triangle problem solving calculator

Triangle Calculator

Please provide 3 values including at least one side to the following 6 fields, and click the "Calculate" button. When radians are selected as the angle unit, it can take values such as pi/2, pi/4, etc.

A triangle is a polygon that has three vertices. A vertex is a point where two or more curves, lines, or edges meet; in the case of a triangle, the three vertices are joined by three line segments called edges. A triangle is usually referred to by its vertices. Hence, a triangle with vertices a, b, and c is typically denoted as Δabc. Furthermore, triangles tend to be described based on the length of their sides, as well as their internal angles. For example, a triangle in which all three sides have equal lengths is called an equilateral triangle while a triangle in which two sides have equal lengths is called isosceles. When none of the sides of a triangle have equal lengths, it is referred to as scalene, as depicted below.

Tick marks on the edge of a triangle are a common notation that reflects the length of the side, where the same number of ticks means equal length. Similar notation exists for the internal angles of a triangle, denoted by differing numbers of concentric arcs located at the triangle's vertices. As can be seen from the triangles above, the length and internal angles of a triangle are directly related, so it makes sense that an equilateral triangle has three equal internal angles, and three equal length sides. Note that the triangle provided in the calculator is not shown to scale; while it looks equilateral (and has angle markings that typically would be read as equal), it is not necessarily equilateral and is simply a representation of a triangle. When actual values are entered, the calculator output will reflect what the shape of the input triangle should look like.

Triangles classified based on their internal angles fall into two categories: right or oblique. A right triangle is a triangle in which one of the angles is 90°, and is denoted by two line segments forming a square at the vertex constituting the right angle. The longest edge of a right triangle, which is the edge opposite the right angle, is called the hypotenuse. Any triangle that is not a right triangle is classified as an oblique triangle and can either be obtuse or acute. In an obtuse triangle, one of the angles of the triangle is greater than 90°, while in an acute triangle, all of the angles are less than 90°, as shown below.

Triangle facts, theorems, and laws

• It is not possible for a triangle to have more than one vertex with internal angle greater than or equal to 90°, or it would no longer be a triangle.
• The interior angles of a triangle always add up to 180° while the exterior angles of a triangle are equal to the sum of the two interior angles that are not adjacent to it. Another way to calculate the exterior angle of a triangle is to subtract the angle of the vertex of interest from 180°.
• The sum of the lengths of any two sides of a triangle is always larger than the length of the third side

a 2 + b 2 = c 2

EX: Given a = 3, c = 5, find b:

3 2 + b 2 = 5 2 9 + b 2 = 25 b 2 = 16 b = 4

• Law of sines: the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. Using the law of sines makes it possible to find unknown angles and sides of a triangle given enough information. Where sides a, b, c, and angles A, B, C are as depicted in the above calculator, the law of sines can be written as shown below. Thus, if b, B and C are known, it is possible to find c by relating b/sin(B) and c/sin(C). Note that there exist cases when a triangle meets certain conditions, where two different triangle configurations are possible given the same set of data.
• Given the lengths of all three sides of any triangle, each angle can be calculated using the following equation. Refer to the triangle above, assuming that a, b, and c are known values.

Area of a Triangle

There are multiple different equations for calculating the area of a triangle, dependent on what information is known. Likely the most commonly known equation for calculating the area of a triangle involves its base, b , and height, h . The "base" refers to any side of the triangle where the height is represented by the length of the line segment drawn from the vertex opposite the base, to a point on the base that forms a perpendicular.

Another method for calculating the area of a triangle uses Heron's formula. Unlike the previous equations, Heron's formula does not require an arbitrary choice of a side as a base, or a vertex as an origin. However, it does require that the lengths of the three sides are known. Again, in reference to the triangle provided in the calculator, if a = 3, b = 4, and c = 5:

Median, inradius, and circumradius

The median of a triangle is defined as the length of a line segment that extends from a vertex of the triangle to the midpoint of the opposing side. A triangle can have three medians, all of which will intersect at the centroid (the arithmetic mean position of all the points in the triangle) of the triangle. Refer to the figure provided below for clarification.

