Rectangle

A rectangle is a quadrilateral with four right angles. It is one of the simplest shapes, and calculating its area only requires that its length and width are known (or can be measured). A quadrilateral by definition is a polygon that has four edges and vertices. In the case of a rectangle, the length typically refers to the longer two edges of the quadrilateral, while the width refers to the shorter of the two edges. When the length and width of a rectangle are equal, the shape is a special case of a rectangle, called a square. The equation for calculating the area of a rectangle is as follows:

area = length × width

The Farmer and his Daughter – Unsold Land

Imagine a farmer trying to sell a piece of land that happens to be perfectly rectangular. Because he owns some cows that he did not want frolicking freely, he fenced the piece of land and knew the exact length and width of each edge. The farmer also lives in the United States, and being unfamiliar with the use of SI units, still measures his plot of land in terms of feet. The foot was defined to be exactly 0.3048 meters in 1959 after having changed over an extensive period of time, as historically, the human body was often used to provide a basis for units of length, and unsurprisingly, was inconsistent based on time and location. Tangent aside, the farmer's plot of land has a length of 220 feet, and a width of 99 feet. Using this information:

area = 220 × 99 = 21780 sq ft

The farmer's plot of land, which has an area of 21,780 square feet, equates to half an acre, where an acre is defined as the area of 1 chain by 1 furlong, which is defined by something else, and so on, and is why SI now exists. Unfortunately for the farmer, he lives in an area predominated by foreign investors with smaller feet, who felt that they should be getting more square feet for their money, and his land remains unsold today.

Triangle

There are many equations for calculating the area of a triangle based on what information is available. As mentioned in the calculator above, please use the Triangle Unit Calculator for further details and equations for calculating the area of a triangle, as well as determining the sides of a triangle using whatever information is available. Briefly, the equation used in the calculator provided above is known as Heron's formula (sometimes called Hero's formula), referring to the Hero of Alexandria, a Greek mathematician and engineer considered by some to be the greatest experimenter of ancient times. The formula is as follows:

The Farmer and his Daughter - Triangle Daze

At this point in time, through extreme effort and perseverance, the farmer has finally sold his 21,780 sq ft plot of land and has decided to use some of the money earned to build a pool for his family. Unfortunately for the farmer, he does not consider the fact that the maintenance costs of a pool for one year alone could likely pay for his children to visit any pool or water theme park for years to come. Even more unfortunately for the farmer, his 7-year-old daughter who has recently traveled to Egypt vicariously through Dora the Explorer, has fallen in love with triangles, and insists that the pool not only be triangular in shape, but also that the measurements must only include the number 7, to represent her age and immortalize this point of her life in the form of a triangular pool. Being a doting father, the farmer acquiesces to his daughter's request and proceeds to plan the construction of his triangular pool. The farmer must now determine whether he has sufficient area in his backyard to house a pool. While the farmer has begun to learn more about SI units, he is as yet uncomfortable with their use and decides that his only viable option is to construct a pool in the form of an equilateral triangle with sides 77 ft in length, since any other variation would either be too large or small. Given these dimensions, the farmer determines the necessary area as follows:

Since the longest distance between any two points of an equilateral triangle is the length of the edge of the triangle, the farmer reserves the edges of the pool for swimming "laps" in his triangular pool with a maximum length approximately half that of an Olympic pool, but with double the area – all under the watchful eyes of the presiding queen of the pool, his daughter, and the disapproving glare of his wife.

Trapezoid

A trapezoid is a simple convex quadrilateral that has at least one pair of parallel sides. The property of being convex means that a trapezoid's angle does not exceed 180° (in contrast, a concave quadrilateral would), while being simple reflects that trapezoids are not self-intersecting, meaning two non-adjacent sides do not cross. In a trapezoid, the parallel sides are referred to as the bases of the trapezoid, and the other two sides are called the legs. There exist more distinctions and classifications for different types of trapezoids, but their areas are still calculated in the same manner using the following equation:

area = b1+ b22× h

where

 b1andb2 are the bases.

 h is the height, or perpendicular distance between the bases

The Farmer and his Daughter – Ramping Endeavors

Two years have passed since the farmer's pool was completed, and his daughter has grown and matured. While her love for triangles still persists, she eventually came to the realization that no matter how well-"triangled" she was, triangles alone cannot make the world go round, and that Santa's workshop could not plausibly balance on the North Pole, were the world a pyramid rather than a sphere. Slowly, she has begun to accept other shapes into her life and pursues her myriad different interests – currently freestyle BMX. As such, she requires a ramp, but unfortunately for the farmer, not just any ramp. The ramp must be comprised of only shapes that can be formed using multiple triangles, since, like her rap idol B.o.B, the farmer's daughter still has difficulty accepting the reality of curved surfaces. It must, of course, also only use the number 9 in its measurements to reflect her age. The farmer decides that his best option is to build a ramp comprised of multiple rectangles, with the side face of the ramp being in the shape of a trapezoid. As the farmer has now become more comfortable with SI, he is able to be more creative with his use of units, and can build a more reasonably sized ramp while adhering to his daughter's demands. He decides to build a ramp with a trapezoidal face with a height of 9 ft, a bottom base of length 29.528 ft (9 m), and a top base of 9 ft. The area of the trapezoid is calculated as follows:

area = 

9 + 29.528

2

× 9 = 173.376 sq ft