How do you find the length of the sides of a square when you only know of its diagonal measure

A Geometry review book I'm reading wants me to find the area of a square that has a diagonal measure of 10 units. However, as you know I need to find the length and square that to find the area. How do I find the length from the square's diagonal measure alone?

there all wrong. the answer is tablecloth

You do it by applying the Pythagorean theorem.

The diagonal is the hypotenuse of a triangle formed by two sides of the square. In this case, the square of the hypotenuse is 100 (square) units. The sum of the squares of the other two sides is also 100 units. The other two sides are equal because we're dealing with a square, so the square of each side is 50 units, and each side's length is the square root of 50 units (about 7.07 units).

The diagonal of a square forms a 45°-45°-90° triangle with the sides of the square being the legs of the triangle and the diagonal being the hypotenuse. Since sin45° = cos45° = √2/2, the diagonal of the square is √2 times the side of the square.

If the diagonal is 10, the side is 10/√2 = 5√2 and the area is (5√2)² = 50 sq units.

if you draw a diagonal through a square you are left with two triangles. IF all sides of a square are the same then the two legs of the triangle of going to be equal.


triangle is a 45-45-90 triangle.

leg 1=x leg2=x hypotentuse=sqrt2*x

since we know that the diagonal or hypotenuse is 10 units then we set up an equation to find x.

10=sqrt2*x
x=10/sqrt2

x=5sqrt2 (I rationalized the deonminator by multiplying the numerator and denominator by sqrt2/sqrt2).

So the length of the legs of the triangle is 5sqrt2. Since the legs of the triangle are the sides of the square then the sides of the the square is 5sqrt2.

Area of a square= l*w

area of a square=5sqrt2*5sqrt2=(5sqrt2)^2= 5^2*(sqrt2^2)

area=50units.

thats a case of pythagoras. as we know the length of the diagonale, and know that diagonals are bisecting each others as well as forming a rectangular triangle with any of the sides in a square, we know already two sides of the triangle, sooo
c²= a² + b², so c= sqrt(a²+b²), while a and b are both at the same length, e/2 (e is the diagonale).
so c = sqrt(25+25) ~7.071 units long.
while c is standing for any of the sides

If the side of a square is L, then the diagonal is sqrt(L^2 + L^2)

...think Pythagoras

ie L*sqrt(2)

So L*sqrt(2) = 10

L = 10/sqrt(2) <<<

(and the area of the square = L^2 = (10^2)/2 = 50)

side= a
then
diag of a square = a√2
So ,
side of square = diag / √2 = 10 / √2
area = 100 /2 = 50 sq. units.