Triangle Calculator

Triangle Facts:

• A triangle cannot have more than one vertex (a point at which or more line segments or edges or curves met) with an inner attitude greater than 90 ranges
• The sum of the interior angles is always same to a hundred and eighty tiers
• The sum of the two interior triangles is equal to the outdoors angles of the triangle
• depending on the indoors angles and length of aspects, we have six styles of triangles as right attitude Triangle, the intense perspective Triangle, the Obtuse perspective Triangle, the Isosceles Triangle, the Scalene Triangle, the equilateral triangle, and many others..

How to Calculate the height and region of a Triangle?

peak of the Triangle:

The formula to calculate the height of a triangle is:

\( h = \dfrac{2A}{b} \)

Where:

• A represents the area of the triangle.
• b is the length of the base.

Area of a Triangle:

To calculate the area of a triangle, use the following formula:

\( \text{Area} = \dfrac{1}{2} \times b \times h \)

Where:

• b is the base of the triangle.
• h is the height of the triangle.

Alternatively, you can calculate the area using Heron’s Formula, which only requires the lengths of the three sides:

\( s = \dfrac{a + b + c}{2} \)

\( \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} \)

Where:

• a, b, c are the sides of the triangle.
• s is the semi-perimeter of the triangle.

How to Find the Sides and Angles of a Triangle?

Triangles are polygons with three vertices. You can calculate their sides and angles using the Pythagorean Theorem, the Law of Sines, and the Law of Cosines.

The Law of Sines:

Using the Law of Sines, you can find the missing side or angle of a triangle:

\( \dfrac{a}{\sin(A)} = \dfrac{b}{\sin(B)} = \dfrac{c}{\sin(C)} \)

For a right triangle with one angle and the hypotenuse given:

a = c \cdot \sin(\alpha) or a = c \cdot \cos(\beta)

b = c \cdot \sin(\beta) or b = c \cdot \cos(\alpha)

Pythagorean Theorem:

For right triangles, the Pythagorean theorem helps find the third side:

\( a^2 + b^2 = c^2 \)

If one side is missing, rearrange the formula:

To find side a:

\( a = \sqrt{c^2 - b^2} \)

To find the hypotenuse c:

\( c = \sqrt{a^2 + b^2} \)

The Law of Cosines:

The Law of Cosines is used to find a side or angle when other values are known:

\( a^2 = b^2 + c^2 - 2bc \cdot \cos(A) \)

Solving for \( \cos(A): \)

\( \cos(A) = \dfrac{b^2 + c^2 - a^2}{2bc} \)

Similarly:

\( b^2 = a^2 + c^2 - 2ac \cdot \cos(B) \)

\( \cos(B) = \dfrac{a^2 + c^2 - b^2}{2ac} \)

\( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \)

\( \cos(C) = \dfrac{a^2 + b^2 - c^2}{2ab} \)

Using Trigonometric Functions to Find Missing Sides:

If one angle and one side are given:

a = b \cdot \tan(\alpha)

b = a \cdot \tan(\beta)

If the area and one side are given:

\( \text{Area} = \dfrac{a \cdot b}{2} \)

To find side b:

\( b = \dfrac{2 \cdot \text{Area}}{a} \)

To find the hypotenuse c:

\( c = \sqrt{a^2 + \left( \dfrac{2 \cdot \text{Area}}{a} \right)^2} \)

How To Calculate The Inradius And Circumradius?

Inradius:

It is the radius of the inscribed circle. In general, it's the circle that can fit inside the triangle and is perpendicular to each side of the polygon. 

Use the below-mentioned formula to calculate the inradius of a triangle:

\(\ r=\dfrac{Area}{semiperimeter}\)

Circumradius:

As the name shows it is the radius of the circumscribed circle. It is the minimum size of the circle that can fit inside the triangle.

