How many square meters are there on land, how much paint goes on a wall, how much fabric goes on a flag? All these questions are about area — and almost all the formulas come from just one: the rectangle.

Who discovered

The first area calculations that we have records of are from scribes from Egypt and Mesopotamia: the Rhind papyrus (c. 1650 BC) calculates areas of rectangular, triangular and even circular fields, and Babylonian tablets show areas of trapezoids. The Greeks gave these rules a justification: in The Elements, Euclid (c. 300 BC) proves that parallelograms with the same base and the same height have the same area (proposition I.35) and that the triangle is half of the parallelogram (I.41). The circle formula was demonstrated by Archimedes (3rd century BC).

The story

The Greek idea was to compare figures by cutting and pasting, without numbers: two figures have the same area if one can be broken down and reassembled into the other. It is the same method as the figures below.

The Egyptians used, for the circle, the rule “take $\tfrac{1}{9}$ from the diameter and square it”, which is equivalent to using $\pi \approx 3{,}16$. Archimedes showed that the area of ​​a circle is equal to that of a triangle with a base equal to the circumference and height equal to the radius — that is, $\pi r^2$.

In everyday life

  • Properties. The price of an apartment is given per square meter; the plan shows the area of ​​each room.
  • Paint and floor. The amount of paint and ceramics depends on the area of ​​the walls and floor.
  • Agriculture. Production and seeds are calculated per hectare (10 000 m²).
  • Pizza. A pizza measuring 40 cm in diameter has almost twice the area of ​​one measuring 30 cm (1 257 cm² versus 707 cm²).

From the rectangle to the other figures

ResumoArea formulas

Figure Area Idea
Rectangle $A = b \cdot h$ lines of squares
Parallelogram $A = b \cdot h$ Cut out a triangle and turn it into a rectangle
Triangle $A = \dfrac{b \cdot h}{2}$ half of a parallelogram
Trapezium $A = \dfrac{(B + b)\, h}{2}$ two trapezoids form a parallelogram
Diamond $A = \dfrac{D \cdot d}{2}$ half of the rectangle of diagonals
Circle $A = \pi r^2$ sectors form an “almost rectangle”
Choose the figure and drag the yellow vertex. Height is always measured perpendicular to the base (dashed line).

AtençãoThe height is perpendicular to the base

In the parallelogram and the triangle, the height is not the inclined side: it is the distance between the base and the opposite vertex, measured at right angles. Tilting the figure without changing the base and height does not change the area.

The area of ​​the circle

Cut the circle into sectors and fit them alternating up and down. With many sectors, the figure becomes a rectangle with base πr and height r.

Exercises

1

A rectangular piece of land measures 12 m by 5 m. What is your area?

Ver resposta

$A = 12 \cdot 5 = 60\ \text{m}^2$

2

A triangle has a base of 10 cm and a height of 6 cm. Calculate the area.

Ver resposta

$A = \dfrac{10 \cdot 6}{2} = 30\ \text{cm}^2$

3

The section of a channel is a trapezoid with bases 9 m and 5 m and height 4 m. What is the section area?

Ver resposta

$A = \dfrac{(9 + 5) \cdot 4}{2} = 28\ \text{m}^2$

4

A diamond-shaped kite has diagonals of 16 cm and 10 cm. How much paper does it use?

Ver resposta

$A = \dfrac{16 \cdot 10}{2} = 80\ \text{cm}^2$

5

A circular flowerbed has 3 m in radius. Calculate its area (use $\pi \approx 3{,}14$).

Ver resposta

$A = \pi \cdot 3^2 = 9\pi \approx 28{,}3\ \text{m}^2$

True or false?

  1. A parallelogram and a rectangle with the same base and height have the same area.
  2. Doubling the radius of a circle doubles the area.
  3. The area of ​​a triangle is half the area of ​​a parallelogram with the same base and height.
  4. In a slanted parallelogram, the height is the slanted side.

Sources: Elements, proposition I.35 · Elements, proposition I.41 · Archimedes (MacTutor).