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Mathable Smart Lab

GRADE 8 • GEOMETRY Strand: Area, Right Triangles and Square Roots

The Baudhāyana-Pythagoras Theorem Smart Lab

Double and halve squares, explore \(\sqrt2\), combine squares, apply \(a^2+b^2=c^2\), and discover Baudhāyana triples.

What is the Baudhāyana-Pythagoras Theorem?

Baudhāyana described how the square on the diagonal of a right triangle relates to the squares on its two perpendicular sides. In modern notation, this is \(a^2+b^2=c^2\), where \(c\) is the hypotenuse.

Doubling a square: A square constructed on the diagonal of a given square has double the area of the original square.
Halving a square: Joining the midpoints of a square’s sides forms a smaller tilted square with half the original area.
Right triangle theorem: For a right triangle, area of square on hypotenuse = sum of areas of squares on the other two sides.

Move the side length. The diagonal square has twice the area of the original square.

Rules Hub

Square Area

If side length is \(s\), area is \(s^2\).

Doubling Square

The square on a square’s diagonal has area \(2s^2\).

Isosceles Right Triangle

If equal sides are \(a\), then \(c^2=2a^2\), so \(c=a\sqrt2\).

\(\sqrt2\)

\(\sqrt2\approx1.41421356...\), a non-terminating, non-fractional number.

Baudhāyana Theorem

For a right triangle: \(a^2+b^2=c^2\).

Baudhāyana Triples

Integer triples such as \((3,4,5)\), \((5,12,13)\), \((8,15,17)\) satisfy \(a^2+b^2=c^2\).

Guided Solve Lab

Score: 0Current Streak: 🔥 0Badge: Theorem Explorer


Activity Zone

🎛️ Right Triangle Calculator

Enter two shorter sides. The lab computes the hypotenuse using \(a^2+b^2=c^2\).

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🔍 Baudhāyana Triple Checker

Check whether three integers can be the sides of a right triangle.

⬛ Square Area Combiner

Choose two square side lengths. The larger square side is the hypotenuse.

🧩 Rule Identifier

Square on the diagonal of a square has double area.

Worksheet Generator

Generate printable practice on doubling/halving squares, \(\sqrt2\), right-triangle calculations, missing sides, and Baudhāyana triples.



Real-Life Lab

Teacher Tools

Learning Outcomes

  • Explain why a square on a diagonal doubles area.
  • Construct or reason about a square with half area.
  • Relate an isosceles right triangle to \(\sqrt2\).
  • Apply \(a^2+b^2=c^2\) to find missing sides of right triangles.
  • Identify and generate Baudhāyana triples.

Exit Ticket Prompts

  • Why does doubling the side of a square create four times, not double, the area?
  • Find the hypotenuse when sides are 5 cm and 12 cm.
  • Check whether \((8,15,17)\) is a Baudhāyana triple.