The medians of the triangle are represented by the line segments m a , m b , and m c . The length of each median can be calculated as follows:

Where a, b, and c represent the length of the side of the triangle as shown in the figure above.

As an example, given that a=2, b=3, and c=4, the median m a can be calculated as follows:

The inradius is the radius of the largest circle that will fit inside the given polygon, in this case, a triangle. The inradius is perpendicular to each side of the polygon. In a triangle, the inradius can be determined by constructing two angle bisectors to determine the incenter of the triangle. The inradius is the perpendicular distance between the incenter and one of the sides of the triangle. Any side of the triangle can be used as long as the perpendicular distance between the side and the incenter is determined, since the incenter, by definition, is equidistant from each side of the triangle.

For the purposes of this calculator, the inradius is calculated using the area (Area) and semiperimeter (s) of the triangle along with the following formulas:

where a, b, and c are the sides of the triangle

Circumradius

The circumradius is defined as the radius of a circle that passes through all the vertices of a polygon, in this case, a triangle. The center of this circle, where all the perpendicular bisectors of each side of the triangle meet, is the circumcenter of the triangle, and is the point from which the circumradius is measured. The circumcenter of the triangle does not necessarily have to be within the triangle. It is worth noting that all triangles have a circumcircle (circle that passes through each vertex), and therefore a circumradius.

For the purposes of this calculator, the circumradius is calculated using the following formula:

Where a is a side of the triangle, and A is the angle opposite of side a

Although side a and angle A are being used, any of the sides and their respective opposite angles can be used in the formula.

Triangle calculator

How does this calculator solve a triangle.

• The expert phase is different for different tasks. The calculator tries to calculate the sizes of three sides of the triangle from the entered data. He gradually applies the knowledge base to the entered data, which is represented in particular by the relationships between individual triangle parameters. These are successively applied and combined, and the triangle parameters are calculated. Calculator iterates until the triangle has calculated all three sides. For example, the appropriate height is calculated from the given area of the triangle and the corresponding side. From the known height and angle, the adjacent side, etc., can be calculated. The calculator uses use knowledge, e.g., formulas and relations like the Pythagorean theorem, Sine theorem, Cosine theorem, and Heron's formula.
• The second stage is the calculation of the properties of the triangle from the available lengths of its three sides.

Examples of how to enter a triangle:

Triangles in word problems:.

Look also at our friend's collection of math problems and questions:

• right triangle
• Heron's formula
• The Law of Sines
• The Law of Cosines
• Pythagorean theorem
• triangle inequality
• similarity of triangles
• The right triangle altitude theorem

• Calculate Δ by 3 sides SSS
• Δ SAS by 2 sides and 1 angle
• Δ ASA by 1 side and 2 angles
• Scalene triangle
• Right-angled Δ
• Equilateral Δ
• Isosceles Δ
• Δ by coordinates
• List of triangles

• Calculators
• Triangle...

Triangle Calculator

Please provide values for any three of the six fields below. At least one of those values must be a side length.

Hello there!

About the Triangle Calculator

This triangle calculator lets you solve a triangle. It calculates the missing measurements of a triangle if you know any one side and any two from the remaing five mesurements.

The calculator will give you not just the answers, but also a step-by-step solution.

Usage Guide

I. valid inputs.

The triangle calculator requires exactly three of the six inputs — one side-length and any two of the remaining inputs.

The inputs themselves must be non-negative real numbers and can be in any format — integers, decimals, fractions, or even mixed numbers. Here are a few examples.

