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Home » Quadrangles » Squares » How to Find the Diagonal of a Square

How to Find the Diagonal of a Square

Last updated: Jul 29, 2020 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

A square is a geometric shape which is fully determined by the lengths of its side, a. If we know the length of the side of a square, we know its perimeter, its area, the length of its diagonals, etc.

In today's lesson, we will find the length of a diagonal of a square using three simple formulas, derived from the length of the square's side, or its perimeter, or its area.

Problem

ABCD is a square with side length a. Find the length of its diagonal, AC.

Square with diagonals

Strategy

The angles of the square are all 90°, so each one of the triangles formed by the diagonals with the sides (ΔBAD , ΔABD, ΔDCA, ΔBDC) is a right triangle.

Both legs of each one of the triangles are sides of the square so their side is known and equal to 'a'. So, pick one of these triangles, and use the Pythagorean theorem to find the length of the diagonal from the length of the side.

Proof

(1) ABCD is a square //Given
(2) ∠BAD = 90° //definition of a square
(3) BA=AD='a' //given, defintion of a square
(4) AC2=a2+a2 //(2),(3) ,Pythagorean theorem
(5) AC=√(2a2)
(6) AC=a ·√2

Having done that, let's derive similar formulas using the square's perimeter and area.

The perimeter of a square, P, is 4 times the length of the side: P=4a. So the side is P/4, and we can plug that into the formula above: AC=P/4 ·√2

And similarly, the area of a square, A, is the side squared, or a2. A=a2. So the side is √A and we can plug that into the formula above: AC=√A ·√2=√(2A)

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About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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About

Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

  • Area of Geometric Shapes
  • Circles
    • Arcs, Angles, and Sectors
    • Chords
    • Inscribed Shapes
    • Tangent Lines
  • Lines and Angles
    • Intersecting Lines and Angles
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  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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