Halving a square
Draw a tilted square inside, joining the middle points.
Halve the side?
Now we want a square with half the area of a given square. A first idea: halve the side.
A square has side 8 and area 64. Its side is halved to 4.
What part of the old area is the new square?
What this lesson covers
The idea
A square with half the area of a given square is constructed by reversing the doubling construction, drawing a tilted smaller square inside the given one.
Halve the side?
Now we want a square with half the area of a given square. A first idea: halve the side.
A square has side 8 and area 64. Its side is halved to 4.
What part of the old area is the new square?
- half of it
- a quarter of it
- an eighth of it
Draw the tilted square
Baudhāyana reverses the doubling picture: draw a tilted square inside the big one. Four points sit on the four sides. Move them and watch the area of the tilted square.
Join the middle points
A square with half the area of a given square is made by reversing the doubling construction: draw a tilted smaller square inside the given one. Join the middle points of the four sides.
The tilted square is made of 4 triangles. The big square is made of 8 of the same triangles. So the tilted square is half.
It is the picture of the last lesson, run backwards: the old square is now the tilted one.
Notes
Join the middle points of the sides of a square. The tilted square inside has half the area. 4 triangles inside, 8 triangles in all.
Check yourself
A square has area 40 sq. cm. Join the middle points of its sides to make a tilted square inside. What is the area of the tilted square in sq. cm?
Answer: 20
Half of 40 is 20 sq. cm. Big square: 8 triangles. Tilted square: 4 of them.
Which one makes a square with half the area of a square?
The tilted square inside is made of 4 triangles. The big square is made of ... of the same triangles.
A square has area 72 sq. cm. Join the middle points to get a tilted square. Then do the same inside the tilted square. What is the area of the smallest square in sq. cm?
Answer: 18
72, then 36, then 18. Two halvings make a quarter, the same as halving the side of the first square.
- Halve the length of the side. That makes 1/2 × 1/2 = a quarter of the area.
- Cut the square into two by a straight line through the middle. Each piece has half the area, but it is a rectangle, not a square.
- Join the middle points of the four sides — correct. Yes! The tilted square has 4 triangles out of 8.
- 6 triangles. Count again: 4 triangles in the middle and 4 corner triangles outside.
- 8 triangles — correct. Yes! 4 in the tilted square and 4 at the corners.
- 16 triangles. That is too many. The 4 corners are the same size as the 4 inside.