MathsCBSE Class 8 · The Baudhayana-Pythagoras Theorem

Doubling and Halving Squares and the Isosceles Right Triangle for Class 8

Q: Who wrote the Sulba-sutras?

The chapter calls the doubling of a square a result of Baudhayana in his Sulba-sutra (c. 800 BCE), with verse 1.9 for doubling a square.

Q: How do I write the hypotenuse if I cannot find root 2 exactly?

Write it as a root 2, and give bounds like 14.1 and 14.2 for a numerical value.

Q: Why does the square on a diagonal pass through the vertices?

Because the diagonals of a square bisect its angles, the sides of the original square, extended, pass through the vertices of the new square.

Updated 17 October 2026 · Fact-checked

The square drawn on the diagonal of a square has double its area, so the diagonal of a square of side a is a times root 2 and the hypotenuse of an isosceles right triangle with equal sides a is c = a root 2, with c squared = 2 a squared. Root 2 is about 1.414 but is neither a terminating decimal nor a fraction. Exams ask for hypotenuses, areas, bounds between decimals and short reasoning proofs.

The tilted square on the diagonal c has double the area of the square on side a, so c² = 2a² and c = a√2.

Illustration drawn for QuestionBuddy.

Understand Doubling and Halving Squares, and the Isosceles Right Triangle

Doubling a square. If each side of a square is doubled, the area becomes $2 \times 2 = 4$ times the original, not 2 times. Baudhayana's *Sulba-sutra* (the chapter dates it c. 800 BCE, verse 1.9) gives the answer: the diagonal of a square produces a square of double the area. Draw a square on the diagonal. The original square is made of 2 congruent right triangles; the new square (drawn with the original's sides extended through its vertices, which works because the diagonals of a square bisect its angles) is made of 4 such triangles. Adding more east-west and north-south lines shows squares made of 2, 4 and 8 small triangles, each double the previous one.

Halving a square. Reverse the construction: join the midpoints of the sides of a square to get a tilted square of half the area. Folding paper so that creases pass through the midpoints of the sides gives the same square $P QR S$ . A square with half the side has only a quarter of the area, so four such squares fill the original.

The isosceles right triangle. Cut a square along its diagonal: two congruent right triangles with two equal sides $a$ and hypotenuse $c$ (the side opposite the right angle) equal to the diagonal. The square on the hypotenuse has area $c^{2}$ and is double the square on a side, so
$ $c^{2} = 2 a^{2}, c = a 2, a = \frac{c}{2}, a^{2} = \frac{c ^{2}}{2} .$ $
Examples: equal sides 12 give $c^{2} = 288$ and $1 6^{2} = 256 < 288 < 289 = 1 7^{2}$ , so $c$ lies between 16 and 17. Hypotenuse $62$ gives $a = 6$ .

What is root 2? It is the positive number whose square is 2: $1^{2} = 1 < 2 < 4 = 2^{2}$ , so $1 < 2 < 2$ . Testing decimals narrows it down: $1.41 1^{2} = 1.990921$ , $1.41 2^{2} = 1.993744$ , $1.41 3^{2} = 1.996569$ , $1.41 4^{2} = 1.999396$ , $1.41 5^{2} = 2.002225$ . So $2$ lies between 1.414 and 1.415, and the process can go on forever.

*It is not a terminating decimal.* If a terminating decimal $x$ squared equalled 2, its last non-zero digit $d$ (one of 1 to 9) would make the last digit of $x^{2}$ , which is the last digit of $d^{2}$ , non-zero (it is 1, 4, 5, 6 or 9), but $2 = 2.000 \dots$ ends in zeros. So the decimal expansion of $2$ never ends.
*It is not a fraction.* If $2 = \frac{m}{n}$ then $2 n^{2} = m^{2}$ . In the prime factorisation of a square every prime occurs an even number of times, so 2 occurs an even number of times on the right but an odd number of times on the left. This is impossible, so $2$ is not $\frac{m}{n}$ . The chapter credits Euclid's *Elements* (c. 300 BCE) with a proof that it cannot be a fraction.

Using the formula. Given $a$ , find $c = a 2$ ; given $c$ , find $a = c / 2$ ; the area of an isosceles right triangle with hypotenuse $c$ is $\frac{1}{2} a^{2} = \frac{c ^{2}}{4}$ . When the value is not exact, give bounds with one digit after the decimal point.