Let's see the following formula that is used to calculate the circumradius of the triangle:

\(\ R=\dfrac{a}{2\sin(A)}\)

Example:

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

• \(\ a = 4\)
• \(\ m∠A = 50^{o}\)
• \(\ m∠B = 40^{o}\)

Solution:

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

Step # 01 (Find Angle of Triangle):

\(\ m∠C = 180° - A - B\)

\(\ m∠C = 180^{o} - 50^{o} - 40^{o}\)

\(\ m∠C = 90^{o}\)

Now converting all angles into radians as follows:

\(\ m∠A = 50^{o}\times \frac{\pi}{180}\)

\(\ m∠A = 0.87266\ rad\)

Similarly:

\(\ m∠B = 40^{o}\times \frac{\pi}{180}\)

\(\ m∠B = 0.69813\ rad\)

Likewise:

\(\ m∠C = 90^{o}\times \frac{\pi}{180}\)

\(\ m∠C = 1.5708\ rad\)

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

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

\(\ b = \frac{4 \times \sin(0.69813)}{\sin(0.87266)}\)

\(\ b = 3.23728\)

Similarly:

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

\(\ c = \frac{4 \times \sin(1.5708)}{\sin(0.87266)}\)

\(\ c = 4.58257\)

Step # 03 (Calculating The Triangle Area):

\(\ A = \frac{ab\sin(C)}{2}\)

\(\ A = \frac{4\times3.23728\sin(1.5708)}{2}\)

\(\ A = 6.47456\)

Step # 04 (Calculating Perimeter And Semiperimeter):

\(\ Perimeter=\ p = a + b + c\)

\(\ Perimeter=\ p = 4 + 3.23728 + 4.58257\)

\(\ Perimeter=\ p = 11.81985\)

Similarly:

\(\ Semiperimeter=\ s = \frac{a + b + c}{2}\)

\(\ Semiperimeter=\ s = \frac{4 + 3.23728 + 4.58257}{2}\)

\(\ Semiperimeter=\ s = 5.90993\)

Step # 05 (Calculation of Heights of Triangle Sides):

\(\ Height=\ h_{a}=\frac{2 \times { Area}}{a}\)

\(\ Height=\ h_{a}=\frac{2 \times 6.47456}{4}\)

\(\ Height=\ h_{a} = 3.23728\)

Similarly:

\(\ Height=\ h_{b}=\frac{2 \times { Area}}{b}\)

\(\ Height=\ h_{b}=\frac{2 \times 6.47456}{3.23728}\)

\(\ Height=\ h_{b} = 4.00000\)

And:

\(\ Height=\ h_{c}=\frac{2 \times { Area}}{c}\)

\(\ Height=\ h_{c}=\frac{2 \times 6.47456}{4.58257}\)

\(\ Height=\ h_{c} = 2.82355\)

Step # 06 (Determining Medians Of Each Side):

\(\ Median=\ m_{a}=\sqrt{(\frac{a}{2})^2 + c^2 - ac\cos(B)}\)

\(\ Median=\ m_{a}=\sqrt{(\frac{4}{2})^2 + 4.58257^2 - 4\times4.58257\cos(0.69813)}\)

\(\ Median=\ m_{a} = 2.97665\)

Similarly:

\(\ Median=\ m_{b}=\sqrt{(\frac{b}{2})^2 + a^2 - ab\cos(C)}\)

\(\ Median=\ m_{b}=\sqrt{(\frac{3.23728}{2})^2 + 4^2 - 4\times3.23728\cos(1.5708)}\)

\(\ Median=\ m_{b} = 3.63730\)

Now:

\(\ Median=\ m_{c}=\sqrt{(\frac{c}{2})^2 + b^2 - bc\cos(A)}\)

\(\ Median=\ m_{c}=\sqrt{(\frac{4.58257}{2})^2 + 3.23728^2 - 3.23728\times4.58257\cos(0.87266)}\)

\(\ Median=\ m_{c} = 3.86590\)

Step # 07 (Finding Inradius):

\(\ Inradius=\ r=\frac{Area}{s}\)

\(\ Inradius=\ r=\frac{6.47456}{5.90993}\)

\(\ Inradius\ r=1.0956\)

Step # 08 (Finding Circumradius):

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

\(\ Circumradius\ R=\frac{4}{2 \times \sin(0.87266)}\)

\(\ Circumradius\ R=2.2942\)