• Whole numbers or decimals → 2 \hspace{0.2em} 2 \hspace{0.2em} 2 , − 4.25 \hspace{0.2em} -4.25 \hspace{0.2em} − 4.25 , 0 \hspace{0.2em} 0 \hspace{0.2em} 0 , 0.33 \hspace{0.2em} 0.33 \hspace{0.2em} 0.33
• Fractions → 2 / 3 \hspace{0.2em} 2/3 \hspace{0.2em} 2/3 , − 1 / 5 \hspace{0.2em} -1/5 \hspace{0.2em} − 1/5
• Mixed numbers → 5 1 / 4 \hspace{0.2em} 5 \hspace{0.5em} 1/4 \hspace{0.2em} 5 1/4

Finally, of course, the inputs shouldn't violate any of the properties of triangles. For example, sum of angles must not exceed 180 ° \hspace{0.2em} 180 \degree \hspace{0.2em} 180° .

ii. Example

If you would like to see an example of the calculator's working, just click the "example" button.

iii. Solutions

As mentioned earlier, the calculator won't just tell you the answer but also the steps you can follow to do the calculation yourself. The "show/hide solution" button would be available to you after the calculator has processed your input.

We would love to see you share our calculators with your family, friends, or anyone else who might find it useful.

By checking the "include calculation" checkbox, you can share your calculation as well.

Here's a quick overview of what it means to solve a triangle and a few related concepts to help you make sense of the solutions provided by the triangle calculator.

For those interested, we have a more comprehensive tutorial on solving triangles .

Solving a Triangle

There are six values describing the six parts of a triangle — three sides and three angles. Now, if we know one side and any two of the other five values, we can use that information to find the remaining three.

Finding the unknown measurements of a triangles from what is known is referred to as solving triangles .

Important Concepts

Let's look at a few of the important concepts that help us solve triangles.

Angle Sum Property

The sum of the three internal angles of a triangle is 180 ° \hspace{0.2em} 180 \degree 180° .

The sine rule states that the ratio of side length to the sine of opposite angle is the same for all sides in a triangle.

Cosine Rule

The cosine rule gives the relationship between the side lengths of a triangle and the cosine of any of its angles. It says —

Re-framing the formula for other sides, we have

For cases where we need to find angles using the cosine rule, the three formulas can be rearranged as —

When it comes to solving triangles, there are five different types of problems depending on which three of the triangle's measurements we know.

• S S S \hspace{0.2em} SSS \hspace{0.2em} SSS — all three sides are known
• S A S \hspace{0.2em} SAS \hspace{0.2em} S A S — two sides and the included angle
• S S A \hspace{0.2em} SSA \hspace{0.2em} SS A — two sides and a non-included angle
• A S A \hspace{0.2em} ASA \hspace{0.2em} A S A — two angles and the included side
• A A S \hspace{0.2em} AAS \hspace{0.2em} AA S — two angles and the non-included side

While every problem can be solved using the fundamentals discussed earlier and a basic knowledge of triangles, each type has a sequence of steps that you can use to solve problems of that type.

Let me show you what I mean using an example.

The lengths of the three sides of a triangle are 6 \hspace{0.2em} 6 \hspace{0.2em} 6 , 7 \hspace{0.2em} 7 \hspace{0.2em} 7 , and 8 \hspace{0.2em} 8 \hspace{0.2em} 8 . Solve the triangle.

The question gives us the three sides of the triangle. So the problem is of type S S S \hspace{0.2em} SSS \hspace{0.2em} SSS . Solving the triangle would mean calculating its three angles.

Step 0.  We start by drawing a rough sketch of the triangle and labeling the information given in the question. It’s not necessary but often makes things easier and helps avoid silly mistakes.

Step 1.  Use the Cosine Rule to find the largest angle

When we know all the side lengths, we can use the Cosine Rule to find any of the angles.

It's best to find the largest angle first — the angle opposite to the longest side.

That's because if there is an obtuse angle ( > 90 ° ) \hspace{0.2em} (>90 \degree) \hspace{0.2em} ( > 90° ) in the triangle, it has to be this angle. So in the next step, we don't need to worry about the obtuse solutions when taking sine inverse.

