Area Of An Equilateral Triangle With A Perimeter Of 36 Cm

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Hey everyone! Let's tackle this geometry problem together. We're given an equilateral triangle with a perimeter of 36 centimeters, and our mission is to find its area, rounded to the nearest square centimeter. Sounds like fun, right? Let's dive in!

Understanding Equilateral Triangles

First things first, let's make sure we're all on the same page about what an equilateral triangle is. Equilateral triangles are special because they have three equal sides and three equal angles. Since all angles in a triangle add up to 180 degrees, each angle in an equilateral triangle is always 60 degrees. This symmetry is super helpful when we're trying to calculate things like area.

Now, when you think about finding the area, you might immediately recall the standard formula: Area = (1/2) * base * height. That's a great formula, but to use it, we need to know the base and the height. In our case, we only have the perimeter. So, how do we get the height from the perimeter? That’s the puzzle we need to solve!

The perimeter is simply the total length of all the sides added together. Since our triangle is equilateral, all three sides are the same length. If the total perimeter is 36 cm, we can easily find the length of one side by dividing the perimeter by 3. This gives us 36 cm / 3 = 12 cm. So, each side of our equilateral triangle is 12 cm long.

Finding the Height

Okay, we know the side length, but we still need the height to use the (1/2) * base * height formula. Here's where a little bit of geometry magic comes in. Imagine drawing a line from one vertex (corner) of the triangle straight down to the midpoint of the opposite side. This line is the height of the triangle, and it also splits our equilateral triangle into two identical right-angled triangles.

When we find the height it's important to remember that the height bisects the base, creating two right-angled triangles. This is crucial because it allows us to use the Pythagorean theorem, a² + b² = c², where c is the hypotenuse (the longest side, which is the side of our equilateral triangle), and a and b are the other two sides (one of which is the height, and the other is half the base).

In our right-angled triangle, the hypotenuse is 12 cm (the side of the equilateral triangle), one side (half the base) is 6 cm (half of 12 cm), and the other side is the height (which we'll call h). Plugging these values into the Pythagorean theorem, we get:

6² + h² = 12²

36 + h² = 144

h² = 144 - 36

h² = 108

h = √108

So, the height (h) is the square root of 108, which is approximately 10.39 cm.

Calculating the Area

Now that we have the height, we can finally calculate the area using our formula: Area = (1/2) * base * height.

The base of our triangle is the full side length, which is 12 cm. The height we just calculated is approximately 10.39 cm. Plugging these values in:

Area = (1/2) * 12 cm * 10.39 cm

Area = 6 cm * 10.39 cm

Area ≈ 62.34 cm²

Rounding to the Nearest Square Centimeter

The question asks us to round the area to the nearest square centimeter. Our calculated area is approximately 62.34 cm². Since 0.34 is less than 0.5, we round down to 62 cm².

Therefore, the area of the equilateral triangle, rounded to the nearest square centimeter, is 62 square centimeters. So, the correct answer is C. 62 square centimeters.

Heron's Formula: An Alternative Approach

Now, you might have noticed that the question also mentions Heron's formula. Heron's formula is a neat way to find the area of a triangle when you know the lengths of all three sides, but not necessarily the height. It's especially useful for triangles that aren't right-angled, but it works perfectly well for equilateral triangles too!

Heron's formula looks a bit intimidating at first, but it's not too bad once you break it down. The formula is:

Area = √[s(s - a)(s - b)(s - c)]

Where:

  • Area is the area of the triangle.
  • a, b, and c are the lengths of the sides of the triangle.
  • s is the semi-perimeter of the triangle (half the perimeter), calculated as s = (a + b + c) / 2.

Let's use Heron's formula to solve our problem and see if we get the same answer. This will be a great way to double-check our work!

Applying Heron's Formula

  1. Find the semi-perimeter (s):

    Since our equilateral triangle has sides of 12 cm each, the perimeter is 36 cm. The semi-perimeter is half of that, so:

    s = 36 cm / 2 = 18 cm

  2. Plug the values into Heron's formula:

    Area = √[s(s - a)(s - b)(s - c)]

    In our case, a = 12 cm, b = 12 cm, c = 12 cm, and s = 18 cm. So:

    Area = √[18(18 - 12)(18 - 12)(18 - 12)]

    Area = √[18(6)(6)(6)]

    Area = √[18 * 216]

    Area = √[3888]

  3. Calculate the square root:

    Area ≈ 62.35 cm²

  4. Round to the nearest square centimeter:

    As before, 62.35 cm² rounds down to 62 cm².

Woo-hoo! We got the same answer using Heron's formula! This confirms that our earlier calculation using the standard area formula was correct.

Key Takeaways

Let's recap what we've learned in this problem:

  1. Equilateral triangles have three equal sides and three equal angles (60 degrees each).
  2. The perimeter of a triangle is the sum of its side lengths.
  3. We can find the height of an equilateral triangle by drawing a line from a vertex to the midpoint of the opposite side, creating two right-angled triangles.
  4. The Pythagorean theorem (a² + b² = c²) is a powerful tool for finding missing side lengths in right-angled triangles.
  5. The standard formula for the area of a triangle is Area = (1/2) * base * height.
  6. Heron's formula is an alternative method for finding the area of a triangle when you know the lengths of all three sides.

By understanding these concepts and practicing problems like this, you'll become a geometry whiz in no time! Keep up the great work, guys!

Practice Makes Perfect

Solving math problems can be like learning a new language—the more you practice, the better you get. Try tackling similar problems with different perimeters or side lengths. You could also explore other types of triangles, like isosceles or scalene triangles, and see how the formulas and approaches change.

Remember, math isn't just about memorizing formulas; it's about understanding the underlying concepts and how they connect. The more you explore and experiment, the more confident and skilled you'll become. So keep practicing, keep asking questions, and most importantly, keep having fun with math!

Final Thoughts

So, there you have it! We successfully found the area of an equilateral triangle with a perimeter of 36 centimeters, using both the standard area formula and Heron's formula. We also reinforced some key concepts about equilateral triangles and the Pythagorean theorem. You guys did awesome! Keep up the great work, and I'll see you in the next math adventure!