Q1. The sides of two similar triangles are in the ratio 2:3. If the area of the smaller is 20 cm², find the area of the larger.
Ratio of areas = (ratio of sides)² = (2/3)² = 4/9
Area of larger = 20 × (9/4)
Area of larger = 45 cm²
Triangles — Class 10
Similar triangles have the same shape at different sizes. Master the criteria, then use the calculator to find any side of a right triangle instantly.
Similar trianglesSame shape, proportional sides, equal corresponding angles — written ΔABC ~ ΔPQR.
Basic Proportionality Theorem (Thales)A line parallel to one side of a triangle divides the other two sides in the same ratio.
Similarity criteriaAAA (all angles equal), SSS (all sides proportional), SAS (two sides proportional + included angle equal).
Pythagoras theoremIn a right triangle, hypotenuse² = base² + height² (i.e. c² = a² + b²).
Right-triangle (Pythagoras) calculator Interactive
Enter the two legs of a right triangle to find the hypotenuse — or a leg and the hypotenuse to find the other leg.
Hypotenuse c = √(a²+b²)—
To find a leg instead, use c² − a² = b², i.e. rearrange: b = √(c² − a²).
Numericals practice Interactive
Pythagoras and similar-triangle problems with steps.
Loading…
Solved examples
Formula sheet
| Quantity | Formula |
|---|---|
| Pythagoras theorem | c² = a² + b² |
| Ratio of areas of similar triangles | = (ratio of sides)² |
| Basic proportionality theorem | AD/DB = AE/EC |
Common mistakes & exam wins
- For similar triangles, the ratio of AREAS is the SQUARE of the ratio of corresponding SIDES.
- AAA similarity only needs equal angles — sides can be any proportional size.
- A Pythagorean triple you should recognise: 3-4-5, 5-12-13, 8-15-17.
- The converse of Pythagoras also holds: if c² = a² + b², the triangle is right-angled.
Frequently asked questions
What is the Pythagoras theorem?
In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides: c² = a² + b².
What is the basic proportionality theorem?
If a line is drawn parallel to one side of a triangle intersecting the other two sides, it divides those sides in the same ratio (also called Thales' theorem).
How does the ratio of sides relate to the ratio of areas in similar triangles?
The ratio of the areas equals the square of the ratio of the corresponding sides.