Here the largest angle would be C \hspace{0.2em} C \hspace{0.2em} C . So using the formula for cos ⁡ C \hspace{0.2em} \cos C \hspace{0.2em} cos C , we have

Taking cos inverse on both sides.

Step 2.  Use the Sine Rule for one of the remaining angles

Now that we know the three sides and one angle, we can use the Sine Rule to find any of the remaining two angles. Let's calculate A \hspace{0.2em} A \hspace{0.2em} A .

According to the sine rule

Substituting the known values and solving for B, we have

Step 3.  Use the Angle Sum Property to find the third angle

And we have solved the triangle.

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Triangle Calculator

The triangle calculator finds all triangle measurements – side lengths, triangle angles, area, perimeter, semiperimeter, heights, medians, inradius, and circumradius.

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Table of Contents

Triangle calculator, directions for use, limitations on the input values, calculation example, triangle: definition and important formulas, conditions of existence of a triangle, triangle measurements.

The triangle calculator is an online triangle solver allowing you to find all triangle measurements based on three known measurements quickly. The calculator takes the lengths of the sides of a triangle and triangle angles as inputs and calculates the following measurements:

• missing side lengths,
• missing triangle angles,
• semiperimeter,
• heights to all sides of the triangle,
• medians to all sides of the triangle,
• circumradius.

The calculator also provides the coordinates of the vertices, the centroid, the inscribed circle center, and the circumscribed circle center, assuming that the coordinates of vertex A are [0, 0].

To use this triangle calculator, enter any three values into the input fields. You can enter the values of any angles or any side lengths. Note that at least one of the values has to represent a side length; otherwise, a triangle will have infinite solutions.

After entering the values, select the units for triangle angles. You can choose between degrees or radians. When selecting radians, use "pi" to represent π. For example, if the angle value is \$\frac{π}{3}\$, enter "pi/3." After inserting the known values, press "Calculate." The calculator will return all missing values from the list above and the schematic view of the triangle, which will help you better visualize it.

After the answer, you can expand the following field - Show Calculation Steps – to get familiar with the solution algorithm and the formulas used to find the answer.

At least one of the known values must be a side length.

When entering the following combination of values – two angles and one side length – note that the sum of the angle values has to be less than 180° or π.

When entering three side lengths, note that the sum of any two side lengths should be greater than the length of the remaining side.

Imagine you are moving and want to borrow a truck from a friend. You will need to load and unload the truck, but it doesn't have a built-in ramp. You have a portable ramp, but you must ensure its dimensions fit the truck's height. Your ramp is not adjustable, and you have measured that its two sides measure as 1 m and 0.8 m, and the angle opposite to the side of 1 m is 85 degrees (see the image). You know that you can adjust the truck's height from 0.5 m to 1 m. Does your ramp fit?

• side b = 1;
• side c = 0.8;
• angle B = 85 degrees.

To determine whether your ramp fits the truck, you need to solve the triangle above and estimate whether the length of side A fits the given range for the truck's height: 0.5< a < 1 .

Inserting the values presented above into the triangle calculator, you get the following answer in the task, we will only need the missing side length.

So the rest of the answers are not demonstrated in this practical example, while the triangle solver still calculates them:

Side a = 0.67376

Side c = 0.8

angle A = 42.16° = 42°9'35" = 0.73582 rad

angle B = 85° = 1.48353 rad

angle C = 52.84° = 52°50'25" = 0.92224 rad

The ramp looks something like this:

We see that a ≈ 0.674, and we know that the height of the truck can be adjusted in the range 0.5 < a < 1 . This means that the ramp height fits the adjustable height of the truck, and you can borrow the truck from your friend instead of renting one!

In geometry, a triangle is a plane figure made by the intersection of three straight non-parallel lines. A triangle can also be described as a polygon with three vertices and three edges. The edges of the triangle are usually called sides.

