Area of a Triangle — by Heron’s Formula

Today we're going to discuss Heron's formula. It allows us to find the area of a triangle when we only know the lengths of its sides. Why do you think this might be useful?

It might be helpful for odd-shaped triangles where we can't easily find the height!

Exactly! Let's say we have a triangle with sides measuring 40 m, 32 m, and 24 m. How would you find its area without knowing the height?

Maybe we can just find the height using some other triangles?

While that could be one approach, we can directly use Heron's formula, which saves time! Let's break down how to calculate it.

To use Heron's formula, the first step is to calculate the semi-perimeter, defined as $s = \frac{a + b + c}{2}$. Why do you think it's important to find the semi-perimeter?

I guess it helps to arrange the area calculations based on the triangle's sides?

Good thinking! For our triangle with sides 40, 32, and 24, what is $s$?

It's $(40 + 32 + 24) / 2 = 48$ m!

Exactly! Now with the semi-perimeter calculated, we can proceed to use it in our area formula.

Now that we have our semi-perimeter, let's apply Heron's formula. The formula is: $Area = \sqrt{s(s-a)(s-b)(s-c)}$. Who can help us plug in the values for a, b, c, and s?

So using $s = 48$, $a = 40$, $b = 32$, and $c = 24$, it would be $Area = \sqrt{48(48-40)(48-32)(48-24)}$?

Yes! Can someone now calculate the area using those numbers?

It would be $\sqrt{48 * 8 * 16 * 24} = 384$ m²!

Excellent job! The area is indeed 384 m². Now, let’s see how this relates to finding height in right triangles.

We calculated the area using Heron's formula, but how might we verify it?

We could calculate it using the base and height!

But we don't have the height!

That’s true! However, we can check if this triangle is right-angled. If so, we can use base and height directly. Can anyone find if this triangle is right-angled?

I remember the Pythagorean theorem! We check $32^2 + 24^2 = 40^2$ and it holds true!

Absolutely! So, we could use the base of 32 and height of 24 to calculate the area and find that it matches! This shows our approach is correct.

Heron's formula is versatile! Let’s consider other types of triangles, for example, an equilateral triangle. Do we remember how to apply Heron's formula for this case?

For an equilateral triangle with sides of length a, all sides are the same!

Correct! What would the semi-perimeter and area be if the side length is 10 cm?

Semi-perimeter would be 15 cm, and I think the area is 25√3 cm²?

Well done! You can find the areas for many types of triangles using this formula. Always remember: right triangles, isosceles triangles, and more!