You know the circumference of a circle and the area of a parallelogram. Cutting a circle into slices connects the two.
Idea clave
$$A = \pi r^2$$ Square the radius, then multiply by $\pi$. Area is measured in square units.
Por qué funciona
Cut a circle into many thin slices, like a pizza, and line them up, points up and down. They make a shape that is almost a parallelogram, and nearly a rectangle when the slices are thin. Its height is the radius $r$. Its length is half the circumference, $\frac{1}{2} \times 2\pi r = \pi r$, because half the crust is on top and half on the bottom. Area $=$ length $\times$ height $= \pi r \times r = \pi r^2$.
Worked examples
Ejemplo resuelto: From the radius
Find the area of a circle with radius 5 cm.
- $A = \pi r^2 = \pi \times 5^2 = 25\pi$ square cm (exact).
- Using $3.14$: $25 \times 3.14 = 78.5$ square cm.
Respuesta: $25\pi \approx 78.5$ square cm
Ejemplo resuelto: From the diameter
A round rug has a diameter of 12 ft. What is its area? Use $3.14$.
- The radius is half the diameter: $r = 6$ ft.
- $A \approx 3.14 \times 6^2 = 3.14 \times 36 = 113.04$ square ft.
Respuesta: $113.04$ square ft
Ejemplo resuelto: Working backward
A circle has an area of $49\pi$ square inches. What is its radius?
- $\pi r^2 = 49\pi$, so $r^2 = 49$.
- $7 \times 7 = 49$, so $r = 7$ inches.
Respuesta: 7 inches
Error común
Mixing up $2\pi r$ (the distance around, in cm) and $\pi r^2$ (the space inside, in square cm), or squaring the diameter. Squaring the diameter gives an area 4 times too big: $12^2 = 144$, but $6^2 = 36$. Check the units: area must be in square units.