The general theorem for any right triangle and Baudhayana triples are on the next topic page of this chapter.

Doubling and Halving Squares, and the Isosceles Right Triangle: formulas to remember

Square on the diagonal: $area on diagonal = 2 \times area of the square$; Baudhayana, verse 1.9 in the chapter.
Hypotenuse of an isosceles right triangle: $c = a 2, c^{2} = 2 a^{2}$; a is each of the equal sides.
Equal side from hypotenuse: $a = \frac{c}{2}, a^{2} = \frac{c ^{2}}{2}$; Useful when c is given.
Area of an isosceles right triangle: $area = \frac{1}{2} a^{2} = \frac{c ^{2}}{4}$; Using the hypotenuse c.
Diagonal of a square from its area: $d^{2} = 2 \times area$; So d = side x root 2.

How to solve Doubling and Halving Squares, and the Isosceles Right Triangle questions

Use this routine for diagonal and hypotenuse questions.

1Find the area of the given square, or use the side a of the isosceles right triangle.
2Use c squared = 2 times a squared (the square on the diagonal has double the area).
3If an exact root is not required, bracket c between two perfect squares, then refine to one decimal place using squares of decimals.
4To find a side from the hypotenuse, use a squared = c squared divided by 2.
5For areas of the triangle, remember area = (1/2) a squared, which can be written without finding a.
6State the unit and, for bounds, give both ends.

Doubling and Halving Squares, and the Isosceles Right Triangle in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Diagonal of a square

Suppose a square park has an area of 450 square metres and a footpath will run along one diagonal.

Find the length of the path exactly, and the side of the park to one decimal place.

Try it first, then show the solutionSolution

The square on the diagonal has double the area: $d^{2} = 2 \times 450 = 900$ , so $d = 30$ m.
Side $^{2} = 450$ . Since $21. 2^{2} = 449.44$ and $21. 3^{2} = 453.69$ , the side lies between 21.2 m and 21.3 m.

Answer: Path 30 m; side between 21.2 m and 21.3 m.

Memory hook: Double the area gives the diagonal squared; you do not need the side.

Halving a square

Imagine a square piece of paper of side 12 cm that you fold so that the creases join the midpoints of the sides.

What is the area of the tilted square you make, and between which two lengths does its side lie, correct to one decimal place?

Try it first, then show the solutionSolution

Area of the paper $= 144$ cm $^{2}$ , so the inner square has area $72$ cm $^{2}$ .
Side $= 72$ ; since $8. 4^{2} = 70.56$ and $8. 5^{2} = 72.25$ , the side lies between 8.4 cm and 8.5 cm.

Answer: 72 square cm; side between 8.4 cm and 8.5 cm

Memory hook: Joining midpoints halves the area, not the side.

Tricks to remember Doubling and Halving Squares, and the Isosceles Right Triangle

Short memory aids. Say each one aloud once, then use it on a question the same day.

Diagonal squared is twice the area
For a square: diagonal squared = 2 x side squared = 2 x area.
Use it when: Any diagonal-of-a-square question.
Half the area, not half the side
Half the side makes a quarter of the area; the midpoint square makes half the area.
Use it when: Halving squares questions.
Root 2 is 1.414
Remember 1.414 squared is 1.999396, so root 2 is between 1.414 and 1.415.
Use it when: Bounds on hypotenuse values.
Even powers of primes in squares
In a square number every prime appears an even number of times; 2 n squared has an odd number of 2s.
Use it when: Proving root 2 is not a fraction.

Doubling and Halving Squares, and the Isosceles Right Triangle: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Square on a diagonal · 1 mark · MCQBoard level

The area of the square drawn on the diagonal of a square of side 6 cm is: (a) 36 sq cm (b) 60 sq cm (c) 72 sq cm (d) 144 sq cm

Try it first, then show the solutionSolution

Area of the original square $= 36$ .
The square on its diagonal has double the area: $2 \times 36 = 72$ .

Answer: (c) 72 sq cm

Example 2: Hypotenuse with bounds · 2 marks · Short answerBoard level

The equal sides of an isosceles right triangle are 10 cm each. Find the hypotenuse exactly and give two integers and then two decimals (one digit after the point) between which it lies.