Two conditions define the existence of a triangle; one condition is applied on the sides, and the other – on the angles. The condition on the sides is based on the triangle inequality. It states that the sum of the lengths of any two sides of the triangle must be greater than or equal to the length of the remaining third side. If the sum of the lengths of the two sides equals the length of the third side, the triangle is called degenerate.

A degenerate triangle is a triangle where all three vertices lie on the same straight line. It's a very special triangle case, usually not discussed in elementary geometry, and is, therefore, not considered here.

The condition on the angles states that the sum of the three angles of any triangle always equals 180° or π radians.

Let's define the most crucial triangle measurements and look at the formulas for calculating their values.

The perimeter of a triangle is the sum of the lengths of all its sides and can be found as follows:

p = a + b + c

The semiperimeter of a triangle – is half of the length of the perimeter of the triangle:

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

The area of a triangle – is a property describing how much space the triangle takes up on a plane. If the lengths of the two sides of the triangle and the angle between these two sides are known, the area of a triangle can be calculated as follows:

$$A=\frac{1}{2}a× b×\sin{C}$$

A triangle's height, or altitude, is perpendicular from one of the angles to the opposite side. Since any triangle has three sides, any triangle will also have three perpendiculars. A height perpendicular to side A is usually denoted as hₐ . Similarly, the other two heights are denoted as \$h_b\$ and h꜀ . The easiest way to find the height of a triangle is through its area:

$$A=\frac{1}{2}× a× h_a=\frac{1}{2}× b× h_b=\frac{1}{2}× c× h_c$$

$$h_a=\frac{2A}{a}, h_b=\frac{2A}{b}, h_c=\frac{2A}{c}$$

Median to a side of a triangle – is the line from a vertex of the triangle to the middle of the opposite side. Any triangle has three medians.

A median to side a is usually denoted as mₐ . Similarly, the other two medians are denoted as \$m_b\$ and m꜀ . We can find the lengths of the medians with the following formula:

$$m_a=\frac{1}{2}\sqrt{2b²+2c^2-a^2}$$

The inradius of a triangle – is the radius of a circle inscribed inside the triangle and touching all of its sides.

The length of the inradius r can be found as follows:

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

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

We can find the length of the circumradius R from the sine rule:

$$2R=\frac{a}{\sin{A}}=\frac{b}{\sin{B}}=\frac{c}{\sin{C}}$$

The sine rule is also beneficial for finding the missing values of side lengths or angles of a triangle. Another helpful rule is the cosine rule:

$$a=\sqrt{b²+c^2-2bc\cos{A}}$$

$$b=\sqrt{a^2+c^2-2ac\cos{B}}$$

$$c=\sqrt{a^2+b²-2ab\cos{C}}$$

The formulas listed above allow calculating all triangle measurements. The triangle calculator uses these formulas to find the missing values.

Online Triangle Calculator

Enter any valid input (3 side lengths, 2 sides and an angle or 2 angle and a 1 side) and our calculator will do the rest.

• Triangle App
• Triangle Animated Gifs

This is the acute triangle in Quadrant I, for more information on this topic, check out the law of sines ambiguous case .

This is the obtuse triangle in Quadrant II, for more information on this topic, check out the law of sines ambiguous case .

Status: Calculator waiting for input

Why is the calculator saying there's an error when there shouldn't be?

The most frequent reason for this is because you are rounding the sides and angles which can, at times, lead to results that seem inaccurate. In these cases, in actuality , the calculator is really producing correct results. However, it is then rounding them for you- which leads to seemingly inaccurate results and possible error warnings. To see if that is your problem, set the rounding to maximum accuracy .

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Triangle Calculator

Solve triangles step by step.

The calculator will try to find all sides and angles of the triangle (right triangle, obtuse, acute, isosceles, equilateral), as well as its perimeter and area, with steps shown.

