10.1.2 Application of Heron's formula to a triangular park

Description

Quick Overview

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.

Standard

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

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 mΒ². 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.

Key Concepts

  • 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.

Memory Aids

🎡 Rhymes Time

  • To find triangle area with sides three, add them and divide by two, you'll see.

πŸ“– Fascinating Stories

  • Imagine a triangle wondering how to measure its area without height. It discovers Heron’s formula and becomes famous for its versatility.

🧠 Other Memory Gems

  • SAS (semi-perimeter, area, sides) to remember the steps clearly.

🎯 Super Acronyms

HERO - Height? Easily Replaced by Other methods!

Examples

  • {'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': '\[ s = \frac{10 + 10 + 10}{2} = 15 \text{ cm}, \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{15(15-10)(15-10)(15-10)} = 25\sqrt{3} \text{ cm}^2 \]'}

Glossary of 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 the sum of the lengths of the sides of a triangle, used in calculating area via Heron's formula.

  • Term: Scalene Triangle

    Definition:

    A triangle with all sides of different lengths.

  • Term: Perimeter

    Definition:

    The total length of the sides of a polygon.