The Baudhāyana-Pythagoras Theorem
Class 8 · Mathematics
5 ideas in Class 8 Mathematics, explained in plain words, with games and quizzes to practise.
Doubling and Halving a Square
Part of: Baudhāyana's theorem and its uses
Doubling the side of a square makes its area 4 times as big, not 2 times. To get a square with double the area, draw a new square on the diagonal of the first one. To get a square with half the area, join the midpoints of the sides of the first square. Both facts can be seen by cutting the figures into small equal triangles.
Why it matters. Children often think that doubling a length doubles the area. Seeing the diagonal trick, found in an ancient Indian text, shows how length and area are linked and prepares them for the theorem that follows.
Try it in two minutes
- Draw a square on squared paper. Ask your child to draw a square with twice the side and count the small squares inside.
- Draw the diagonal of the first square and a square on it. Count the small triangles in each square.
- Join the midpoints of the sides of a square and ask what fraction of the area the tilted square covers.
Try at home. Fold a square piece of paper so the crease lines pass through the midpoints of the sides, and ask your child what fraction of the sheet is left in the middle.
Hypotenuse of an Isosceles Right Triangle
Part of: Baudhāyana's theorem and its uses
A right triangle with two equal sides is half of a square. If the equal sides are a and the hypotenuse is c, the square on the hypotenuse has double the area of the square on a, so c² is 2 × a². For equal sides of 1 unit, c² is 2 and c is the square root of 2, which lies between 1.414 and 1.415. This number goes on forever without repeating and cannot be written as a fraction m/n.
Why it matters. Square roots turn up in measuring diagonals and distances. Children see here that some lengths are exact in geometry but not exact as decimals.
Try it in two minutes
- Draw a square of side 1 on squared paper and its diagonal. Say that the diagonal has length root 2.
- Ask: is root 2 more than 1? Is it less than 2? Square 1.4 and 1.5 to see which side of 2 they fall.
- Ask your child to find the square of the hypotenuse for legs of 5 units.
Try at home. Use a calculator to square 1.41, 1.414 and 1.4142 and see how close each gets to 2 without ever being exactly 2.
The Baudhāyana-Pythagoras Theorem
Part of: Baudhāyana's theorem and its uses
Baudhāyana found that the square on the hypotenuse of a right triangle has area equal to the sum of the areas of the squares on the other two sides. In symbols, if the shorter sides are a and b and the hypotenuse is c, then a² + b² = c². For legs 3 and 4, c² is 9 + 16 = 25, so c is 5. The same fact is known as the Pythagorean theorem.
Why it matters. This is one of the most useful facts in geometry. It lets children find an unknown length without measuring and is used in building, maps and later maths.
Try it in two minutes
- Draw a right triangle with sides 3 cm and 4 cm. Measure the third side and check it is 5 cm.
- Draw squares on all three sides on squared paper and count the unit squares to see that the two small ones add up to the big one.
- Ask: if the shorter sides are 6 and 8, what should the hypotenuse be?
Try at home. Ask your child to find the hypotenuse of a triangle with sides 9 and 12 and check it with a ruler.
Using the Theorem to Find Lengths
Part of: Baudhāyana's theorem and its uses
The theorem can find a missing side of any right triangle. For a missing hypotenuse, add the squares of the shorter sides and take the root. For a missing shorter side, subtract the square of the known side from the square of the hypotenuse. A rectangle's diagonal, the side of a rhombus from its diagonals, and a ladder against a wall are all right triangle problems.
Why it matters. Real objects like ladders, screens and fields have right angles. Children who can spot the right triangle can find lengths without measuring.
Try it in two minutes
- Ask: a ladder 5 m long leans on a wall with its foot 3 m away. How high does it reach?
- Draw a rectangle 6 by 8 and ask for its diagonal.
- Ask your child to explain whether to add or subtract the squares when the hypotenuse is known.
Try at home. Measure the sides of a book or screen and check the diagonal with the theorem and a ruler.
Baudhāyana Triples
Part of: Baudhāyana's theorem and its uses
Three whole numbers (a, b, c) with a² + b² equal to c² are called a Baudhāyana triple, for example (3, 4, 5) and (5, 12, 13). Multiplying every number of a triple by the same whole number gives another triple, so there are infinitely many. A triple with no common factor above 1 is primitive. Using (n − 1)² + (2n − 1) = n², an odd square such as 49 gives n of 25 and the triple (7, 24, 25).
Why it matters. Spotting patterns and proving that they always work is the heart of mathematics. Triples also give neat whole-number lengths for building and carpentry.
Try it in two minutes
- Ask your child to check that (5, 12, 13) works by squaring each number.
- Multiply 3, 4, 5 by 2, by 3 and by 10 and check each result.
- Ask whether (6, 8, 10) can be cut down to a smaller triple, and what that tells us.
Try at home. Ask your child to list all the right-triangle triples they can find where every number is at most 20.
What your child will practise
- Baudhāyana Triples: Number Sort (Sort It Out)
- Doubling and Halving a Square: Mission Control Quest (Collect the Answer)
- Hypotenuse of an Isosceles Right Triangle: Desert Quick Fire (Quick Fire)
- The Baudhāyana-Pythagoras Theorem: Ocean Puzzle (Clue Puzzle)
- Using the Theorem to Find Lengths: Ocean Quick Fire (Quick Fire)
- Quiz: Baudhāyana Triples (8 questions)
- Quiz: Doubling and Halving a Square (8 questions)
- Quiz: Hypotenuse of an Isosceles Right Triangle (8 questions)
- Quiz: The Baudhāyana-Pythagoras Theorem (8 questions)