Calculating the Perimeter of a 30-60-90 Triangle with a Hypotenuse of 8 cm

The 30-60-90 triangle is a special right triangle with well-defined ratios for its sides. This article will explain how to find the perimeter of such a triangle when the hypotenuse is given, using a step-by-step approach and relevant examples.

Understanding the Properties of a 30-60-90 Triangle

The sides of a 30-60-90 triangle adhere to a specific ratio:

The side opposite the 30-degree angle (shortest side) is x. The side opposite the 60-degree angle is xsqrt;3. The hypotenuse opposite the 90-degree angle is 2x.

Given that the hypotenuse is 8 cm, we can set up an equation to solve for x.

Setting Up and Solving the Equation

Given that the hypotenuse is 8 cm, we can set up the equation:

2x 8

Solving for x:

x 8 / 2 4 cm

With x known, we can determine the lengths of the other two sides:

The side opposite the 30-degree angle (shortest side): x 4 cm The side opposite the 60-degree angle: xsqrt;3 4sqrt;3 cm

Calculating the Perimeter of the Triangle

Now that we have the lengths of all three sides, we can calculate the perimeter P by adding them together:

P 4 4sqrt;3 8

Letrsquo;s simplify the expression:

P 12 4sqrt;3 cm

To get an approximate decimal value, we can replace sqrt;3 with its approximate value (1.732):

P asymp; 12 4(1.732) 12 6.93 18.93 cm

An Alternative Method Using Trigonometry

We can also use trigonometric functions to solve the problem. For a 30-60-90 triangle with a hypotenuse of 8 cm:

Shorter leg SL opposite the 30° angle: 1/2 * 8 4 cm Longer leg LG opposite the 60° angle: 8sqrt;3 / 2 4sqrt;3 cm

The perimeter can then be calculated as:

P 8 4 4sqrt;3 12 4sqrt;3 cm asymp; 18.93 cm

Conclusion

Understanding the properties and ratios of a 30-60-90 triangle simplifies the process of finding the perimeter when the hypotenuse is given. By following the steps outlined above, you can efficiently calculate the perimeter and obtain an accurate result.

Key Takeaways:

The ratios of the sides in a 30-60-90 triangle are 1 : sqrt;3 : 2. The hypotenuse is always twice the length of the shortest side. Using the known side and the properties of this special triangle helps in finding the perimeter accurately.

For further exploration, you can experiment with different values of the hypotenuse in a 30-60-90 triangle and calculate their perimeters to deepen your understanding.