Sizu Riatto
Algebra · Chapter 17

Squares, Roots and Right Triangles

A square root undoes a square. Between perfect squares you can bracket it, and the right-triangle rule a² + b² = c² finds a missing side.

Home › Calculators › Squares, Roots and Right Triangles

Opens on Chapter 17’s example: a square patio of 196 square feet; then √50 by bracketing; then a triangle with legs 9 and 12.

1. A square root
2. Bracket a root
3. A right triangle
Enter your numbers, then press Work it out.

How it works

√n is the number that, squared, gives n; x² = n has two answers, √n and −√n, though a length only uses the positive one. If n isn’t a perfect square, it sits between two that are: 49 < 50 < 64, so √50 is between 7 and 8. One divide-and-average step sharpens it: divide n by your guess and average the two. In a right triangle the side opposite the right angle (the hypotenuse) is longest, and a² + b² = c²: add the squares of the legs and take the root, or, for a missing leg, take the known leg’s square from the hypotenuse’s.

From the book, Chapter 17. A 196 sq ft square patio is 14 ft on a side. √50 is between 7 and 8, about 7.07. Legs of 9 and 12 give c² = 225, so 15 (not √441); a 13 ft ladder 5 ft out reaches 12 ft up; a 48 by 27 inch screen is about 55.1 inches across. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. The algebra is done with exact fractions, so 1/3 stays 1/3; decimals are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.