Circle (with diameter)- perimeter and area

A circle is a two-dimensional shape made up of all the points in a plane that lie at the same fixed distance from a single central point. That distance is called the radius, and every radius drawn from the center to the edge has exactly the same length. A straight line that crosses the circle through its center is the diameter, and it is always twice the radius, splitting the circle into two equal halves. The boundary of the circle itself is called the circumference, and it encloses the flat region we measure as its area. One of the circle's most remarkable features is that the ratio of its circumference to its diameter is always the same number, the constant pi (about 3.14), no matter how large or small the circle is. This perfect, uniform symmetry makes the circle one of the most efficient shapes in nature, enclosing the greatest area for a given boundary length. From wheels, gears, and pipes to lenses, clocks, orbits, and the ripples on water, circles appear everywhere in mathematics, engineering, design, and the natural world.