Understanding 30°-60°-90° Triangle Perimeter and Area

A 30°-60°-90° triangle is a special right triangle with angles of 30°, 60°, and 90°. The sides of this triangle are in the ratio 1:sqrt{3}:2. Given that the hypotenuse is 28 centimeters, how can we determine the perimeter and area of this triangle?

Using Trigonometry to Find Sides

The hypotenuse of a 30°-60°-90° triangle is twice the length of the smallest side. Using the sine function, we can find the lengths of the other two sides.

Let's start by finding the side opposite the 30° angle. Using sin(30°) 0.5 and the formula opposite side / hypotenuse 0.5, we calculate:

Side opposite to 30° 0.5 * 28 14 cm

Next, we need to find the side opposite the 60° angle. We use the Pythagorean theorem a^2 b^2 c^2, where c 28 cm (hypotenuse), a 14 cm (opposite to 30°), and b ? (opposite to 60°).

Substitute the sides into the formula:

b^2 c^2 - a^2

b^2 28^2 - 14^2

b^2 784 - 196

b^2 588

b sqrt{588} 24.25 cm

Now that we have the side lengths, we can determine the perimeter and area of the triangle.

Calculating the Perimeter

The perimeter of a triangle is the sum of its sides. For the 30°-60°-90° triangle, the perimeter is calculated as follows:

Perimeter 28 cm 14 cm 24.25 cm 66.25 cm

Calculating the Area

The area of a triangle can be found using the formula 1/2 * base * height. In a 30°-60°-90° triangle, the area is calculated as follows:

Area 1/2 * 14 cm * 24.25 cm

Area 169.75 square centimeters

Alternative Method Using the Ratio

Another way to find the perimeter and area is to use the special ratio of a 30°-60°-90° triangle. The sides are in the ratio 1:sqrt{3}:2. With the hypotenuse known, we can directly calculate the other sides:

Short leg 28 * sin(30°) 14 cm Long leg 28 * sin(60°) 28 * sqrt{3}/2 14 sqrt{3} cm

Using these side lengths, we can again calculate the perimeter and area:

Perimeter 28 cm 14 cm 14 * sqrt{3} cm ≈ 66.25 cm Area 1/2 * 14 cm * 14 * sqrt{3} cm ≈ 169.75 square centimeters

Understanding these calculations is crucial when dealing with 30°-60°-90° triangles. Whether you use trigonometric functions or the specific ratio, the perimeter and area of such triangles can be accurately determined.