Triangle Calculator

What Is a Triangle Calculator?

A triangle calculator is an online geometry tool that calculates the missing measurements of a triangle from known sides and angles. It works as both a triangle side calculator and a triangle angle calculator, making it useful when you need to solve a triangle without performing every calculation manually.

You can enter values for:

• Side a

• Side b

• Side c

• Angle A

• Angle B

• Angle C

Each side is opposite its corresponding angle. For example, side a is opposite Angle A, side b is opposite Angle B, and side c is opposite Angle C.

Enter any three known values, with at least one of them being a side. The triangle solver uses the available information to determine the remaining measurements.

In addition to sides and angles, the results can provide detailed geometric properties such as area, perimeter, semiperimeter, inradius, circumradius, heights, medians, angle bisectors, coordinates, and triangle centers.

How to Use the Triangle Calculator

Using the calculator takes only a few steps.

1. Enter your known sides: Add Side a, Side b, or Side c if you know their lengths.

2. Enter your known angles: Add Angle A, B, or C in the corresponding fields.

3. Provide three known values: Enter any three measurements, including at least one side.

4. Select the angle unit: Choose the appropriate angle unit, such as degrees or radians.

5. Click Calculate: The triangle angle calculator will determine the missing triangle measurements.

6. Review your results: Check the solved sides, angles, area, perimeter, diagram, and other geometric properties.

7. Start again if needed: Click Clear to remove the entered values and solve another triangle.

Always make sure that you match each side with its opposite angle correctly.

How Does the Triangle Calculator Work?

The triangle calculator first checks which sides and angles you entered. It then identifies the triangle-solving method that fits those known measurements.

Depending on your inputs, the calculator may solve the triangle using:

SSS – Side-Side-Side

If all three sides are known, the calculator can use the Law of Cosines to determine the angles.

SAS – Side-Angle-Side

When two sides and the angle between them are known, the triangle angle calculator can first calculate the missing side using the Law of Cosines and then solve the remaining angles.

ASA – Angle-Side-Angle

If two angles and the side between them are known, the third angle can be found using the triangle angle sum. The remaining sides can then be calculated using the Law of Sines.

AAS – Angle-Angle-Side

When two angles and a non-included side are known, the missing angle is calculated first. The Law of Sines can then determine the other sides.

SSA – Side-Side-Angle

SSA means two sides and an angle that is not between them are known. This situation needs special attention because it can sometimes produce no triangle, one triangle, or two possible triangles. This is known as the SSA ambiguous case.

Triangle Formulas Used in Triangle Calculator

Different formulas are needed depending on the information available.

Triangle Angle Sum

The three interior angles of every triangle add up to 180°.

$A+B+C=180^\circ$

For example, if Angle A is 105° and Angle B is 45°:

$C=180^\circ-105^\circ-45^\circ=30^\circ$

Triangle Perimeter

The perimeter is the total length around the triangle.

$P=a+b+c$

If the sides are 32, 23.42585, and 18.58442:

$P=32+23.42585+18.58442$

$P\approx74.01027$

Semiperimeter

The semiperimeter is half of the triangle's perimeter.

$s=\frac{a+b+c}{2}$

It is commonly used in Heron's formula and several other triangle calculations.

Law of Sines

The Law of Sines connects each side to the sine of its opposite angle.

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$

It is especially useful when solving ASA, AAS, and certain SSA triangles.

Law of Cosines

The Law of Cosines is useful when three sides are known or when you know two sides and the included angle.

For side c:

$c^2=a^2+b^2-2ab\cos C$

Similar versions of the formula can be used to find sides a and b.

Heron's Formula for Triangle Area

If all three side lengths are known, the area can be calculated using Heron's formula:

$K=\sqrt{s(s-a)(s-b)(s-c)}$

where (s) represents the semiperimeter.

Another common area formula, when the base and corresponding height are known, is:

$K=\frac{1}{2}bh$

Inradius

The inradius is the radius of the circle that fits inside the triangle and touches all three sides.

$r=\frac{K}{s}$

where (K) is the area and (s) is the semiperimeter.

Circumradius

The circumradius is the radius of the circle passing through all three vertices of the triangle.

$R=\frac{abc}{4K}$

Triangle Heights

A triangle has three heights, one corresponding to each side.

$h_a=\frac{2K}{a}$

$h_b=\frac{2K}{b}$

$h_c=\frac{2K}{c}$

The height is the perpendicular distance from a vertex to its opposite side.

Triangle Medians

A median connects a vertex to the midpoint of the opposite side.

For the median from vertex A:

$m_a=\frac{1}{2}\sqrt{2b^2+2c^2-a^2}$

Similar formulas can calculate (m_b) and (m_c).

Angle Bisectors

An angle bisector divides one of the triangle's angles into two equal angles.

For Angle A, the length of the internal angle bisector can be calculated as:

$l_a= \frac{\sqrt{bc[(b+c)^2-a^2]}}{b+c}$

Equivalent formulas can be used for the bisectors from vertices B and C.

How to Solve a Triangle

There is no single formula for every triangle. The correct method depends on which measurements you already know.

Solve a Triangle Using Three Sides (SSS)

If you know Side a, Side b, and Side c, first confirm that the measurements can form a valid triangle.

