Double the side
The area becomes 2 × 2 = 4 times as big.
A first guess
Baudhāyana (about 800 BCE) asked: How can we make a square with double the area of a given square?
A first guess: double the length of every side. A square has side 2 and area 4. We double its side to 4.
What will the area of the new square be?
What this lesson covers
The idea
Doubling the length of each side of a square does not double its area: the new square has 2 × 2 = 4 times the area of the original.
A first guess
Baudhāyana (about 800 BCE) asked: How can we make a square with double the area of a given square?
A first guess: double the length of every side. A square has side 2 and area 4. We double its side to 4.
What will the area of the new square be?
- 8 sq. units
- 12 sq. units
- 16 sq. units
Fill the big square
The blue square is the old square. The dashed square has double the side. Tap an empty corner to put in another copy of the blue square. How many copies fit?
2 across and 2 down
Doubling the length of each side of a square does not double its area. The new square has 2 × 2 = 4 times the area of the original.
The side doubles across and it doubles down. That is 2 × 2 = 4 copies of the old square.
3 × 3 = 9 becomes 6 × 6 = 36. 36 = 4 × 9. Four times, not two times.
Notes
Double the side of a square and the area becomes 2 × 2 = 4 times as big. To get a square with double the area we need something other than doubling the side.
Check yourself
A square floor has side 6 m. A new floor has double the side. What is the area of the new floor in square metres?
Answer: 144
New side 12 m. Area 12 × 12 = 144. Check: old area 6 × 6 = 36 and 4 × 36 = 144.
A square has area 25 sq. cm. A new square has twice the side. What is its area?
The side of a square becomes 3 times as long. The area becomes ...
A new square has 4 times the area of an old square. How long is its side compared with the old side?
- 50 sq. cm. That doubles the area. But the side doubles across and down: 2 × 2 = 4 times.
- 100 sq. cm — correct. Yes! The side 5 becomes 10, and 10 × 10 = 100 = 4 × 25.
- 625 sq. cm. That is 25 × 25. The area grows 2 × 2 = 4 times, not 25 times.
- 3 times as big. The area is side × side. Both the width and the height grow 3 times.
- 6 times as big. You added 3 + 3. The two growths multiply: 3 × 3.
- 9 times as big — correct. Yes! 3 × 3 = 9. Double the side gives 2 × 2 = 4 times, triple the side gives 3 × 3 = 9 times.
- 2 times as long — correct. Yes! 2 × 2 = 4. A side twice as long gives an area 4 times as big.
- 4 times as long. Then the area would be 4 × 4 = 16 times as big.
- 8 times as long. Then the area would be 8 × 8 = 64 times as big.