10.1.1 - Introduction to Heron's formula

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Introduction to Triangles

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Teacher
Teacher

Today, we are exploring how we can find the area of a triangle when we only know the lengths of its sides. Do you remember how we usually find the area of a triangle?

Student 1
Student 1

Yes! We use the formula for area as 1/2 times the base times the height.

Student 2
Student 2

But what if we don’t know the height?

Teacher
Teacher

Excellent question! This is where Heron’s formula comes into play. It allows us to calculate the area without needing the height. Do you want to hear how this works?

Understanding Heron's Formula

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Teacher
Teacher

"Heron's formula states that the area \( A \) of a triangle with sides \( a \), \( b \), and \( c \) can be calculated using:

Applying Heron's Formula

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Teacher
Teacher

Now, using our earlier example of a triangle with sides 40 m, 32 m, and 24 m, we have \( s = 48 \) m. Can we find \( A \)?

Student 1
Student 1

Let’s calculate \( s-a, s-b, \) and \( s-c \). We get 8 m, 24 m, and 16 m respectively.

Student 2
Student 2

So then \( A = \sqrt{48(8)(24)(16)}\)?

Teacher
Teacher

Yes! And calculating that gives you the area of the triangle. Can someone show me how?

Student 3
Student 3

When we plug in the numbers, we get \( A = \sqrt{48 \times 8 \times 24 \times 16} = 384 \) m²!

Teacher
Teacher

Perfect! And you see how it matches our earlier result using height as well. Great teamwork!

Verifying the Formula

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Teacher
Teacher

We have calculated the area using both Heron’s formula and by calculating the height. Why is it helpful to verify our results in different ways?

Student 4
Student 4

It makes sure our calculations are correct!

Student 1
Student 1

And it helps us understand the right triangle property too.

Teacher
Teacher

Exactly! Verifying through different methods reinforces our understanding. Remember the importance of cross-checking!

Introduction & Overview

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Quick Overview

Heron's formula allows for the calculation of the area of a triangle using only the lengths of its sides.

Standard

This section introduces Heron's formula for calculating the area of a triangle given the lengths of its sides. It emphasizes the significance of this formula when determining area through height is complex, explaining the derivation of the formula, its components, and providing illustrative examples.

Detailed

Detailed Summary

Heron's formula provides a method to calculate the area of a triangle when the height is not known. The formula states:

\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]

where \(s\) is the semi-perimeter defined as:
\[ s = \frac{a + b + c}{2} \]
This section begins with a scenario of finding the area of a triangular park with known sides (40 m, 32 m, 24 m), highlighting the inadequacy of using height for area calculation.

It further discusses the historical context, noting the contributions of Heron of Alexandria who formulated this equation between 10 C.E. – 75 C.E.

Three examples demonstrate calculating the area using Heron's formula, validating results through methods such as the right triangle area calculation, and verifying through multiple triangle types (equilateral and isosceles). Exercises at the end reinforce learning by encouraging students to apply this formula to both given and constructed problems.

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Heron's Formula
Heron's Formula

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Defining the Area of a Triangle

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We know that the area of a triangle when its height is given, is
\[ \text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \] . Now suppose that we know the lengths of the sides of a scalene triangle and not the height. Can you still find its area? For instance, you have a triangular park whose sides are 40 m, 32 m, and 24 m. How will you calculate its area? Definitely if you want to apply the formula, you will have to calculate its height. But we do not have a clue to calculate the height. Try doing so. If you are not able to get it, then go to the next section.

Detailed Explanation

The area of a triangle can typically be found using the formula that involves its base and height. However, if we only know the sides of the triangle (like with a scalene triangle), calculating the height may not be straightforward or possible. This introduces a problem for calculating the area based solely on side lengths, which is where Heron's formula comes into play. It allows us to find the triangle's area without needing to know its height.

Examples & Analogies

Imagine a triangular park where you want to put in flower beds. You know the lengths of the triangle's sides but not how high it is from the base to the top point where the flowers would grow. If you only have the side lengths, you might feel stuck. Heron's formula acts like a special tool that lets you calculate the area even when you don’t know how tall the triangle is.

Heron's Background

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Heron was born in about 10AD possibly in Alexandria in Egypt. He worked in applied mathematics. His works on mathematical and physical subjects are so numerous and varied that he is considered to be an encyclopedic writer in these fields. His geometrical works deal largely with problems on mensuration written in three books.