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Our Triangle Calculator easily finds the sides and angles of a triangle with just a few inputs. If you want to deepen your understanding of triangles, our online tool is here.

How to Use the Triangle Calculator?

Enter the known measurements of the triangle. This could include side lengths or angles.

Calculation

Once you've entered all the available information, click the "Calculate" button.

The calculator will instantly provide results based on your input. This will include the missing side lengths, angles, perimeter, and area.

What Is a Triangle?

• Equilateral Triangle: All three sides are of equal length, and all three angles are 60 degrees.
• Isosceles Triangle: It has two sides of equal length, and the angles opposite those sides are also equal.
• Scalene Triangle: All sides have different lengths and all angles have different measures.

• Acute Triangle: All three angles are less than 90 degrees.
• Obtuse Triangle: One of the angles is greater than 90 degrees.
• Right-Angled Triangle: One of the angles is exactly 90 degrees, making it a right angle.

Triangles are fundamental shapes in geometry and have various properties and theorems associated with them.

What Are Some Key Facts, Theorems, and Laws Related to Triangles?

Facts about Triangles:

• Angle Sum: The sum of the interior angles of any triangle is always 180 degrees.
• Exterior Angle: An exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Area Calculation: The area $$$ A $$$ of a triangle with the side $$$ b $$$ and the height $$$ h $$$ dropped to this side can be found using the following formula:

Triangle Theorems:

Pythagorean Theorem (For Right Triangles): In a right triangle, the square of the length of the hypotenuse $$$ c $$$ is equal to the sum of the squares of the lengths of the other two sides (called legs) $$$ a $$$ and $$$ b $$$ . This can be represented as

• Isosceles Triangle Theorem: In an isosceles triangle, the angles opposite the equal sides are also equal.
• Base Angles Theorem: The base angles are congruent in an isosceles triangle.
• Converse of the Base Angles Theorem: If two angles of a triangle are congruent, then the sides opposite those angles are also congruent, making it an isosceles triangle.

Triangle Laws

Law of Sines: For any triangle, the ratio of the length of a side to the sine of its opposite angle is constant. This can be written as

Law of Cosines: This rule relates the lengths of the sides of a triangle to the cosine of one of its angles. It can be used for any triangle, not just right triangles. Mathematically, it can be represented as

Law of Tangents: For any triangle,

Understanding and applying these facts, theorems, and laws is critical to anyone who studies geometry. They provide insight into the properties of triangles and provide fundamental knowledge for more complex geometric concepts and real-world applications.

Why Choose Our Triangle Calculator?

Our calculator uses advanced algorithms to guarantee accuracy every time, giving you confidence in the results.

User-Friendly Interface

Built with user preferences in mind, our platform makes it easy to solve triangles, even for people new to geometry.

Versatility

Regardless of the type of triangle (scalene, isosceles, acute, or obtuse), our calculator will help you.

Fast Results

Our calculator provides answers almost instantly, making it much faster to solve multiple problems.

How accurate are the results from the Triangle Calculator?

Our calculator uses advanced algorithms to ensure accurate and correct results.

Which types of triangles can this calculator handle?

Our Triangle Calculator is versatile and can handle various types of triangles, including scalene, isosceles, right, acute, and obtuse triangles.

What formulas does the Triangle Calculator use?

The calculator uses a variety of geometric formulas depending on the input, such as the Pythagorean theorem for right triangles, and the law of sines and cosines for others.

What is the Pythagorean Theorem?

The Pythagorean Theorem is a fundamental theorem in Euclidean geometry that describes the relationship between the three sides of a right triangle. It states that the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Capital-letter variable names correspond to angle measures, opposite from each side length named by the lowercase-letter counterpart.  All angle measure inputs are in degrees (pre-multiplied by 180:pi).  The side lengths a , b and c are in arbitrary units and do not at all affect the triangle size or drawing in any way, other than to help compute any unknown angle measures.