Try it first, then show the solutionSolution

$c^{2} = 2 \times 1 0^{2} = 200$ , so $c = 102$ . (1 mark)
$1 4^{2} = 196 < 200 < 225 = 1 5^{2}$ ; also $14. 1^{2} = 198.81 < 200 < 201.64 = 14. 2^{2}$ . So $14 < c < 15$ and $14.1 < c < 14.2$ . (1 mark)

Answer: c = 10 root 2, between 14 and 15, and between 14.1 and 14.2

Example 3: Finding the equal sides · 3 marks · Short answerHarder board level

The hypotenuse of an isosceles right triangle is 16 cm. Find the area of the triangle, and give bounds with one decimal digit for each equal side.

Try it first, then show the solutionSolution

$a^{2} = \frac{c ^{2}}{2} = \frac{256}{2} = 128$ . (1 mark)
Area $= \frac{1}{2} a^{2} = 64$ sq cm. (1 mark)
$11. 3^{2} = 127.69 < 128 < 129.96 = 11. 4^{2}$ , so each equal side is between 11.3 cm and 11.4 cm. (1 mark)

Answer: Area 64 sq cm; each equal side between 11.3 cm and 11.4 cm.

Example 4: Diagonal from the area · 3 marks · Short answerHarder board level

A square has area 98 square cm. Find the length of its diagonal. Is the side a whole number?

Try it first, then show the solutionSolution

The square on the diagonal has area $2 \times 98 = 196$ . (1 mark)
So the diagonal $= 196 = 14$ cm. (1 mark)
Side $= 98$ ; $9^{2} = 81 < 98 < 100 = 1 0^{2}$ , so the side is not a whole number (it equals $72$ ). (1 mark)

Answer: Diagonal 14 cm; the side is not a whole number.

Example 5: Why root 2 is not a fraction · 4 marks · Long answerHOTS / competency

Explain, using prime factorisation, why there are no counting numbers m and n with m / n equal to root 2. Why can root 2 not be a terminating decimal?

Try it first, then show the solutionSolution

Suppose $2 = \frac{m}{n}$ . Squaring, $2 n^{2} = m^{2}$ . (1 mark)
In the prime factorisation of any square every prime occurs an even number of times, so 2 occurs an even number of times in $m^{2}$ . (1 mark)
In $2 n^{2}$ the prime 2 occurs one more time than in $n^{2}$ , an odd number of times. A number cannot have both an odd and an even number of 2s in its factorisation, a contradiction. (1 mark)
If $2$ were a terminating decimal ending in a non-zero digit $d$ , its square would end in a non-zero digit (the last digit of $d^{2}$ is 1, 4, 5, 6 or 9), but 2 has only zeros after the decimal point. So it is not terminating. (1 mark)

Answer: Contradiction in the powers of 2, and a non-zero last digit of the square: root 2 is neither a fraction nor a terminating decimal.

Where students lose marks in Doubling and Halving Squares, and the Isosceles Right Triangle

The slips examiners see most often, and what to write instead.

Mistake: Doubling the side to double the area
Doubling the side gives four times the area.
Wrong working
"To double the area of a square of side 5, take side 10."
Right working
Side 10 has area 100, which is 4 times 25. Use side $52$ (the diagonal), area 50.
Do this: Say "area scales as the square of the side".
Usually costs: The whole part.
Mistake: Writing the hypotenuse as 2a
The hypotenuse of an isosceles right triangle is a root 2, which is about 1.414a, not 2a.
Wrong working
"Equal sides 12, so hypotenuse 24."
Right working
$c = 122$ , and $c^{2} = 288$ lies between $1 6^{2}$ and $1 7^{2}$ .
Do this: Use c squared = 2 a squared.
Usually costs: The answer.
Mistake: Calling root 2 equal to 1.414
1.414 squared is 1.999396, not 2. The decimal of root 2 never ends.
Wrong working
" $2 = 1.414$ exactly."
Right working
$1.414 < 2 < 1.415$ .
Do this: Use the sign for between and give both bounds.
Usually costs: 1 mark.
Mistake: Assuming the side of the half-area square is half the side
Half the area has side equal to the original side divided by root 2.
Wrong working
"Side 12, half area, so side 6."
Right working
Area 72, side $72 \approx 8.5$ .
Do this: Compute the area first.
Usually costs: The answer.

Exam tips

Start with an area equation: c squared = 2 a squared. It removes the need to find root 2.
When bounds are asked, show two perfect squares around the number, then the one-decimal refinement.
Draw the square and its diagonal-square; a labelled sketch earns the explanation mark.
In a proof by contradiction, state the assumption, the contradiction and the conclusion.
Write units (cm, square cm) on every line with measurements.

