Triangle Calculator

How to Calculate Triangle Properties

Triangles are fundamental shapes in geometry with unique properties that make them essential in various mathematical and real-world applications. Here's a comprehensive guide on how to calculate triangle properties:

Triangle Formulas

1. Law of Sines:

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

   Where a, b, and c are the side lengths, and A, B, and C are the angles opposite to these sides, respectively.

2. Law of Cosines:

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

   This formula relates the lengths of the sides to the cosine of one of the angles.

3. Area (Heron's formula):

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

   Where s is the semiperimeter: \( s = \frac{a + b + c}{2} \)

4. Perimeter:

   \( Perimeter = a + b + c \)

5. Inradius:

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

   Where r is the radius of the inscribed circle.

6. Circumradius:

   \( R = \frac{abc}{4 \cdot Area} \)

   Where R is the radius of the circumscribed circle.

Calculation Steps

1. Identify the known values (side lengths and/or angles).
2. Determine which case applies (SSS, SAS, ASA, AAS, etc.).
3. Use the appropriate formula(s) to find unknown sides or angles:
   • For SSS, use the Law of Cosines to find angles.
   • For SAS, use the Law of Cosines to find the third side, then Law of Sines for remaining angles.
   • For ASA or AAS, use the Law of Sines to find unknown sides.
4. Calculate the perimeter by adding all side lengths.
5. Calculate the semiperimeter (s).
6. Use Heron's formula to calculate the area.
7. Calculate the inradius and circumradius using their respective formulas.

Example Calculation

Let's solve a triangle with side a = 4 units, side b = 5 units, and angle C = 60°.

1. Given: a = 4 units, b = 5 units, C = 60°
2. Use the Law of Cosines to find side c:

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

   \( c^2 = 4^2 + 5^2 - 2(4)(5) \cos 60° \)

   \( c^2 = 16 + 25 - 40(0.5) = 21 \)

   \( c = \sqrt{21} \approx 4.58 \) units

3. Use the Law of Sines to find angles A and B:

   \( \sin A = \frac{a \sin C}{c} = \frac{4 \sin 60°}{4.58} \approx 0.7544 \)

   \( A = \arcsin(0.7544) \approx 48.89° \)

   \( B = 180° - A - C = 180° - 48.89° - 60° \approx 71.11° \)

4. Calculate perimeter:

   \( Perimeter = 4 + 5 + 4.58 = 13.58 \) units

5. Calculate semiperimeter:

   \( s = \frac{13.58}{2} = 6.79 \) units

6. Calculate area using Heron's formula:

   \( Area = \sqrt{6.79(6.79-4)(6.79-5)(6.79-4.58)} \approx 9.93 \) square units

7. Calculate inradius:

   \( r = \frac{9.93}{6.79} \approx 1.46 \) units

8. Calculate circumradius:

   \( R = \frac{4 \cdot 5 \cdot 4.58}{4 \cdot 9.93} \approx 2.31 \) units

Visual Representation

This diagram illustrates the triangle from our example, showing how the Law of Sines and Law of Cosines relate the sides and angles of the triangle.