How to Calculate the Area of Triangle When One Side Is Given

How to Calculate the Area of Triangle When One Side Is Given
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Geometry is the study of shapes and figures that take up a given space. Geometric problems try to identify the size and scope of those shapes by solving mathematical equations. Geometry problems have two types of information: "givens" and "unknowns." The givens represent the information in the problem that is given to you. The unknowns are the pieces of the equation you must solve. It is possible to find the area of a triangle with only one side length given. However, to solve the problem, you also need to know two of the interior angles.

TL;DR (Too Long; Didn't Read)

To calculate the area of a triangle given one side and two angles, solve for another side using the Law of Sines, then find the area with the formula: area = 1/2 × ​b​ × ​c​ × sin(A).

Find Third Angle

Determine the third angle of the triangle. For example, the sample problem has a triangle where side ​B​ is 10 units. Both angle ​A​ and angle ​B​ are 50 degrees. Solve for angle ​C​. Math law states that the angles of a triangle add up to 180 degrees, therefore

\text{Angle} A + \text{Angle} B + \text{Angle} C = 180.

Insert the given angles into the equation.

50 + 50 + C = 180

Solve for ​C​ by adding the first two angles and subtracting from 180.

180 - 100 = 80

Angle ​C​ is 80 degrees.

Set up Rule of Sines

Use the sine rule to re-write the equation. The sine rule is a mathematical rule that aids in solving unknown angles and lengths. It states:

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

In the equation the small ​a​, ​b​ and ​c​ represent the lengths, while the capital ​A​, ​B​ and ​C​ represent the internal angles of the triangle. Because all portions of the equation equal each other, you can use any two portions. Use the portion for the side you were given. In the sample problem this is side ​B​, 10 units.

Following the laws of math re-write the equation as:

c = \frac{b \sin C}{\sin B}

The small ​c​ represents the side you are solving for. The capital ​C​ is moved to the numerator on the opposite side of the equation because according to the laws of math you must isolate ​c​ in order to solve for it. When moving a denominator, it goes to the numerator so you can later multiply it.

Solve Rule of Sines

Insert the givens into your new equation.

c = \frac{10 × \sin (100)}{\sin (50)}

Place this into your geometry calculator to return a result of:

c = 12.86

Find Triangle Area

Solve for the area of the triangle. To find the area of a triangle you need two side lengths which you have now obtained. One equation for the area of a triangle is

\text{area} = \frac{1}{2} × b × c × \sin(A)

The "​b​" and "​c​" represent two sides and ​A​ is the angle between them.

Therefore:

\begin{aligned} \text{area} &= 0.5 × 10 × 12.86 × \sin(50) \\ &= 49.26 \text{ units}^2 \end{aligned}

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