Calculation of Triangle Side Lengths using Area and Angles

When you have the area and all three angles of a triangle, it's possible to calculate the side lengths efficiently. This method involves using the area formula combined with trigonometric identities such as the Law of Sines and Heron's Formula to find the desired side lengths. Let's break down the process step by step.

Step-by-Step Guide to Finding Triangle Side Lengths

Given:

Step 1: Use the Area Formula

The area of a triangle can be expressed as:

A frac{1}{2}ab sin C

where a and b are the lengths of two sides, and C is the angle between them.

Step 2: Use the Law of Sines

The Law of Sines states that:

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

where R is the circumradius of the triangle.

Step 3: Calculate the Circumradius R

The circumradius can be calculated using the area and the sine of one of the angles:

R frac{abc}{4A}

Alternatively, you can find R using:

R frac{A}{frac{1}{2}ab sin C}

Step 4: Determine the Side Lengths

From the Law of Sines, you can express the sides in terms of R and the angles:

a 2R sin A

b 2R sin B

c 2R sin C

Step 5: Combine the Formulas

Substitute the value of R back into the equations for a, b, and c:

a frac{2A sin A}{A sin C}

b frac{2A sin B}{A sin C}

c frac{2A sin C}{A sin C} 2R sin C

Example Calculation

Suppose the area A 30 square units and the angles are A 30^{circ}, B 60^{circ}, and C 90^{circ}.

Calculate the circumradius R using the Law of Sines:

R frac{A}{frac{1}{2}ab sin C}

Find the sides:

a 2R sin 30^{circ}

b 2R sin 60^{circ}

c 2R sin 90^{circ}

Alternatively, you can use the area to find R and substitute it to find a, b, and c.

Advanced Approach: Using Heron’s Formula

Let's denote these angles as alpha, beta, and gamma clearly their sum is pi.

Let the unknown sides opposite these angles be a, b, and c.

Apply the sine law to compute the sides b and c:

b frac{a sin [beta]}{sin [alpha]}

c frac{a sin [gamma]}{sin [alpha]}

Thus, the only unknown value is the length of side a since the other sides b and c are expressed through a.

Next, find the area using Heron's formula:

area^2 p (p - a) (p - b) (p - c)

where p is the semi-perimeter, p frac{a b c}{2}.

Using this formula, we can find that:

a^2 frac{2 cdot A cdot sin [alpha]}{sin [beta] sin [gamma]}

b^2 frac{2 cdot A cdot sin [beta]}{sin [alpha] sin [gamma]}

c^2 frac{2 cdot A cdot sin [gamma]}{sin [alpha] sin [beta]}

Conclusion

Using the area and the angles, you can derive the side lengths of the triangle through these formulas. If you have specific values for the area and angles, I can help you calculate the exact side lengths!