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Math Calculators ▶ Triangle Calculator

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**Table of Content**

An online triangle calculator is exclusively programmed to determine each and every term associated with this geometrical figure. This content is arranged for you people so that you may not face any difficulty in using this free triangle solver. Also, you will get a detailed context regarding all terms to understand them better.

Simply stay focused!

In geometrical analysis:

**“A polygon having only three sides is considered a triangle”**

Let us understand some basics first:

**Vertex:**

**“A particular point where two sides meet with each other is called vertex”.**

In a triangle, we have three sides that are connected to each other. In this way, we have three vertices **A**, **B**, and **C** for every triangle which are also known as **‘edges’**

**Angle:**

**“An angle is formed by joining an incident ray with the reflected ray”.**

We have three triangle angles designated as **∠A**, **∠B**, and **∠C**, respectively. These angles differ for every different types of triangles. With the help of a free angle finder calculator, you can easily find angle of triangle in a fraction of seconds.

**Sides:**

**“The line segments joining to form a triangle are termed sides of a triangle.”**

In a triangle, we have three sides as **a**, **b,** and **c**, respectively.

You can easily solve the triangle with all above terms known and fetching them to a free triangle side calculator.

This sections falls into two categories that are properly explained below:

Depending on the interior angles, we have three types of triangle described as follows:

**Right-Angled Triangle:**

**“A triangle having at least one right angle is called a right-angled triangle.”**

Mathematically, we have:

$$ m∠C = 90^\text{o} $$

A right triangle calculator can be used to analyze a right-angled triangle thoroughly.

**Acute-Angled Triangle:**

**“A triangle whose measure of all angles is acute is called an acute-angled triangle.”**

Mathematically, we have:

$$ m∠A < 90^\text{o} $$

$$ m∠B < 90^\text{o} $$

$$ m∠C < 90^\text{o} $$

**Obtuse-Angled Triangle:**

**“A triangle having at least one angle greater than 90 degree and other two acute angles is known as an obtuse-angled triangle.”**

Mathematically, we have:

$$ m∠A >90^\text{o} $$

$$ m∠B < 90^\text{o} $$

$$ m∠C < 90^\text{o} $$

Subjecting to a free online triangle calculator assists you in determining all the angles for any above of the triangles.

On the basis of triangle sides, we have following types of triangles:

**Equilateral Triangle:**

**“If all three sides of a triangle are equal, then it is said to be an equilateral triangle”.**

Mathematically, we have:

$$ m∠A = m∠B = m∠C = 60^\text{o} $$

**Isosceles Triangle:**

**“It is a triangle whose at least two sides are equal to each other”.**

The two equal sides of the isosceles triangle have a vertex angle among them. The only side that is different from two of the sides is its base.

Mathematically, we have:

$$ m∠A = 38^\text{o} $$

$$ m∠B = m∠C = 71^\text{o} $$

**Scalene Triangle:**

**“It is a triangle in which all the three sides and angles are not congruent.”**

Mathematically, we have:

$$ Side A ≠ Side B ≠ Side C $$

$$ m∠A ≠ m∠B ≠ m∠C $$

Following are some key features of triangles that you need to keep in mind for sure:

**Sum Of Interior Angles:**

No matter what is the individual measure of all the angles of a triangle. The only fact to be considered is that their sum is always supplementary.

$$ m∠A + m∠B + m∠C = 180^\text{o} $$

**\(90^\text{o}\) Angle Or Greater:**

In case if the interior angles become equal to or greater than 90 degrees, then the resulting image will not be considered a triangle.

**Sum Of Lengths:**

If we add any two sides of the triangle, then their sum is always larger than the third side.

$$ Side A + Side B > Side C $$

**The Law Of Sine:**

By using the law of sine, you can easily find the missing side of a triangle or missing angle of a triangle without any hurdle.

$$ \frac{a}{sin\left(A\right)} = \frac{b}{sin\left(B\right)} = \frac{c}{sin\left(C\right)} $$

For a proper understanding of the law of sine, subject to the free law of sine calculator.

In this section, we will be discussing and mentioning the formula against each single term in a triangle that are very crucial. What you need to do is to stay focused.

Here we have different formulas defined for different cases.

