We have a figure, including a square and a semicircle. We need to find out the area of the shaded part in the picture. Suppose the side length of a square is one centimeter. Because the diameter of the semicircle is equal to the side length of the square, the radius of the semicircle is a/2 cm. According to the topic, we can establish the following equation:

The area of a square is 2 square centimeters. The area of a semicircle is π× (a/2) 2/2 = π a 2/8 square centimeters. Shaded area = square area-semicircle area.

Using mathematical equations, we can express it as: shadow area = a 2-π a 2/8. Now we need to solve this equation and find out the area of the shadow part. The calculation result is: 16-2pi cm2. So the shadow area in the figure is: 16-2pi cm2.

Expansive materials: development history

Area of a circle

In the 5th century BC, Hippocrates in Hiosburg was the first person to show that the area of a disk (the area surrounded by a circle) was directly proportional to the square of its diameter, which was part of the orthogonality in Hippocrates' era, but the proportionality constant was not determined. In the 5th century BC, eudoxus of Cornydos also found that the area of a disk is directly proportional to the square of its radius.

1794, the French mathematician Adrian-Marie Legendre proved that π2 is irrational. This also proves that π is an irrational number. 1882, the German mathematician Ferdinand von Lin Deman proved that π is a transcendental number (not the solution of any polynomial equation with rational coefficients), which confirmed the speculation of Legendre and Euler.

Triangular region

Heron (or hero) of Alexandria found the so-called Heron formula in the triangle, and a proof in Metrica's book written about 60 years ago can be found in his book. Some people think that Archimedes knew this formula two centuries ago. Because Metrica is a collection of mathematical knowledge available in the ancient world, it is possible that this formula is earlier than the references in this book.

In 499, aryabhata, a great mathematician and astronomer in the classical times of Indian mathematics and Indian astronomy, expressed the area of a triangle as half the height of Aryabhatiya. China discovered the formula equivalent to heron independently of the Greeks. Published in 1247 "Nine Chapters Publishing" in Shu Qi (referred to as "Nine Chapters Mathematics") by Qin.