Cut and slide

About this video

A flower bed between two streets that cross at a slant has sides that lean, so counting squares is hard: its edges cut right across them. This animation shows the trick that finds its area. Cut straight down from a top corner of the parallelogram, at a right angle to the bottom side. A triangle comes off one end; slide it to the other end and it fits the gap exactly. Now the shape is a rectangle, and since nothing was added or thrown away it has the same area. The rectangle is as long as the parallelogram's base and as tall as the cut, its height, so the area of a parallelogram is base × height.

Then comes the common mistake. The height is the distance straight up from the base, not the slanted side. A parallelogram with a base of 6 cm, a slanted side of 5 cm and a height of 4 cm has an area of 6 × 4 = 24 cm², not 6 × 5 = 30. On a grid the video pushes a parallelogram over further and further: with the same base and height it covers the same 12 squares however far it leans.

Triangles come next. Cut a rectangle 6 by 4 from corner to corner and you get two equal right-angled triangles of 12 cm² each. For any triangle, turn a copy upside down and the two fit together into a parallelogram with the same base and height, so the triangle is half of base × height. A sail 8 m along the bottom and 5 m high needs half of 40, which is 20 m² of cloth. This video goes with the lessons The Area of a Parallelogram and The Area of a Triangle of Geometry Level 1 on Knaph.

Transcript

1. 0:00A flower bed between two streets that cross at a slant. Its sides lean. How do you count the squares?
2. 0:08Cut straight down from a top corner, at a right angle. A triangle comes off.
3. 0:14Slide it to the other end. It fits the gap exactly, and now it's a rectangle.
4. 0:22Nothing added, nothing thrown away. So the area is the same, base times height.
5. 0:30The mistake to watch. The slanted side is five, but the height, straight up, is four.
6. 0:37Six times four is twenty-four. Six times five would be too much.
7. 0:44Push it over more and more. Same base, same height, the same twelve squares.
8. 0:56Now a triangle, like a sail. Start with a rectangle and cut it corner to corner.
9. 1:02Two equal halves. Six times four is twenty-four, so each triangle is twelve.
10. 1:12Any triangle? Take a copy, turn it upside down, and the two make a parallelogram.
11. 1:19Same base, same height. So a triangle is half of base times height.
12. 1:26A sail eight metres along the bottom and five metres high. Eight times five is forty. Half is twenty square metres.
13. 1:36Cut, slide, or double it. Base times height, and half of it for a triangle.