Triangle Calculation: Find C

Preface

Learn about triangle calculation to find H, triangle area calculator and right angled triangle calculation in the concept of triangle calculation to find C. Check out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

Become a champ of triangle calculation to find C in just 4 minutes!

Introduction

The Law of Sines was provided by Persian mathematicians in the 10th century.

Table of Contents 

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General case

"C" generally represents the third side in a triangle.

Right-angled triangle

In case of a right-angled triangle, it is referred to as "c" or the hypotenuse (h).

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General case

To calculate "c" in any triangle we use sine and cosine formulas given as below:

Law of sines:

The Law of Sines is given by the following formula:

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

It is given as the ratio of side to the sine of the opposite angle.

Law of cosines:

The Law of Cosines is given by the following formula:

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

It is given as the ratio of subtraction of length of opposite side squared from the  sum of the lengths of  adjacted sides squared to the product of two times the lengths of adjacent sides.

    

To find the area of triangle having sides A,B and C, we use Heron's formula.

Perimeter of triangle = ( A+B+C ) \( S = \dfrac{A+B+C}{2}\) \(Area = \sqrt{S(S-A)(S-B)(S-C)}\)

Let's try to apply our learning on Heron's formula using the below given simulation.

Special Case: Right-angled triangle

In case of right angled triangle, the side "c" (or hypotenuse) is calculated using the Pythagorean Theorem. According to the theorem, the sum of the squares of base and perpendicular is always equal to the square of the hypotenuse. 

It is given by the following formula:

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

Area of triangle as per the above figure will be given by formula,

A = \( \dfrac{1}{2} \times b \times a \)

Try your hand at the simulation to find the area of triangle.

 

Let's learn how to calculate value of c in a triangle.

Example 1

Claire drew a triangle with 2 sides and 3 angles as given below. Help her do the triangle calculation to find C and angles A and C. 

To find the third side, Claire uses the Law of Sines which is given as follows:

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

Given the value of sides a and b, 8 units and 12 units respectively and A = 30^∘. On applying the Law of Sines she gets,

\( \dfrac{8}{\sin 30} \)= \( \dfrac{12}{\sin B} \)

Hence, \( \sin B \) = \( \dfrac{12 \times \sin 30}{8} \)

\( \sin B = \dfrac{6}{8} \)           

sin 30^∘ = \(\dfrac{1}{2} \), as refererred from trigonometric values.

B = \( \sin^{ - 1} 0.75 \)

B =  48.6^∘ 

Sum of all angles in a triangle is 180^∘

Hence, A + B + C = 180^∘ 

30^∘ + 48.6^∘ + C = 180^∘ 

C = 180^∘ - (30^∘ + 48.6^∘) = 101.4^∘ 

Hence, by the Law of Sines,

\( \dfrac{12}{\sin 48.6} = \dfrac{c}{\sin 101.4} \)

c = \( \dfrac{12 \times \sin 101.4}{\sin 48.6} \)

c = 15.8 units

Example 2

Isabella took an example of a right-angled triangle. The area of triangle was 30 sq.unit having length of base equal to 5 units. Help Isabella do the triangle calculation to find H. 

Isabella knows area of triangle = \dfrac{1}{2} \times b \times a.

Hence, 

\( 30 = \dfrac{1}{2} \times 5 \times a \)

\( a = \dfrac{30 \times 2}{5} = 12 \)

To find the third side "c" Pythogoras Theorem will be used which is given by the following formula,

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

\( 12^2+5^2 = c^2 \)

\( c^2 = 144 + 25 = 169 \)

\( c = \sqrt{169} = 13 \)

Hence, c = 13 units.

Example 2

 

 

How will Leonard do the triangle calculation to find hypotenuse of a right angled triangle if the area of triangle is 84 unit² and the perpendicular is 24 units?   

Solution

Leonard knows area of triangle = \( \dfrac{1}{2} \times b \times a \).

Hence, 

\( 84 = \dfrac{1}{2} \times b \times 24 \)

\( b = \dfrac{84 \times 2}{24} = 7 \)

To find the third side c Pythogoras Theorem will be used which is given by the following formula,

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

\( 24^2 + 7^2 = c^2 \)
\( c^2 = 576 + 49 = 625 \)

\( c = \sqrt{625} = 25 \) units

\(\therefore\) Leonard found the length of third side to be 25 units.

Example 3

 

 

How will Jenny prove that when the Law of Cosines is applied to the right angled triangle, it'll result in formula of Pythagorean Theorem?

Solution

Let's say Jenny assumes angle C  = 90^∘ 

The Law of Cosines for angle C will be written as,

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

As, cos 90^∘ = 0 

Applying in the above formula she will get, 

\( \dfrac{a^2+b^2-c^2}{2ab} = 0 \)

On cross multiplying 2ab she'll get,

\( a^2+b^2-c^2 = 0 \)

This final equation is the formula of Pythagorean Theorem.

Hence, the final equation obtained is \( a^2+b^2 = c^2 \).

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Conclusion

We hope you enjoyed learning about triangle calculation to find C with the simulations and practice questions. Now, you will be able to easily solve problems on triangle calculation to find H, triangle area calculator and right angled triangle calculation.

You can also simplify this topic with our Math Experts in Cuemath’s LIVE and interactive online classes. Book a FREE trial class today!