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10.1.2. Application of Heron's formula to a triangular park
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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.
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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!
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Create a free accountSo 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!
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Create a free accountLet’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.
Overview
Short Summary
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.
Medium Summary
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.
Detailed Summary
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.
Reference YouTube Videos
Audio Book
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Create a free accountThe formula given by Heron about the area of a triangle, is also known as Hero’s formula. It is stated as:
Area of a triangle = √(s(s − a)(s − b)(s − c))
where a, b and c are the sides of the triangle, and s = semi-perimeter, i.e., half the perimeter of the triangle = (a + b + c)/2.
Detailed Explanation
Heron's formula allows us to calculate the area of a triangle when we know the lengths of all three sides. The semi-perimeter 's' is calculated first, by summing the lengths of the sides and dividing by two. Then the area can be calculated using the formula, where 'a', 'b', 'c' are the side lengths of the triangle.
Examples & Analogies
Imagine you have a triangular garden, but you don't want to measure the height. Instead, you just measure the three sides. Using Heron's formula is like having a cheat sheet that helps you find the area without needing to calculate the height, just like measuring and dividing the ingredients for a recipe instead of having to estimate how much to use visually.
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Create a free accountLet us apply it to calculate the area of the triangular park ABC, mentioned above. Let us take a = 40 m, b = 24 m, c = 32 m,
so that we have s = (40 + 24 + 32) / 2 = 48 m.
s − a = (48 − 40) = 8 m, s − b = (48 − 24) = 24 m, s − c = (48 − 32) = 16 m.
Therefore, area of the park ABC = √(s(s − a)(s − b)(s − c)) = √(48 × 8 × 24 × 16) = 384 m².
Detailed Explanation
To find the area of park ABC, we first identify the sides: a = 40 m, b = 24 m, and c = 32 m. We calculate the semi-perimeter 's' as 48 m. By determining how much shorter each side is than 's', we find three additional values. Plugging these values into Heron's formula gives us the area of the triangle, which turns out to be 384 m².
Examples & Analogies
Think of it like solving a puzzle where each side of the triangle gives you a piece that you need. First, you put together the edges (the lengths of the sides). Then, by using Heron's formula, you can find the size of the triangle's 'shadow' (area) even if you don’t see it directly!
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Create a free accountWe see that 32² + 24² = 1024 + 576 = 1600 = 40². Therefore, the sides of the park make a right triangle. The largest side, i.e., BC which is 40 m will be the hypotenuse and the angle between the sides AB and AC will be 90°.
Detailed Explanation
By applying the Pythagorean theorem, we can check whether the triangle is a right triangle. Here, the squares of the shorter sides (32 and 24) add up to the square of the longest side (40), confirming it's a right triangle with a 90° angle between the two shorter sides.
Examples & Analogies
Imagine you have a right-angled triangular slice of cake. By measuring the base and height correctly (the lengths of two sides), you can confirm that the longest cut represents a straight slice through the middle, creating a perfect right angle.
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Create a free accountWe can check that the area of the park is (1/2) × 32 × 24 m² = 384 m². We find that the area we have got is the same as we found by using Heron’s formula.
Detailed Explanation
Using a more traditional method to find the area of the triangle, we use the formula for area based on base and height. This method also leads us to an area of 384 m², confirming that Heron's formula calculations are correct.
Examples & Analogies
Think back to our cake analogy. Sometimes slicing it in a straight manner (using base times height) can confirm the same amount of cake you can serve as estimating by measuring the three sides and calculating the area with Heron’s formula.
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Create a free accountNow using Heron's formula, you verify this fact by finding the areas of other triangles discussed earlier viz., (i) equilateral triangle with side 10 cm. (ii) isosceles triangle with unequal side as 8 cm and each equal side as 5 cm.
Detailed Explanation
In the examples that follow, you can practice using Heron’s formula to find the area of various other types of triangles. By applying the same method of finding the semi-perimeter and calculating the area using the set sides, you'll become more comfortable with the formula's application.
Examples & Analogies
Just like following a recipe with different ingredients, you can use the same formula (like a recipe for areas) for differently shaped triangles. Each shape can yield the same satisfaction of understanding how to calculate area just like baking leads to delicious results!
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Key Concepts
Core takeaways and short definitions to help you quickly recall the key ideas from this section.
Heron's Formula: A way to calculate the area of a triangle without height.
Semi-perimeter: Important in the calculation of area using Heron's formula.
Verification of area through different methods: Including right triangle calculations.
Examples
Step-by-step examples to apply the section's ideas and test your understanding.
{'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': '$$ s = \frac{10 + 10 + 10}{2} = 15 \text{ cm}, \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{15(15-10)(15-10)(15-10)} = 25\sqrt{3} \text{ cm}^2 $$'}
Memory Aids
Interactive tools to help you remember key concepts
Stories
Flash Cards
Glossary
Heron's Formula
A formula to calculate the area of a triangle when the lengths of all three sides are known.
Semiperimeter
Half the sum of the lengths of the sides of a triangle, used in calculating area via Heron's formula.
Scalene Triangle
A triangle with all sides of different lengths.
Perimeter
The total length of the sides of a polygon.