − a)(s − b)(s − c))

This is especially useful for SSS problems.

Heron’s formula example

For the:

3-4-5 triangle

first calculate:

s = (3 + 4 + 5)/2

= 6

Then:

Area = √(6(6 − 3)(6 − 4)(6 − 5))

= √(6 × 3 × 2 × 1)

= √36

= 6

So:

Area = 6 square units

This agrees with:

3 × 4 ÷ 2 = 6

because the legs are perpendicular.

Area from two sides and the included angle

Another useful formula is:

Area = ab sin(C) / 2

where:

C

is the included angle between sides:

a

and:

b

For example:

a = 8

b = 10

C = 30°

Then:

Area = 8 × 10 × sin(30°) ÷ 2

Since:

sin(30°) = 0.5

then:

Area = 80 × 0.5 ÷ 2

= 20

Multiple area methods should agree

If enough information is available, the triangle area can sometimes be calculated in several ways.

For example:

• base-height formula;
• Heron’s formula;
• two sides and included-angle formula.

For the same valid triangle, these methods should agree apart from rounding.

Comparing two methods can be a strong independent check.

Triangle perimeter

The perimeter is simply:

P = a + b + c

For a:

3-4-5

triangle:

P = 3 + 4 + 5

= 12

Perimeter uses linear units.

Area uses square units.

Do not confuse the two.

The longest side faces the largest angle

This provides a useful consistency check.

If:

A > B

then:

a > b

because side a is opposite angle A and side b is opposite angle B.

For example, in a triangle with angles:

90°, 60°, 30°

the side opposite:

90°

is longest.

The side opposite:

30°

is shortest.

If the calculator produces the opposite ordering, recheck the side-angle labels.

Equal angles have equal opposite sides

If:

A = B

then:

a = b

This describes an isosceles triangle.

Likewise, if:

a = b

then:

A = B

This can provide another quick check on calculated results.

Equilateral triangles have three equal sides

An equilateral triangle satisfies:

a = b = c

and:

A = B = C = 60°

If side length is:

s

then its area can be written:

Area = (√3/4)s²

For example, if:

s = 4

then:

Area = 4√3

or approximately:

6.93 square units

Isosceles triangles contain useful symmetry

If two sides are equal, the opposite angles are also equal.

Suppose:

a = b

Then:

A = B

That can reduce the number of separate calculations required.

The altitude from the vertex between the equal sides also divides the base into two equal parts in the standard isosceles setup.

45-45-90 triangles have a fixed ratio

For a right triangle with acute angles:

45° and 45°

the side ratio is:

1 : 1 : √2

So if each leg is:

5

the hypotenuse is:

5√2

This provides a useful exact check against a calculator result.

30-60-90 triangles also have a fixed ratio

The side lengths opposite:

30°

60°

and:

90°

have the ratio:

1 : √3 : 2

So if the short leg is:

4

then:

long leg = 4√3

and:

hypotenuse = 8

These special triangles can provide quick exact answers without full trigonometric calculations.

Right-triangle area has a shortcut

Because the two legs are perpendicular:

Area = ab/2

where:

a

and:

b

are the legs.

For:

a = 6

and:

b = 8

then:

Area = 6 × 8 ÷ 2

= 24

No separate height calculation is needed.

Rounding too early can change later answers

Suppose you calculate an angle as:

36.869897…°

and immediately round to:

36.9°

Then use that rounded value to calculate another side.

The final side may differ from the result obtained with full precision.

For multi-step triangle solving:

keep guard digits

and:

round only the final reported values

whenever possible.

Angle rounding can break the 180-degree check

Imagine three calculated angles displayed as:

33.3°

66.6°

and:

80.0°

Their rounded sum is:

179.9°

That does not necessarily mean the triangle calculation is wrong.

The unrounded values may still total exactly:

180°

Allow for ordinary rounding when checking displayed values.

Exact forms can be useful

A right triangle may produce a side such as:

√13

A decimal approximation is:

3.6055…

If further calculation follows, keeping:

√13

can preserve exactness.

Use decimals when the task requires numerical reporting.

Units must remain consistent

Suppose one side is:

3 ft

and another is:

24 in

Convert before using formulas.

Since:

24 in = 2 ft

the sides should be entered consistently as:

3 ft and 2 ft

or:

36 in and 24 in

Mixing units directly can invalidate area, perimeter, and side calculations.

Area uses square units

If the sides are measured in:

metres

then perimeter is in:

metres

while area is in:

square metres

For example:

Perimeter = 18 m

but:

Area = 12 m²

The numerical units represent different geometric quantities.

A triangle diagram may not be drawn to scale

A sketch showing one side visually longer than another is not proof of the actual dimensions.

Use:

• labeled measurements;
• calculated relationships;
• angle values;
• and geometric constraints.

Do not infer missing quantities from the appearance of a diagram unless it is explicitly stated to be drawn to scale.

Surveying triangles need more than ideal geometry

Triangle formulas are useful in surveying, but field measurements also involve:

• instrument accuracy;
• coordinate systems;
• slope distance;
• horizontal distance;
• measurement uncertainty;
• datum;
• and closure checks.

