The Baudhayana-Pythagoras Theorem · Lesson 1 of 9
Doubling and Halving a Square
“Use diagonals, congruent triangles, and midpoint constructions to double or halve a square’s area.”
• Distinguish a change in side length from a change in area. • Construct and explain a square with twice a given area. • Construct and explain a square with half a given area. • Use equal triangular pieces to justify an area comparison.
Imagine that you have a square sheet of coloured paper and want to make a new square with exactly twice its area. You may cut another identical sheet and rearrange the pieces, but you must not leave gaps or overlaps. Making a larger shape is easy; making the correct larger square needs more thought. Your first task is to decide what must double: the length of the boundary, or the amount of paper enclosed by it?
Area measures the surface covered by a shape. Side length measures only a distance along its edge. These quantities are connected, but they do not change in the same way. A drawing and a few paper triangles will help you see the difference before you use any new formula.
The constructions in this lesson move in both directions. First you will combine equal areas; then you will work backwards to create half an area. Keep asking two questions: Why is the new shape a square, and why does it have the required area?
Doubling a Square
Start with a square of side 3 units. It contains 3 rows of 3 unit squares, so its area is 9 square units. If you double the side to 6 units, there are now 6 rows of 6 unit squares. The area becomes 36 square units, which is four times 9, not twice 9.
You can also see this without counting every unit square. A square whose side is twice the original side can be divided into four copies of the original square: two across and two down. Doubling affects both dimensions, so the area is multiplied by 2 × 2.
The required area in our example is 18 square units. A side of 6 units is therefore too long. We need a different construction, and the diagonal of the original square provides it.
Problem
A square has side 5 cm. A student doubles its side to obtain twice its area. Is the result correct?
- 1.The original area is 5 × 5 = 25 cm². Twice this area would be 50 cm².
- 2.The proposed side is 10 cm, so its area is 10 × 10 = 100 cm².
- 3.Since 100 ÷ 25 = 4, the proposed square has four times the area.
- 4.The error is doubling both dimensions when the aim is only to double the area.
Building a Square on the Diagonal
Draw a diagonal of your original square. It divides the square into two identical right triangles. Each has two equal perpendicular sides and two angles of 45°. These triangles are congruent: they have the same size and shape and can fit exactly over one another.
Now use the diagonal as one side of a new square. This new square is tilted relative to the original. Tilting does not change the meaning of a square: it still has four equal sides and four right angles.
Extend the horizontal and vertical sides of the original square through the construction. The original square can be counted as two congruent triangular pieces; the square on its diagonal can be counted as four pieces of exactly the same size. Four pieces have twice the area of two pieces. This establishes the area relationship without needing the diagonal’s numerical length.
Each small triangle has equal perpendicular sides. The separate original square has two such pieces; the tilted square has four. Compare pieces of the same size, not how large a shape looks because of its orientation.
Doubling a Square Using Paper
Cut out two identical squares. Leave one whole. Draw both diagonals on the second square and cut along them to obtain four congruent triangles. Place one triangle against each side of the whole square, with the longest edge of each triangle touching that side.
The four outer tips become the vertices of the new square. Each outer side consists of two aligned equal edges from neighbouring triangles. At every outer tip, two 45° angles meet to make a right angle. Thus the arrangement is a square, not merely a four-sided shape that looks square.
No piece has been stretched, lost, or covered by another piece. The new square contains the whole first square and all of the second square. Its area is therefore exactly twice the original area. Its side has the same length as the diagonal of an original square.
Take two identical squares and cut each along just one diagonal. You now have four identical right triangles. Arrange their right-angle vertices together at one point. Their four hypotenuses form the outside boundary. Explain why the four central right angles fill a complete turn and why each outside corner is 45° + 45°. Compare this construction with the whole-square-and-four-triangles construction.
Problem
A square has area 7 cm². You construct a square on its diagonal, then repeat the construction once more. Find the successive areas.
- 1.The first diagonal construction doubles area: 2 × 7 = 14 cm².
- 2.The next construction doubles the new area: 2 × 14 = 28 cm².