**Case 1:**

When two sides **a** and **b** are given with **∠B**, use the formula given below:

$$ ∠A = sin^{-1}(\frac{a\times sin(B)}{b}) $$

**Case 2:**

When two sides **a** and **b** are given with **∠C**, use the following formula:

$$ ∠A = cos^{-1}(\frac{c^2 + b^2 – a^2}{2cb}) $

**Case 3:**

When two angles are given:

Use the following formula to evaluate the third angle:

$$ m∠A + m∠B + m∠C = 180^\text{o} $$

**Case 4:**

When two sides **a** and **b** are given with angle **A**, use the following formula for missing angles:

$$ ∠B = sin^{-1}(\frac{b\times sin(A)}{a}) $$

Like angles calculations, we have the following cases to determine the sides of the triangle:

There can be numerous ways of determining the area of a triangle that depends upon what basis you will be doing that. Here we have two different formulae to find the area of the triangle which are as follows:

**Given Base And Height:**

If you are given base and height of a triangle, then you can easily calculate the area with the help of the following equation:

$$ Area = \frac{1}{2} Base * Height $$

**Given The Sides:**

If you are given all the three sides of a triangle, then you can find the area by using the following equation:

$$ Area Of Triangle = \sqrt(s\left(s – a\right)\left(s – b\right)\left(s – c\right)) $$

You can also work for the area with the help of the formula given below:

$$ Area = \frac{ab.sin(C)}{2} $$

The perimeter of a triangle can be calculated with the help of the following equation below:

$$ P = a + b + c $$

The formula for semiperimeter is as follows:

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

With area and the sides of the triangle known, you can easily work for the heights by using these equations:

$$ \text{Height }h_a=\frac{2 \times \text{ Area}}{a} $$

$$ \text{Height }h_b=\frac{2 \times \text{ Area}}{b} $$

$$ \text{Height }h_c=\frac{2 \times \text{ Area}}{c} $$

**“A line that is drawn from a vertex and extended to cut the opposite side of that vertex from its center point is called median”.**

The median can be determined using the formula given as follows:

$$ \text{Median }m_a=\sqrt{(\frac{a}{2})^2 + c^2 – ac.cos(B)} $$

$$ \text{Median }m_b=\sqrt{(\frac{b}{2})^2 + a^2 – ab.cos(C)} $$

$$ \text{Median }m_c=\sqrt{(\frac{c}{2})^2 + b^2 – bc.cos(A)} $$

You can find the inradius of a triangle by using the formula given below:

$$ \text{Inradius r}=\frac{area}{s} $$

Working for circumradius of a triangle becomes easier if you use the respective equation below:

$$ \text{Circumradius R}=\frac{a}{2.sin(A)} $$

Let us solve some examples so that you may understand the concept better:

**Example # 01:**

Let x be the area of a triangle. If the sides of the triangle are a = 2, b = 6, and c = 4, find the value of x in a triangle.

**Solution:**

First of all, we need to calculate the value of semiperimeter as follows:

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

$$ s = \frac{2 + 3 + 3}{2} $$

$$ s = \frac{8}{2} $$

$$ s = 4 $$

We know that area of the triangle with all of its sides given can be computed as follows:

$$ Area Of Triangle = \sqrt(s\left(s – a\right)\left(s – b\right)\left(s – c\right)) $$

Putting the values of sides and s:

$$ Area Of Triangle = \sqrt(4\left(4 – 2\right)\left(4 – 3\right)\left(4 – 3\right)) $$

$$ Area Of Triangle = \sqrt(4\left(2\right)\left(1\right)\left(1\right)) $$

$$ Area Of Triangle = \sqrt(\left(8\right) $$

$$ Area Of Triangle = 2.828 $$

Here triangle area calculator helps you to find the same answer but without any hurdle and fastly.

**Example # 02:**

How to calculate angle in a triangle whose two respective are given as:

$$ m∠A = 90^\text{o} $$

$$ m∠B = 45^\text{o} $$

**Solution:**

As we know that sum of the interior angles of a triangle is \(180^\text{o}\):

$$ m∠A + m∠B + m∠C = 180^\text{o} $$

Putting the values of both given angles and simplifying:

$$ 90^\text{o} + 45^\text{o} + m∠C = 180^\text{o} $$

$$ m∠C = 180^\text{o} – 90^\text{o} – 45^\text{o} $$

$$ m∠C = 45^\text{o} $$

A triangle angle calculator can also help you out in finding any missing angle certainly.

**Example # 03:**

The measurement of two angles in a triangle are given as:

$$ m∠A = 60^\text{o} $$

$$ m∠B = 20^\text{o} $$

Now if a particular side is **a = 2**, find the missing side of a triangle.

