find the length of the longest diagonal through a cuboid of sides 4cm by 13cm by 16cm. please show working
Bruh I have the exact same question
if the cuboid has sides x,y,z, then the main diagonal d has length
d^2 = x^2+y^2+z^2
To find the length of the longest diagonal through a cuboid, we can use the Pythagorean theorem.
The Pythagorean theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
In a cuboid, the longest diagonal passes through the opposite corners of the cuboid, which creates a right-angled triangle with the three sides being the lengths of the three edges of the cuboid.
Let's label the sides of the cuboid as follows:
Length = 13 cm
Width = 4 cm
Height = 16 cm
Using the Pythagorean theorem, we can find the length of the longest diagonal (d) as follows:
d² = Length² + Width² + Height²
d² = (13cm)² + (4cm)² + (16cm)²
d² = 169cm² + 16cm² + 256cm²
d² = 441cm²
Taking the square root of both sides, we get:
d = sqrt(441cm²)
Therefore, the length of the longest diagonal through the cuboid is approximately 21 cm.
To find the length of the longest diagonal in a cuboid, we can use the 3D Pythagorean theorem.
The 3D Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse (longest side) is equal to the sum of the squares of the other two sides.
In a cuboid, the longest diagonal is the hypotenuse of a right-angled triangle formed using the three dimensions of the cuboid.
Given the sides of the cuboid as 4 cm, 13 cm, and 16 cm, we can find the length of the longest diagonal as follows:
1. Calculate the square of the length of each side:
- Side A: 4^2 = 16
- Side B: 13^2 = 169
- Side C: 16^2 = 256
2. Add up the squares of the sides:
16 + 169 + 256 = 441
3. Finally, take the square root of the sum to find the length of the longest diagonal of the cuboid:
√441 = 21 cm
Therefore, the length of the longest diagonal through a cuboid with sides measuring 4 cm by 13 cm by 16 cm is 21 cm.