Question Video: Finding the Area of a Square given Its Diagonal Length | Nagwa Question Video: Finding the Area of a Square given Its Diagonal Length | Nagwa

# Question Video: Finding the Area of a Square given Its Diagonal Length Mathematics • Second Year of Preparatory School

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Find the area of a square whose diagonal is 9 cm.

02:35

### Video Transcript

Find the area of a square whose diagonal is nine centimeters.

Letโs begin by sketching this square. Weโre told that its diagonal, which is the straight line segment connecting opposite vertices of the square, is nine centimeters long. We can then recall that the two diagonals of the square are of equal length and that we can find the area of a square with the diagonal of length ๐ units using the formula a half ๐ squared.

This is in fact the special case of the formula for finding the area of a rhombus. The area of a rhombus with diagonals of length ๐ one and ๐ two is a half ๐ one multiplied by ๐ two. A square is just a special type of rhombus in which both diagonals are the same length. So replacing ๐ one and ๐ two in this formula with ๐ gives the formula for the area of a square, using the length of its diagonal.

Returning to our earlier formula and substituting nine for the length of the squareโs diagonal gives that the area of this square is a half multiplied by nine squared. Nine squared is 81, so this simplifies to 81 over two. The units for area are square units, so we have that the area of the square is 81 over two square centimeters.

An alternative approach, which we wonโt go through in great detail here, would involve calculating the side length of the square. We recall that each diagonal of a square divides it into two congruent right triangles. Labeling the side length of the square as ๐  centimeters, we could then use the Pythagorean Theorem to write down the equation ๐  squared plus ๐  squared is equal to nine squared. This simplifies to two ๐  squared equals 81, and then dividing both sides of the equation by two gives ๐  squared equals 81 over two.

In fact, we can stop here and donโt actually need to find the side length of the square at all. The area of a square is its side length squared, which is exactly what we have here. This alternative method confirms our answer, which is that the area of a square with a diagonal of nine centimeters is 81 over two squared centimeters.

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