Practice questions on Doubling and Halving Squares, and the Isosceles Right Triangle

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · Assertion-reasonBoard level
Assertion (A): The square drawn on the diagonal of a square has double the area of the square.
Reason (R): The original square is made of 2 congruent right triangles and the square on its diagonal is made of 4 of them.
(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.
Hint
Count the small congruent triangles.
Answer
(a). 4 triangles against 2 triangles, all congruent, so double the area.
Q2 · 2 marks · Very short answerBoard level
A square piece of paper has side 12 cm. A square is made by joining the midpoints of its sides. Find its area, and say between which two numbers (to one decimal place) its side lies.
Hint
The tilted square has half the area.
Answer
Area $= \frac{144}{2} = 72$ cm $^{2}$ . Side $= 72$ , and $8. 4^{2} = 70.56 < 72 < 72.25 = 8. 5^{2}$ , so the side is between 8.4 cm and 8.5 cm.
Q3 · 3 marks · Short answerHarder board level
Show that a number with a terminating decimal expansion cannot have 2 as its square. (Hint: look at the last non-zero digit of the decimal.)
Hint
The square of d (1 to 9) never ends in 0.
Answer
Suppose $x$ has $k$ digits after the decimal point and the last digit is $d \neq = 0$ . Then $x^{2}$ has $2 k$ digits after the point, and its last digit is the last digit of $d^{2}$ . For $d = 1, \dots, 9$ the squares $1, 4, 9, 16, 25, 36, 49, 64, 81$ end in $1, 4, 9, 6, 5, 6, 9, 4, 1$ , never 0. So $x^{2}$ has a non-zero digit in the $2 k$ th place, while $2 = 2.000 \dots$ has none. Therefore $x^{2} \neq = 2$ .
Q4 · 4 marks · Case-basedHOTS / competency
Case study: a square plot has area 450 square metres. A path will run along both diagonals.
(i) Between which two numbers (one decimal place) does the side of the plot lie? (1 mark)
(ii) Find the length of each diagonal exactly. (1 mark)
(iii) Laying the path costs Rs 120 per metre. Find the cost for both diagonals. (1 mark)
(iv) A fence is built by joining the midpoints of the sides of the plot. Find the area inside the fence. (1 mark)
Hint
The square on the diagonal has area 900.
Answer
(i) $21. 2^{2} = 449.44 < 450 < 453.69 = 21. 3^{2}$ , so between 21.2 m and 21.3 m. (ii) $d^{2} = 2 \times 450 = 900$ , so $d = 30$ m. (iii) Length $= 2 \times 30 = 60$ m; cost $= 60 \times 120 = 7200$ rupees. (iv) The midpoint square has half the area: 225 square metres.
Q5 · 3 marks · Short answerHarder board level
The hypotenuse of an isosceles right triangle is 20 cm. Find its area without finding the length of the equal sides.
Hint
a squared = c squared divided by 2 and area = half of a squared.
Answer
$a^{2} = \frac{c ^{2}}{2} = \frac{400}{2} = 200$ . Area $= \frac{1}{2} a^{2} = 100$ sq cm. (Equivalently, area $= c^{2} /4 = 100$ .)

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.What area does the square on the diagonal have?Show answer
Double the area of the original square.
2.What is the hypotenuse of an isosceles right triangle with equal sides a?Show answer
a root 2.
3.Between which two decimals does root 2 lie?Show answer
1.414 and 1.415.
4.Is root 2 a fraction or a terminating decimal?Show answer
Neither.
5.Who is credited with the first proof that root 2 is not a fraction in the chapter?Show answer
Euclid, in his Elements (c. 300 BCE).
6.What is the hypotenuse?Show answer
The side opposite the right angle.

Doubling and Halving Squares, and the Isosceles Right Triangle: frequently asked questions

Who wrote the Sulba-sutras?

The chapter calls the doubling of a square a result of Baudhayana in his Sulba-sutra (c. 800 BCE), with verse 1.9 for doubling a square.

How do I write the hypotenuse if I cannot find root 2 exactly?

Write it as a root 2, and give bounds like 14.1 and 14.2 for a numerical value.

Why does the square on a diagonal pass through the vertices?

Because the diagonals of a square bisect its angles, the sides of the original square, extended, pass through the vertices of the new square.

Is the area of the isosceles right triangle half of the square on a side?

Yes: area = (1/2) a squared.