- 3.The areas are 7, 14, and 28 cm². The final area is four times the starting area.
- 4.The operation doubles the current area each time; it does not add a fixed 7 cm² each time.
Halving a Square
Now reverse the question. Suppose you have one square and want a smaller square with half its area. Halving each side will not work: two shorter sides fit along each original side, so four of those tiny squares fill the original. Each tiny square has one quarter of the area.
Instead, mark the midpoint of every side and join neighbouring midpoints. The four joining segments enclose a tilted square. Around it are four equal corner triangles. If the original side is s, each corner triangle has perpendicular sides s/2 and s/2.
Each corner triangle therefore has area ½ × (s/2) × (s/2) = s²/8. Together the four corners have area s²/2. Removing them from the original area s² leaves s²/2 inside. This is the required half-area square.
Halving a Square Using Paper
Fold a square so that each corner moves to the centre. The fold line for each corner passes through the midpoints of its two adjacent sides. The four crease lines outline the smaller square. Unfold the paper and trace these lines so that you can compare the areas without hiding any regions.
Why is the middle shape a square? The corner triangles are congruent, so their hypotenuses—the sides of the inner shape—are equal. Each corner triangle has two 45° angles. At a midpoint, the inner angle is 180° − 45° − 45° = 90°. Equal sides and right angles together establish that the inner figure is a square.
Joining opposite midpoints divides the inner square into four congruent triangles. The whole original square contains eight triangles of that same size. This gives a second area explanation: the inner square uses four out of eight equal pieces.
Problem
A square has side 8 cm. Compare the midpoint square with a square whose side is 4 cm.
- 1.The original area is 8 × 8 = 64 cm².
- 2.The midpoint construction has area 64 ÷ 2 = 32 cm².
- 3.The square with half the original side has area 4 × 4 = 16 cm².
- 4.Thus the midpoint square has twice the area of the half-side square. They are different constructions.
Problem
A square formed by joining the midpoints has area 18 square units. Find the area of the original square.
- 1.The inner square has half the original area.
- 2.Therefore the original area is 2 × 18 = 36 square units.
- 3.If you also want the original side, find the positive number whose square is 36: it is 6 units.
- 4.Check: half of 6 × 6 is 18, as required.
Quiz
If a square’s side is doubled, what happens to its area?
A square has area 11 cm². What is the area of the square constructed on its diagonal?
Which construction gives a square with half the original area?
Why is counting triangles useful in these constructions?
Which evidence proves that the inner midpoint figure is a square?
Practice Problems
- A square has area 24 cm². Find the area of the square on its diagonal. Solution: 1. The diagonal construction doubles area. 2. Calculate 2 × 24 = 48 cm².
- A square has side 10 cm. Find the area enclosed by its side midpoints. Solution: 1. The original area is 10² = 100 cm². 2. The midpoint square has half this area: 100 ÷ 2 = 50 cm².
- Starting with area 6 square units, double the area three times. Solution: 1. After the first construction the area is 12. 2. After the second it is 24; after the third it is 48 square units. 3. Three doublings multiply the starting area by 8.
- A student says that halving the side halves the area. Explain the error using a square of side 6 units. Solution: 1. Its area is 36 square units. 2. Half its side is 3 units, giving area 9 square units. 3. Since 9 is one quarter of 36, both dimensions have been halved. A midpoint construction would instead give area 18.
- A square’s midpoint square has area 20 cm². A new square is then built on a diagonal of the original square. Find the new area. Solution: 1. The original area is twice 20, or 40 cm². 2. The diagonal construction doubles the original area again. 3. The new area is 80 cm². Keep track of which square each operation refers to.
Key Takeaways
• Area measures surface coverage; side length measures distance. • Doubling a square’s side multiplies its area by four. • The square on a diagonal has twice the original square’s area. • Joining the four side midpoints gives a square with half the original area. • Halving the side gives one quarter of the original area. • Congruent pieces and right-angle reasoning explain why the constructions work.
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Hypotenuse of an Isosceles Right Triangle