How to find the diagonal for a cube and cuboid?

Diagonal for a Cube:
To find the diagonal of a cube, which is a special type of cuboid with all sides equal, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Step 1: Measure the length of one side of the cube. Let's call this value "a."

Step 2: Apply the Pythagorean theorem. Since a cube is made up of six congruent squares, one face diagonal can be considered as the hypotenuse of a right-angled triangle, with the two sides equal to "a."

The formula to find the diagonal of a face, also known as the space diagonal of the cube, is:

D = a√3

where D represents the diagonal and "a" represents the length of one side.

Diagonal for a Cuboid:
Finding the diagonal of a cuboid involves a similar process of using the Pythagorean theorem. A cuboid is a three-dimensional figure with six rectangular faces.

Step 1: Measure the length (l), width (w), and height (h) of the cuboid.

Step 2: Apply the Pythagorean theorem to find the diagonal. The diagonal of a cuboid, also known as the space diagonal, can be found using the formula:

D = √(l^2 + w^2 + h^2)

where D represents the diagonal, "l" represents the length, "w" represents the width, and "h" represents the height.

This formula calculates the length of the diagonal by finding the square root of the sum of the squares of the three dimensions.

Example:
Let's consider a cuboid with dimensions: length (l) = 4 cm, width (w) = 3 cm, and height (h) = 5 cm.

Using the formula D = √(l^2 + w^2 + h^2):

D = √(4^2 + 3^2 + 5^2)
D = √(16 + 9 + 25)
D = √50
D ≈ 7.07 cm

Therefore, the diagonal of the given cuboid is approximately 7.07 cm.

Note: It is important to remember that the units used for measuring the sides should be consistent throughout the calculation to obtain accurate results.