Introduction

Area, a fundamental concept in geometry, defines the size or extent of a two-dimensional figure or shape in a plane. It’s akin to imagining the quantity of paint needed to cover a surface. This article explores the calculation of areas for different geometric shapes and sheds light on intriguing scenarios involving a farmer and his daughter.

Rectangle: The Farmer’s Plight

A rectangle, with its four right angles, is among the simplest shapes. Calculating its area requires knowledge of length and width. For a rectangle:

\[ \text{area} = \text{length} \times \text{width} \]

In a whimsical tale, a farmer in the United States, unfamiliar with SI units, measures his rectangular plot in feet. With a length of 220 feet and a width of 99 feet, his unsold land covers an area of 21,780 square feet, equivalent to half an acre. Alas, foreign investors with different measurement standards hinder his land from finding a buyer.

Triangle: A Daughter’s Demand

Triangles, with their diverse area formulas, add complexity to the farmer’s life. Her daughter, fascinated by triangles, insists on a triangular pool with sides measuring 77 feet. Using Heron’s formula:

\[ \text{area} = \sqrt{s(s – x)(s – y)(s – z)} \]

The farmer calculates the area as 2567.33 square feet. The triangular pool becomes a unique spectacle in the backyard, a testament to the farmer’s dedication to his daughter’s whims.

Trapezoid: A Ramp for Adventure

A trapezoid, a convex quadrilateral with one pair of parallel sides, involves the bases and height in its area calculation:

\[ \text{area} = \frac{b_1 + b_2}{2} \times h \]

The farmer, adapting to his daughter’s evolving interests, constructs a BMX ramp using rectangles and a trapezoidal face. With a height of 9 feet, a bottom base of 29.528 feet, and a top base of 9 feet, the trapezoidal area is 173.376 square feet.

Circle: Crop Circles and Pranks

Circles, defined by a center point and radius, have a simple area formula:

\[ \text{area} = \pi r^2 \]

The farmer’s daughter, now a teenager with unconventional hobbies, creates a crop circle with an outer radius of 15 feet, covering an area of 706.858 square feet. The prank, however, attracts unwanted attention and damages to the farmer’s crops from alien enthusiasts.