I also maintain a separate research project to derive many exact trigonometric ratios so that many of these answers can be expressed in exact form.

All JavaScript on this website is freedom-respecting software. Learn more.

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On behalf of our dedicated team, we thank you for your continued support. It's fulfilling to see so many people using Voovers to find solutions to their problems. Thanks again and we look forward to continue helping you along your journey!

Nikkolas and Alex Founders and Owners of Voovers

Triangle Calculator

Lesson on Solving Triangles

Lesson contents, how to calculate the sides of a triangle.

If solving for a side length of a right triangle where know two side lengths, we may use the Pythagorean theorem. The Pythagorean theorem is given as:

Where  a and  b are the legs and  c is the hypotenuse.

For non-right triangles, we must know three parameters of the triangle. The three known parameters may either be two side lengths and an angle or two angles and a side length.

There are several formulas we may use for solving side lengths. The most common and versatile are the law of cosines and the  law of sines .

The law of cosines is split into three formulas. They are given as: a 2 = b 2 + c 2 – 2bc×cos(A) b 2 = a 2 + c 2 – 2ac×cos(B) c 2 = a 2 + b 2 – 2ab×cos(C) Where  a ,  b , and c are the side lengths and  A ,  B , and  C are the internal angles.

The law of sines is given as: sin(A) ⁄ a = sin(B) ⁄ b = sin(C) ⁄ c Where  a ,  b , and c are the side lengths and  A ,  B , and  C are the internal angles.

There are other formulas available for solving triangle sides, but the law of cosines and law of sines may be leveraged in combination to solve any triangle and therefore will most commonly be used.

How to Calculate the Angles of a Triangle

When solving for a triangle’s angles, a common and versatile formula for use is called the sum of angles . It is given as: A + B + C = 180 Where  A ,  B , and  C are the internal angles of a triangle.

If two angles are known and the third is desired, simply apply the sum of angles formula given above. If three side lengths are known, use the law of cosines. If an angle and two sides are known, use either the law of cosines or the law of sines, depending on the combination of angle and sides.

How the Calculator Works

The calculator on this page is written in the programming language JavaScript (JS) which allows it to run in your device’s internet browser JS engine. The JS engine running the code allows for instant answers at the click of a button.

When the calculate button is pressed, your selected “solving for” parameter is inputted, and the calculator determines which set of formulas will be used to solve. Then, your inputted side/angle parameters are converted to standard units and fed into the applicable formulas. This calculator is powered by the law of cosines and other basic triangle identities.

Once the desired output parameter has been calculated, the solution’s units are converted if the selected unit requires so. The final answer is rounded and then printed to the applicable output area on the calculator.

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Triangle Calculator to Solve SSS, SAS, SSA, ASA, and AAS Triangles

This triangle solver will take three known triangle measurements and solve for the other three.

The calculator will also solve for the area of the triangle, the perimeter, the semi-perimeter, the radius of the circumcircle and the inscribed circle, the medians, and the heights.

Plus, unlike other online triangle calculators, this calculator will show its work by detailing each of the steps it took to solve the formulas for finding the missing values.

Finally, the triangle calculator will also calculate the coordinates of the vertices, the centroid, and the circumcenter, and draw the solved triangle based on those coordinates (requires latest version of your web browser software).

Triangle Calculator

Solve triangles given SSS, SAS, SSA, ASA, or AAS.

Selected Data Record:

A Data Record is a set of calculator entries that are stored in your web browser's Local Storage. If a Data Record is currently selected in the "Data" tab, this line will list the name you gave to that data record. If no data record is selected, or you have no entries stored for this calculator, the line will display "None".

Side a length:

Enter the length of side a .

Side b length:

Enter the length of side b .

Side c length:

Enter the length of side c .

Angle B length:

Enter the angle of vertex B .

Angle A length:

Enter the angle of vertex A .