**Solution:**

We know that:

$$ m∠A + m∠B + m∠C = 180^\text{o} $$

$$ 60^\text{o} + 20^\text{o} + m∠C = 180^\text{o} $$

$$ m∠C = 180^\text{o} – 60^\text{o} – 20^\text{o} $$

$$ m∠C = 100^\text{o} $$

To convert this angle into radians, we have:

$$ m∠C = 100^\text{o} * \frac{π}{180^\text{o} $$

$$ m∠C = 100^\text{o} * \frac{3.14}{180^\text{o} $$

$$ m∠C = \frac{5 * 3.14}{9} $$

$$ m∠C = 1.744 rad $$

The same way we have to convert the given angles A and B:

$$ m∠A = 60^\text{o} * \frac{π}{180^\text{o} $$

$$ m∠A = 60^\text{o} * \frac{3.14}{180^\text{o} $$

$$ m∠A = 1.046 rad $$

$$ m∠B = 20^\text{o} * \frac{π}{180^\text{o} $$

$$ m∠B = 20^\text{o} * \frac{3.14}{180^\text{o} $$

$$ m∠B = 0.34 rad $$

Heading towards triangle measurements for sides b and c:

$$ b = \frac{a\times sin(B)}{sin(A)} $$

$$ b = \frac{2 \times sin(0.34907)}{sin(1.0472)} $$

$$ b = 0.78986 $$

Now we have:

$$ c = \frac{a\times sin(C)}{sin(A)} $$

$$ c = \frac{2 \times sin(1.74533)}{sin(1.0472)} $$

$$ c = 2.27432 $$

It may seem tricky enough but solving triangles with triangle calculator is the best option considered because it is very easy to use it.

**Example # 04:**

Let x be the area in a triangle. The base of a triangle is 3cm and its height is 4cm. How to find the value of x in a triangle?

**Solution:**

The area of triangle is calculated as follows:

$$ Area = \frac{1}{2} Base * Height $$

$$ Area = \frac{1}{2} 3 * 4 $$

$$ Area = \frac{12}{2} $$

$$ Area = 6cm^{2} $$

An area of triangle calculator is the most optimized way of solving triangles precisely.

**Example # 05:**

In a right triangle, the measure of an angle is \(60^\text{o}\).

**Solution:**

As we know that in a right triangle, one angle is always\(90^\text{o}\). So we have:

$$ m∠A + m∠B + m∠C = 180^\text{o} $$

$$ m∠A + 60^\text{o} + 90^\text{o}= 180^\text{o} $$

$$ m∠A = 180^\text{o} – 60^\text{o} + 90^\text{o} $$

$$ m∠A = 30^\text{o} $$

As the given triangle is the right triangle, you can use the right triangle angle calculator to find the missing angler of the triangle.

**Example # 05:**

Predict a triangle along with its terms with the following given information:

$$ a = 2 $$

$$ m∠A = 60^\text{o} $$

$$ m∠B = 20^\text{o} $$

**Solution:**

As we are given two angles and one side, let us start!

**Step # 01 (Calculating The Missing Angle):**

$$ m∠C = 180° – A – B $$

$$ m∠C = 180^\text{o} – 60^\text{o} – 20^\text{o} $$

$$ m∠C = 100^\text{o} $$

Now converting all angles into radians as follows:

$$ m∠A = 60^\text{o} *\frac{π}{180} $$

$$ m∠A = 60^\text{o} *\frac{3.14}{180} $$

$$ m∠A = \frac{188.4}{180} $$

$$ m∠A = 1.0472 rad $$

Similarly:

$$ m∠B = 20^\text{o} *\frac{π}{180} $$

$$ m∠B = 20^\text{o} *\frac{3.14}{180} $$

$$ m∠B = \frac{62.8}{180} $$

$$ m∠B = 0.34907 rad $$

Likewise:

$$ m∠C = 100^\text{o} *\frac{π}{180} $$

$$ m∠C = 100^\text{o} *\frac{3.14}{180} $$

$$ m∠C = \frac{314}{180} $$

$$ m∠C = 1.74533 rad $$

**Step # 02 (How To Find The Side Of A Triangle?):**

As we are given only one side, we need to find the length of triangle sides as follows:

$$ b = \frac{a\times sin(B)}{sin(A)} $$

$$ b = \frac{2 \times sin(0.34907)}{sin(1.0472)} $$

$$ b = 0.78986 $$

Similarly, we have:

$$ c = \frac{a\times sin(C)}{sin(A)} $$

$$ c = \frac{2 \times sin(1.74533)}{sin(1.0472)} $$

$$ c = 2.27432 $$

Here a triangle length calculator can be used to depict the unknown sides of the triangle exactly.

**Step # 03 ( Determining The Area Of The Triangle):**

$$ Area = \frac{ab.sin(C)}{2} $$

$$ Area = \frac{2\times0.78986.sin(1.74533)}{2} $$

$$ Area = 0.77786 $$

Here you can also use area of a triangle calculator to determine more precise outputs.

