10.1.3 Verification of areas using Heron’s formula

Description

Quick Overview

Heron's formula provides a method to calculate the area of a triangle when the lengths of all three sides are known, without needing the height.

Standard

This section introduces Heron's formula for finding the area of a triangle using the lengths of its sides. It illustrates how to apply the formula through various triangle examples, emphasizing verification of computed areas.

Detailed

Verification of Areas Using Heron’s Formula

In this section, we explore the application of Heron’s formula, which enables the calculation of the area of a triangle given the lengths of its sides. The formula states that

\[
\text{Area} = \sqrt{s(s-a)(s-b)(s-c)}
\]
where \(s\) is the semi-perimeter defined as \(s = \frac{a + b + c}{2}\), with \(a\), \(b\), and \(c\) representing the sides of the triangle.

We begin with an example triangle park with sides measuring 40m, 32m, and 24m. Calculating the semi-perimeter gives \(s = 48m\) and using Heron's formula, we find the area matches our verification through the traditional base-height method, showcasing the consistency and effectiveness of the formula. We then apply Heron’s formula to additional triangles, examining an equilateral triangle with a side of 10 cm, and an isosceles triangle with sides measuring 5 cm and 8 cm, thus verifying versatility in various triangle types.

This section reinforces the significance of Heron’s formula in practical applications where traditional height-based area calculations may not be feasible.

Key Concepts

  • Heron's Formula: A method to calculate the area of a triangle using its side lengths.

  • Semi-perimeter: The half of the perimeter of a triangle, critical for using Heron's formula.

  • Right Triangle: A triangle in which one angle is exactly 90 degrees.

Memory Aids

🎵 Rhymes Time

  • Heron’s way to find the space, use the sides to set the pace.

📖 Fascinating Stories

  • Imagine a triangle trying to find a way to show off its area by revealing the secrets of its sides to calculate the space inside.

🧠 Other Memory Gems

  • Remember the formula: Sway! Area = √(s(s-a)(s-b)(s-c))!

🎯 Super Acronyms

HAVE

  • Heights aren't vital
  • all sides equal.

Examples

  • {'example': 'Find the area of a triangle with sides 40m, 32m, and 24m.', 'solution': '\[ s = \frac{40 + 32 + 24}{2} = 48m \] \n\[ \text{Area} = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{48\cdot(48-40)(48-32)(48-24)} = \sqrt{48\cdot8\cdot16\cdot24} = 384 m^2 \]'}

  • {'example': 'Calculate the area of an equilateral triangle with side 10cm.', 'solution': '\[ s = \frac{10 + 10 + 10}{2} = 15cm \] \n\[ \text{Area} = \sqrt{15\cdot(15-10)(15-10)(15-10)} = \sqrt{15\cdot5\cdot5\cdot5} = 25\sqrt{3} cm^2 \]'}

  • {'example': 'Area of an isosceles triangle with side lengths 5cm, 5cm, and 8cm.', 'solution': '\[ s = \frac{5 + 5 + 8}{2} = 9cm \] \n\[ \text{Area} = \sqrt{9\cdot(9-5)(9-5)(9-8)} = \sqrt{9\cdot4\cdot4\cdot1} = 12 cm^2 \]'}

Glossary of Terms

  • Term: Heron's Formula

    Definition:

    A formula that gives the area of a triangle when the lengths of all three sides are known.

  • Term: Semiperimeter

    Definition:

    Half of the perimeter of a triangle, calculated as s = (a + b + c) / 2.

  • Term: Area

    Definition:

    The amount of space inside a two-dimensional shape, measured in square units.