If you would like to save the current entries to the secure online database, tap or click on the Data tab, select "New Data Record", give the data record a name, then tap or click the Save button. To save changes to previously saved entries, simply tap the Save button. Please select and "Clear" any data records you no longer need.

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If the tools panel becomes "Unstuck" on its own, try clicking "Unstick" and then "Stick" to re-stick the panel.

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Right triangle calculator

Input any two values for a right triangle, and the calculator will find the unknown element. The calculator provides a step-by-step explanation on how to calculate missing elements. The calculator gives you a step-by-step guide on how to find the missing value.

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Right triangle calculations

The calculator uses the following formulas to find the missing values of a right triangle:

Example 01 :

Find hypotenuse $ c $ of a right triangle if $ a = 4\,cm $ and $ b = 8\,cm $.

When we know two sides, we use the Pythagorean theorem to find the third one.

Example 02 :

Find the angle $\alpha$ of a right triangle if hypotenuse $ c = 14~cm$ and leg $ a = 8~cm$.

In order to find missing angle we can use the sine function

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Welcome to MathPortal. This website's owner is mathematician Miloš Petrović. I designed this website and wrote all the calculators, lessons, and formulas .

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Solving Triangles

"Solving" means finding missing sides and angles.

Six Different Types

If you need to solve a triangle right now choose one of the six options below:

Which Sides or Angles do you know already? (Click on the image or link)

... or read on to find out how you can become an expert triangle solver :

Your Solving Toolbox

Want to learn to solve triangles?

Imagine you are " The Solver " ... ... the one they ask for when a triangle needs solving!

In your solving toolbox (along with your pen, paper and calculator) you have these 3 equations:

1. Angles Add to 180° :

A + B + C = 180°

When you know two angles you can find the third.

2. Law of Sines (the Sine Rule):

When there is an angle opposite a side, this equation comes to the rescue.

Note: angle A is opposite side a, B is opposite b, and C is opposite c.

3. Law of Cosines (the Cosine Rule):

This is the hardest to use (and remember) but it is sometimes needed to get you out of difficult situations.

It is an enhanced version of the Pythagoras Theorem that works on any triangle.

Six Different Types (More Detail)

There are SIX different types of puzzles you may need to solve. Get familiar with them:

This means we are given all three angles of a triangle, but no sides.

AAA triangles are impossible to solve further since there are is nothing to show us size ... we know the shape but not how big it is.

We need to know at least one side to go further. See Solving "AAA" Triangles .

This mean we are given two angles of a triangle and one side, which is not the side adjacent to the two given angles.

Such a triangle can be solved by using Angles of a Triangle to find the other angle, and The Law of Sines to find each of the other two sides. See Solving "AAS" Triangles .

This means we are given two angles of a triangle and one side, which is the side adjacent to the two given angles.

In this case we find the third angle by using Angles of a Triangle , then use The Law of Sines to find each of the other two sides. See Solving "ASA" Triangles .

This means we are given two sides and the included angle.

For this type of triangle, we must use The Law of Cosines first to calculate the third side of the triangle; then we can use The Law of Sines to find one of the other two angles, and finally use Angles of a Triangle to find the last angle. See Solving "SAS" Triangles .

This means we are given two sides and one angle that is not the included angle.

In this case, use The Law of Sines first to find either one of the other two angles, then use Angles of a Triangle to find the third angle, then The Law of Sines again to find the final side. See Solving "SSA" Triangles .

This means we are given all three sides of a triangle, but no angles.

In this case, we have no choice. We must use The Law of Cosines first to find any one of the three angles, then we can use The Law of Sines (or use The Law of Cosines again) to find a second angle, and finally Angles of a Triangle to find the third angle. See Solving "SSS" Triangles .

Tips to Solving

Here is some simple advice:

When the triangle has a right angle, then use it, that is usually much simpler.

When two angles are known, work out the third using Angles of a Triangle Add to 180° .

Try The Law of Sines before the The Law of Cosines as it is easier to use.

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