**Step # 04 (Calculating Perimeter And Semiperimeter):**

$$ \text{Perimeter p} = a + b + c $$

$$ \text{Perimeter p} = 2 + 0.78986 + 2.27432 $$

$$ \text{Perimeter p} = 5.06418 $$

Similarly:

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

$$ \text{Semiperimeter s} = \frac{2 + 0.78986 + 2.27432}{2} $$

$$ \text{Semiperimeter s} = 2.53209 $$

**Step # 05 (Calculation Of Heights Of Triangle Sides):**

We can determine the height of each side as given below:

$$ \text{Height }h_a=\frac{2 \times \text{ Area}}{a} $$

$$ \text{Height }h_a=\frac{2 \times 0.77786}{2} $$

$$ \text{Height }h_a = 0.77786 $$

Now we have:

$$ \text{Height }h_b=\frac{2 \times \text{ Area}}{b} $$

$$ \text{Height }h_b=\frac{2 \times 0.77786}{0.78986} = 1.96961 $$

$$ \text{Height }h_b = 1.96961$$

Similarly:

$$ \text{Height }h_c=\frac{2 \times \text{ Area}}{c} $$

$$ \text{Height }h_c=\frac{2 \times 0.77786}{2.27432} = $$

$$ \text{Height }h_c = 0.68404 $$

With the help of trig triangle calculator, you can solve for triangle heights in a fraction of seconds.

**Step # 06 (Determining Medians Of Each Side):**

$$ \text{Median }m_a=\sqrt{(\frac{a}{2})^2 + c^2 – ac.cos(B)} $$

$$ \text{Median }m_a=\sqrt{(\frac{2}{2})^2 + 2.27432^2 – 2\times2.27432.cos(0.34907)} $$ $$ \text{Median }m_a = 1.37775 $$

Similarly:

$$ \text{Median }m_b=\sqrt{(\frac{b}{2})^2 + a^2 – ab.cos(C)} $$

$$ \text{Median }m_b=\sqrt{(\frac{0.78986}{2})^2 + 2^2 – 2\times0.78986.cos(1.74533)} $$

$$ \text{Median }m_b = 2.10482 $$

Now we have:

$$ \text{Median }m_c=\sqrt{(\frac{c}{2})^2 + b^2 – bc.cos(A)} $$

$$ \text{Median }m_c=\sqrt{(\frac{2.27432}{2})^2 + 0.78986^2 – 0.78986\times2.27432.cos(1.0472)} $$

$$ \text{Median }m_c = 1.00936 $$

**Step # 07 (Finding Inradius):**

$$ \text{Inradius r}=\frac{area}{s} $$

$$ \text{Inradius r}=\frac{0.77786}{2.53209} $$

$$ \text{Inradius r}=0.3072 $$

**Step # 08 (Finding Circumcenter):**

$$ \text{Circumradius R}=\frac{a}{2.sin(A)} $$

$$ \text{Circumradius R}=\frac{2}{2 \times sin(1.0472)} $$

$$ \text{Circumradius R}=1.1547 $$

**Prediction About The Triangle:**

As one angle is \(90^\text{o}\), the given triangle is a right-angled triangle.

The free to use triangle maker is the best method so far to determine each and every term that is associated with any type of triangle. Before getting late, let us tell you the proper use of it!

**Input:**

- You just need to give three values as the input. These include two of the three angles and at least one side of the triangle
- After you feed these values, hit the calculate button

**Output:**

The area of a triangle calculator calculates the following terms:

- Sides of the triangle
**a**,**b**, and**c** - Angles of the triangle that are
**∠A**,**∠B,**and**∠C**, respectively. - Area of the triangle
- Perimeter and semiperimeter of the triangle
- Height of each side which are \(h_{a}\), \(h_{b}\), and \(h_{c}\)
- Median of all three sides that are \(m_{a}\), \(m_{b}\), and \(m_{c}\)
- Inradius of the triangle
- Circumradius of the triangle

An oblique triangle is the one that is either acute or obtuse, but not the right triangle.

The basic trigonometric identities are given as follows:

$$ cos\theta = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} $$

$$ sin\theta = \frac{\text{Opposite Side}}{\text{Hypotenuse}} $$

$$ tan\theta = \frac{\text{Opposite Side}}{\text{Adjacent Side}} $$

The following are the trigonometric functions defined below:

$$ Sine\left(sin\right) $$

$$ Cosine\left(cos\right) $$

$$ Tangent\left(tan\right) $$

$$ Secant\left(sec\right) $$

$$ Cosec\left(cosec\right) $$

$$ Cotangent\left(cot\right) $$

A triangle geometry is a branch of mathematics in which trigonometry revolves. We can not solve any triangle without having knowledge of trigonometry. A free triangle calculator also follows the basic laws of trigonometry to solve for triangles. Professional scholars, engineers, students and even aerospace make use of triangle analysis widely to determine various terms.

From the source of wikipedia: Trigonometric functions, Circumradius, inradius, and exradii, Inequalities, Properties

From the source of khan academy: Non-right triangles & trigonometry, Trig word problem

From the source of lumen learning: SSA Triangles, Key Equations