The triangle inequality requires:

$a+b>c$

$a+c>b$

$b+c>a$

Once the triangle is valid, the Law of Cosines can be used to calculate its angles. The area can then be found using Heron's formula.

Solve a Triangle Using Two Sides and an Angle (SAS)

Suppose you know two sides and the included angle. Use the Law of Cosines to find the third side.

Once all three sides are available, calculate the remaining angles using the Law of Cosines or Law of Sines.

Solve a Triangle Using Two Angles and a Side (ASA or AAS)

First find the third angle:

$C=180^\circ-A-B$

Once all three angles are known, use the Law of Sines to calculate the missing sides.

Solve a Triangle Using SSA

An SSA triangle requires extra care.

Depending on the measurements, there may be:

• No valid triangle

• Exactly one triangle

• Two possible triangles

This happens because the same combination of two sides and a non-included angle can sometimes produce two different geometric configurations.

A solve the triangle calculator makes this type of problem easier because it can evaluate the supplied measurements before producing a result.

Triangle Calculator Example

Suppose you enter:

• Side a = 32

• Angle A = 105°

• Angle B = 45°

First, calculate Angle C:

$C=180^\circ-105^\circ-45^\circ$

$C=30^\circ$

Next, use the Law of Sines to calculate Side b:

$\frac{32}{\sin105^\circ} = \frac{b}{\sin45^\circ}$

This gives approximately:

$b\approx23.42585$

Side c can be calculated in the same way:

$\frac{32}{\sin105^\circ} = \frac{c}{\sin30^\circ}$

$c\approx16.56347$

The triangle calculator can then use the completed triangle to determine its area, perimeter, semiperimeter, heights, medians, radii, bisectors, coordinates, and other geometric properties.

Types of Triangles

Triangles can be classified by their sides or their angles.

Triangle Types by Side Length

Equilateral Triangle: All three sides are equal, and every interior angle measures 60°.

Isosceles Triangle: Two sides have equal lengths, and the angles opposite those sides are also equal.

Scalene Triangle: All three sides have different lengths. Its three angles are generally different as well.

Triangle Types by Angles

Acute Triangle: All three angles are less than 90°.

Right Triangle: One angle is exactly 90°. For a right triangle, the relationship between the sides can also be expressed using the Pythagorean theorem:

$a^2+b^2=c^2$

where (c) is the hypotenuse.

Obtuse Triangle: One interior angle is greater than 90°.

Benefits of Using a Triangle Calculator

A triangle solver can save time when a problem requires several formulas. Instead of manually calculating each missing value, you can enter the measurements you already know and obtain the completed triangle.

It can also help you:

• Find missing sides and angles quickly.

• Reduce repetitive manual calculations.

• Calculate advanced geometric properties.

• Visualize the solved triangle.

• Compare side lengths and angles.

• Check manually calculated answers.

When and Where Is a Triangle Calculator Useful?

A triangle calculator is useful whenever you know part of a triangle but need to determine its remaining measurements.

It can help with geometry and trigonometry homework, classroom demonstrations, engineering calculations, architectural planning, construction layouts, surveying, navigation, CAD-related work, and other situations involving triangular geometry.

For professional projects, calculated results should still be checked against the standards, measurements, and tolerances required for the specific application.

Who Should Use a Triangle Solver?

The triangle calculator can be useful for students and teachers learning geometry or trigonometry, as well as engineers, architects, surveyors, designers, and construction professionals who need quick triangle calculations.

It is also useful for anyone who wants to find an unknown triangle side, angle, area, or related geometric measurement without solving several equations manually.

Tips for Accurate Triangle Calculations

• Enter measurements in the correct side and angle fields.

• Remember that Side a is opposite Angle A, and the same rule applies to b/B and c/C.

• Provide three known measurements, including at least one side.

• Select the correct angle unit before calculating.

• Do not enter zero or negative side lengths.

• Remember that the three interior angles must total 180°.

• Make sure three supplied sides satisfy the triangle inequality.

• Use sufficient decimal precision when entering measured values.

• Pay attention to the possible ambiguous case when solving an SSA triangle.

Common Triangle Calculation Mistakes

One common mistake is pairing a side with the wrong opposite angle. This can produce an incorrect result when using the Law of Sines.

Another error is entering angles that cannot form a valid triangle. For example, two angles totaling 180° or more leave no valid measurement for the third angle.

Users may also overlook the triangle inequality when entering three sides or select degrees when their input values are actually in radians.

Finally, avoid rounding intermediate calculations too early. Keeping additional decimal places until the final result generally produces a more accurate answer.

Conclusion

The triangle calculator provides a quick way to solve missing sides, angles, and other geometric properties using the measurements you already know. Whether you need a triangle side calculator, triangle angle calculator, or complete triangle solver, it can simplify SSS, SAS, ASA, AAS, and SSA calculations while providing detailed results for further analysis.

Helpful Resources

• Law of Sines and Cosines - Math is Fun

• Triangle Properties - Khan Academy

• Heron's Formula Explained

Pro Tips

• Always provide at least one side length the triangle cannot be solved with angles alone.

• The sum of all three angles must equal 180°. The calculator will alert you if values are inconsistent.

• If two sides and a non-included angle are given (SSA), there may be two possible triangles or none.

• Use degree mode for everyday calculations and radian mode for scientific or engineering use.

• Slope and side ratios of similar triangles remain constant, useful for scaling problems.