Loading the lesson
Loading the lesson
Search AllRounder.ai
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Search the courses, subjects, tracks, games and features, or jump straight to a page.
Learn it your way
A short class with Sarah and her students. The other ways are under More.
Sign up free to listen. Transcripts are free to read.
Today, we're going to learn about Heron's formula, a handy way to calculate the area of any triangle when you know the lengths of its sides. Can anyone tell me what the formula is?
Is it the one that uses the semi-perimeter?
Exactly! The semi-perimeter $s$ is key to the formula. We calculate it by taking half the sum of all three sides: $s = \frac{a + b + c}{2}$.
How do we use that to find the area?
Great question! The area is calculated using: $$ ext{Area} = \sqrt{s(s - a)(s - b)(s - c)} $$. We'll apply this to a triangular park example.
Let’s apply what we’ve learned to find the area of a triangular park with sides of 40 m, 32 m, and 24 m. Who can help me start with calculating the semi-perimeter?
So, $s = \frac{40 + 32 + 24}{2} = 48 \text{ m}$!
Fantastic! Now let's find the areas, we need to calculate $s - a, s - b,$ and $s - c$. What do we get?
We get \(s - a = 8 \text{ m}, s - b = 16 \text{ m}, s - c = 24 \text{ m}.\
Exactly! Now we can substitute these values into the formula and calculate the area!
So we've calculated the area using Heron's formula and found it to be 384 m². Can anyone remind me how we could verify this?
We could check if the triangle is a right triangle and use the base and height instead!
Correct! We observe that $32^2 + 24^2 = 40^2$, confirming it is a right triangle. The area calculation using $\frac{1}{2} \times \text{base} \times \text{height} = 384 \text{ m}^2$ also checks out.
This method really shows how flexible these formulas can be!
Let’s take a look at another example. Suppose we have an equilateral triangle with each side of 10 cm. Who remembers how to find the area using Heron’s formula?
We calculate the semi-perimeter first, right? So, $s = \frac{10 + 10 + 10}{2} = 15 \text{ cm}$!
Great! Now, what’s next?
We substitute into the area formula: \[ \text{Area} = \sqrt{15(15 - 10)(15 - 10)(15 - 10)} = 25\sqrt{3} \text{ cm}². \
Well done! Let’s try one more to solidify this concept before we move on to exercises.
{'example': 'Calculate the area of a triangular park with sides 40 m, 32 m, and 24 m.', 'solution': '$$ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{48(48-40)(48-32)(48-24)} = \sqrt{48(8)(16)(24)} = 384 \text{ m}^2 $$'}
{'example': 'Find the area of an equilateral triangle with each side 10 cm.', 'solution': '$$ s = \frac{10 + 10 + 10}{2} = 15 \text{ cm}, \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{15(15-10)(15-10)(15-10)} = 25\sqrt{3} \text{ cm}^2 $$'}
To find triangle area with sides three, add them and divide by two, you'll see.
Imagine a triangle wondering how to measure its area without height. It discovers Heron’s formula and becomes famous for its versatility.
SAS (semi-perimeter, area, sides) to remember the steps clearly.
HERO - Height? Easily Replaced by Other methods!
Term
What does Heron's formula calculate?
Click to flip
Term
Define semi-perimeter.
Click to flip
A formula to calculate the area of a triangle when the lengths of all three sides are known.
Half the sum of the lengths of the sides of a triangle, used in calculating area via Heron's formula.
A triangle with all sides of different lengths.
The total length of the sides of a polygon.
Heron's formula allows us to calculate the area of a triangle when only the lengths of its sides are known, without needing to determine the height.
In this section, we learn how to apply Heron's formula to find the area of various triangles, including an example involving a triangular park. We understand how the semi-perimeter is calculated and its role in determining the area, even for scalene triangles where height is not readily available.
In this section, we explore the application of Heron's formula to find the area of a triangle when the side lengths are known, but the height is not. Heron's formula states that the area of a triangle can be calculated using the semi-perimeter $s = \frac{a + b + c}{2}$, where $a, b,$ and $c$ are the lengths of the sides of the triangle. The area is then given by $$ ext{Area} = \sqrt{s(s - a)(s - b)(s - c)} $$. We apply this formula to a triangular park with sides measuring 40 m, 32 m, and 24 m, leading to a calculated area of 384 m². We also verify this with a right triangle calculation. Through examples and exercises, we solidify our understanding of how to use Heron's formula effectively.