Finding the Area of an Equilateral Triangle from Perimeter

Let's find the area of an equilateral triangle when its perimeter is given. We are given that the perimeter of the equilateral triangle is 48 cm.

Understanding Equilateral Triangles

An equilateral triangle is a special type of triangle where all three sides are equal in length, and all three internal angles are equal (each measuring 60 degrees). The perimeter of any polygon is the sum of the lengths of its sides.

Calculating the Side Length

Since all three sides of an equilateral triangle are equal, we can find the length of one side by dividing the perimeter by 3.

Let 'a' be the length of one side of the equilateral triangle.

Perimeter $= a + a + a = 3a$

Given Perimeter $= 48$ cm

So, $3a = 48$ cm

To find the side length 'a', we divide 48 by 3:

$$a = \frac{48}{3}$$

$$a = 16 \text{ cm}$$

The length of each side of the equilateral triangle is 16 cm.

Area Formula for an Equilateral Triangle

The area of an equilateral triangle can be calculated using the formula:

$$\text{Area} = \frac{\sqrt{3}}{4} \times \text{side}^2$$

Where 'side' is the length of one side of the triangle.

Calculating the Area

Now we substitute the side length ($a = 16$ cm) into the area formula:

$$\text{Area} = \frac{\sqrt{3}}{4} \times (16)^2$$

$$\text{Area} = \frac{\sqrt{3}}{4} \times (16 \times 16)$$

$$\text{Area} = \frac{\sqrt{3}}{4} \times 256$$

We can simplify this expression by dividing 256 by 4:

$$256 \div 4 = 64$$

So the area is:

$$\text{Area} = \sqrt{3} \times 64$$

$$\text{Area} = 64\sqrt{3} \text{ cm}^2$$

The area of the equilateral triangle is $64\sqrt{3}$ cm$^2$.

Summary of Steps

• Determine the side length from the perimeter.
• Use the formula for the area of an equilateral triangle.
• Substitute and calculate.

Revision Table: Equilateral Triangle Properties

Additional Information: Geometric Formulas

Understanding basic geometric formulas is key to solving problems involving shapes like triangles. For equilateral triangles, knowing the relationship between side length, perimeter, area, and altitude is very helpful. Remember that $\sqrt{3}$ is an irrational number, approximately equal to 1.732.

This problem specifically uses the perimeter to find the side length, and then uses the side length to find the area. Similar problems might involve using the area to find the side length or perimeter, or using the altitude to find other properties.