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Diagonal of A Square: Definition and Examples | EDU.COM

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Diagonal of A Square: Definition and Examples | EDU.COMEDU.COMResourcesBlogGuidePodcastPlanBackHomesvg]:size-3.5">Math Glossarysvg]:size-3.5">Diagonal of A SquareDiagonal of A Square: Definition and ExamplesTable of ContentsDiagonal of a Square Definition and Properties of a Square's Diagonal

The diagonal of a square is a line segment that connects opposite vertices (corners) of the square. In a square, which is a flat, closed shape with four equal sides and four right angles, there are two diagonals. These diagonals are equal in length and bisect each other at right angles, forming four right triangles within the square.

A square's diagonals have several important properties. They divide the square into two congruent right-angle isosceles triangles. When the diagonals meet the vertices, they form 454545-degree angles, effectively bisecting each pair of opposite angles. The length of a diagonal can be calculated using the Pythagoras' theorem and is expressed by the formula d=a2d = a\sqrt{2}d=a2​, where ddd is the diagonal length and aaa is the side length of the square.

Examples of Finding a Square's Diagonal Example 1: Finding the Diagonal of a Square with Known Side Length Problem:

Find the diagonal length of a square with 181818 inches of side.

Step-by-step solution:

Step 1, We know the side length of the square is a=18a = 18a=18 inches.

Step 2, Apply the diagonal formula. The formula for a square's diagonal is d=a2d = a\sqrt{2}d=a2​.

Step 3, Substitute the value of side length into the formula:

d=18×2d = 18 \times \sqrt{2}d=18×2​ d=18×1.414d = 18 \times 1.414d=18×1.414 d=25.42d = 25.42d=25.42

Step 4, Write the final answer with proper units. The diagonal length of the square is 25.4225.4225.42 inches.

Example 2: Finding Side Length from Diagonal Length Problem:

The length of the diagonal of a square is 15215\sqrt{2}152​ inches. Find the side length of its side.

Step-by-step solution:

Step 1, We know the diagonal length d=152d = 15\sqrt{2}d=152​ inches.

Step 2, Use the diagonal formula and solve for the unknown side length aaa:

d=a2d = a\sqrt{2}d=a2​

Step 3, Substitute the known value of the diagonal:

152=a215\sqrt{2} = a\sqrt{2}152​=a2​

Step 4, Compare both sides of the equation to find the value of aaa:

a=15a = 15a=15

Step 5, Write the final answer with proper units. The side length of the square is 151515 inches.

Example 3: Finding Diagonal Length from the Perimeter Problem:

Find the diagonal length of a square field with a perimeter of 124124124 feet.

Step-by-step solution:

Step 1, Use the perimeter to find the side length. The perimeter of a square equals 4a4a4a where aaa is the side length.

124=4a124 = 4a124=4a

Step 2, Solve for the side length:

a=1244=31a = \frac{124}{4} = 31a=4124​=31 feet

Step 3, Now that we know the side length, apply the diagonal formula:

d=a2d = a\sqrt{2}d=a2​ d=31×2d = 31 \times \sqrt{2}d=31×2​ d=31×1.414d = 31 \times 1.414d=31×1.414 d=43.83d = 43.83d=43.83 feet

Step 4, Write the final answer with proper units. The diagonal length of the square field is 43.8343.8343.83 feet.

Comments(1)MCMs. CarterSeptember 17, 2025I’ve used the diagonal formula from this page to help my kids with their geometry homework. The examples really make it easy to understand, and they’ve started solving problems confidently. Great resource!

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