Dissection

The idea

Dissection is the process of cutting a figure into pieces and rearranging them to get a different figure of equal area.

Same cloth, new shape

A piece of cloth is a rectangle of 8 by 3 squares. A tailor needs a 6 by 4 rectangle. Both cover the same number of squares: 24.

Can the 8 by 3 cloth be cut into two pieces that fit together, with nothing left over, to make the 6 by 4 cloth?

• No, a different shape needs different cloth
• Yes, with a clever cut and no waste
• Only if some cloth is thrown away

Cut and slide

The cloth is cut along steps into two pieces. Drag the top piece, or use the arrows, until the two pieces fill the dashed rectangle.

Cut, slide, same area

The process of cutting a figure into pieces and rearranging them to get a different figure of equal area is called dissection.

The 8 by 3 rectangle was cut along a staircase into two pieces of 12 squares each. Slide the top piece 2 squares right and 1 up and the pieces make a 6 by 4 rectangle. Count: 8 × 3 = 24 and 6 × 4 = 24.

Cutting and sliding adds no cloth and takes none away, so the area stays the same. The shape changed, and so did the edge: it was 2 × (8 + 3) = 22 and is now 2 × (6 + 4) = 20.

This is a very old idea. The builders of altars in the Śulba-Sūtras had to turn one shape into another with exactly the same area, and they used dissection.

Notes

The process of cutting a figure into pieces and rearranging them to get a different figure of equal area is called dissection.

Check yourself

A rectangle is cut into pieces, and the pieces are rearranged into a different shape with no gaps and no overlaps. What stays the same?

A rectangle 9 cm by 4 cm is cut into pieces and rearranged into a square with nothing left over. How long is a side of the square, in cm?

Answer: 6

The area is 9 × 4 = 36 cm². The square has side × side = 36, so the side is 6 cm.

Which figure can be made by dissecting a rectangle 4 cm by 6 cm?

A sheet 12 cm by 5 cm is cut into pieces and rearranged into a rectangle 6 cm wide. How long is the new rectangle, in cm?

Answer: 10

The area is 12 × 5 = 60 cm². The new rectangle has 6 × length = 60, so the length is 10 cm.

• The shape and the area. The shape changes: that is the whole point of a dissection. Only the area is kept.
• The area and the perimeter. The perimeter can change. The 8 by 3 rectangle has edge 22 and the 6 by 4 rectangle made from it has edge 20.
• Only the area — correct. Yes! No part is added or removed, so the area is the same. The shape and the edge can change.
• Nothing stays the same. The area stays the same: the pieces are the same, so they cover the same number of squares.
• A rectangle 4 cm by 5 cm. That has area 20 cm², but the original has 24 cm². Pieces cannot disappear.
• A rectangle 3 cm by 8 cm — correct. Yes! 3 × 8 = 24 cm², the same as 4 × 6 = 24 cm².
• A square 5 cm by 5 cm. That has area 25 cm², one more than 24 cm². A dissection cannot add area.
• A rectangle 3 cm by 9 cm. That has area 27 cm², more than 24 cm².