Detailed Explanation

Heron of Alexandria was a notable figure in the history of mathematics. His contributions laid foundational work in applied mathematics, including geometry. His exploration of different geometrical shapes and their areas demonstrates his extensive knowledge. His book focused on measurements of various shapes, but notably, it contains the formula for the area of a triangle, leading to his prominent legacy in mathematics.

Examples & Analogies

Think of Heron like a pioneer explorer in a vast land of geometry. Just as explorers mapped new territories, Heron mapped out mathematical principles that help us navigate the complexities of shapes, starting with triangles. His writings are like guides we still use today for our own explorations in geometry.

Introduction to Heron's Formula

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The formula given by Heron about the area of a triangle, is also known as Hero’s formula. It is stated as:
\[ \text{Area of a triangle} = \sqrt{s(s-a)(s-b)(s-c)} \]
where a, b and c are the sides of the triangle, and \(s = \frac{a+b+c}{2}\), the semi-perimeter of the triangle.

Detailed Explanation

Heron's formula allows us to calculate the area of any triangle when we know the lengths of all three sides. We first find the semi-perimeter (s) by adding the lengths of the three sides and dividing by 2. Once we have s, we can plug it into the formula using the three sides to find the area. This is particularly useful for triangles where finding the height is complicated.

Examples & Analogies

Imagine you are an architect needing to figure out how much space a triangular rooftop will cover. You have the measurements of the side lengths but no idea of the height. Heron’s formula is like a secret recipe that uses just those side lengths to calculate the area, ensuring you know how much roofing material you'll need without any guesswork.

Practical Application of Heron's Formula

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This formula is helpful where it is not possible to find the height of the triangle easily. Let us apply it to calculate the area of the triangular park ABC, mentioned above (see Fig. 10.2). Let us take a = 40 m, b = 24 m, c = 32 m, so that we have s = \[ \frac{40 + 24 + 32}{2} = 48 \] m. Thus, we can apply Heron's formula to find the area.

Detailed Explanation

When applying Heron's formula to the triangular park, we start by determining the semi-perimeter using the side lengths of the triangle. After calculating s, we then substitute it and the sides into the formula, allowing us to find the area in an effective manner without needing to calculate the height. This process exemplifies the utility of Heron’s formula in practical situations.

Examples & Analogies

Think of the triangular park as a pizza slice. You know the lengths of the crust and the sides (like your measuring tape), but you can’t measure how high it rises in the middle. Heron’s formula lets you calculate the area of that pizza slice just from knowing those crust lengths, helping you decide how many slices you can serve to your friends at a picnic!

Definitions & Key Concepts

Learn essential terms and foundational ideas that form the basis of the topic.

Key Concepts

  • Heron's Formula: Used to find the area of a triangle with known side lengths.

  • Semi-perimeter: Defined as half the perimeter of the triangle.

  • Right Triangle: A triangle where one angle measures 90 degrees.

Examples & Real-Life Applications

See how the concepts apply in real-world scenarios to understand their practical implications.

Examples

  • {'example': 'Find the area of a triangle with sides 40 m, 24 m, and 32 m.', 'solution': '\[ s = \frac{40 + 24 + 32}{2} = 48 \ m \] \n\[ A = \sqrt{48(48-40)(48-24)(48-32)} = \sqrt{48 \times 8 \times 24 \times 16} = 384 \ m^2 \]'}

  • {'example': 'Calculate the area of an equilateral triangle with side length 10 cm.', 'solution': '\[ s = \frac{10 + 10 + 10}{2} = 15 \ cm \] \n\[ A = \sqrt{15(15-10)(15-10)(15-10)} = \sqrt{15 \times 5 \times 5 \times 5} = 25\sqrt{3} \ cm^2 \]'}

Memory Aids

Use mnemonics, acronyms, or visual cues to help remember key information more easily.

🎵 Rhymes Time

  • Heron's way to find the area, sides in play, and math won’t stray.

🧠 Other Memory Gems

  • SAS - Semi-perimeter, Area, Sides.

📖 Fascinating Stories

  • Imagine Heron, a wise man, finding triangles' secrets hidden in the land.

🎯 Super Acronyms

HAS - Height Alternative is Heron’s solution.

Flash Cards

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Glossary of Terms

Review the Definitions for terms.

  • Term: Heron's Formula

    Definition:

    A formula to calculate the area of a triangle when the lengths of all three sides are known.

  • Term: Semiperimeter

    Definition:

    Half of the triangle's perimeter, calculated as \( s = \frac{a + b + c}{2} \).

  • Term: Triangle

    Definition:

    A polygon with three edges and vertices.

  • Term: Scalene Triangle

    Definition:

    A triangle where all sides are of different lengths.