A general Triangle Calculator solves an ideal plane triangle.

It does not replace survey adjustment or professional field methods.

Navigation may use spherical rather than plane geometry

For small local diagrams, plane triangles may be appropriate.

For long distances across the Earth’s surface, ordinary flat-triangle formulas may not represent the geometry accurately.

Navigation and geodesy can require:

• spherical trigonometry;
• ellipsoidal models;
• bearings;
• coordinates;
• or geodesic calculations.

The appropriate geometry depends on the scale of the problem.

Structural triangles require additional analysis

A triangle may appear in:

• trusses;
• braces;
• frames;
• roofs;
• structural layouts.

Knowing the side lengths and angles does not determine:

• member capacity;
• loads;
• forces;
• connections;
• deflection;
• or structural safety.

The calculator solves geometry only.

Common triangle calculation mistakes to avoid

Pairing a side with the wrong angle

Side:

a

must be opposite angle:

A

Assuming SSA always has one solution

It may have zero, one, or two valid triangles.

Using a sloping side as height

Area formulas need perpendicular height when height is specified.

Using radians for degree inputs

Check the calculator’s angle mode.

Forgetting the triangle inequality

Not every three positive lengths can form a triangle.

Rounding an intermediate angle too early

Keep more digits until all sides and angles are solved.

Forgetting the 180-degree angle sum

The interior angles of a plane triangle must total:

180°

Treating AAA as enough information

Three angles determine shape but not size.

Quick triangle examples

Known values	Useful method
3 sides	Law of cosines, then law of sines
2 sides + included angle	Law of cosines
2 angles + 1 side	Angle sum + law of sines
2 sides + nonincluded angle	Law of sines + ambiguity check
2 right-triangle sides	Pythagorean theorem
1 right-triangle side + acute angle	Trigonometric ratios

Choosing the appropriate method can reduce both calculation time and rounding error.

How to check a solved triangle

After calculating all values, verify:

Angle sum

A + B + C ≈ 180°

Triangle inequality

Each pair of sides sums to more than the third.

Side-angle ordering

The longest side faces the largest angle.

Law of sines or cosines

Substitute the results back into one of the governing relationships.

Area

The area should be positive.

A result that fails one of these checks deserves another review.

When to use another calculator

Use the Area Calculator when you only need the area of a triangle or another geometric shape.

Use the Scientific Calculator for manual sine, cosine, tangent, inverse trigonometric, square-root, and power calculations.

For ordinary arithmetic and order-of-operations checks, use the Basic Calculator.

FAQ

How many measurements are needed to solve a triangle?

Usually three independent measurements are needed, including at least one side.

Can three angles determine a triangle completely?

No.

AAA determines shape but not scale.

At least one side is required to determine the actual size.

What is the triangle inequality?

For a valid nondegenerate triangle, the sum of any two side lengths must be greater than the third side.

What is the angle sum of a triangle?

For an ordinary plane triangle:

A + B + C = 180°

When should I use the law of sines?

It is commonly useful for:

ASA

AAS

and some:

SSA

problems.

When should I use the law of cosines?

It is especially useful for:

SSS

and:

SAS

problems.

What is the SSA ambiguous case?

When two sides and a nonincluded angle are known, the law of sines can sometimes produce two possible angles with the same sine.

The measurements may therefore produce zero, one, or two valid triangles.

How do I calculate triangle area from three sides?

Use Heron’s formula:

s = (a + b + c)/2

then:

Area = √(s(s−a)(s−b)(s−c))

How do I calculate area from two sides and an angle?

If the known angle is included between sides a and b:

Area = ab sin(C)/2

How do I solve a right triangle?

Use the Pythagorean theorem for sides and sine, cosine, or tangent for angles and side relationships.

Does the calculator use degrees or radians?

Use the mode required by the problem.

Most ordinary geometry problems use degrees unless radians are explicitly specified.

Why do my rounded angles total 179.99° instead of 180°?

Displayed values may have been rounded individually.

The unrounded internal values can still sum to 180°.

Can a triangle have two 90-degree angles?

No.

Two 90° angles would already total 180°, leaving no positive third interior angle.

Can the longest side face the smallest angle?

No.

In a valid triangle, the longest side is opposite the largest angle.

Does the calculator solve structural or surveying problems?

It solves ideal triangle geometry.

Structural design, professional surveying, navigation, and other applied fields can require additional methods and uncertainty analysis.

Mathematics reference

For additional explanations and worked examples involving triangles, geometry, trigonometry, and related mathematics, see:

OpenStax — Prealgebra 2e

Match the method to the information you actually know

A Triangle Calculator works best when the known values are labeled before any formula is chosen.

SSS and SAS naturally lead to the law of cosines.

ASA and AAS usually begin with the 180° angle sum and then the law of sines.

Right triangles often use the Pythagorean theorem and basic trigonometry.

SSA deserves special care because two different triangles can sometimes satisfy the same measurements.

After solving, check the angle sum, triangle inequality, side-angle ordering, and one independent formula before accepting the result.

That combination of correct labeling, the right solving method, and a final validity check prevents most triangle-calculation errors.