A cuboid has it's longest diagonal as 18cm and a square base of side 8cm, calculate the surface area and volume of the cuboid
I don't no how to solve this problem
if the height is h, then
8^2 + 8^2 + h^2 = 18^2
h = 14
So the area is 2(8*8 + 8*14 + 8*14)
and the volume is 8^2 * 14
If the dimensions of a rectangular cuboid are a, b and c,
then the length of the space diagonal is:
d = √ ( a² + b² + c² )
Since the base is square then:
a = b = 8 cm
d = √ ( a² + b² + c² )
d = √ ( 8² + 8² + c² )
18 = √ ( 64 + 64 + c² )
18 = √ ( 128 + c² )
Rise both sides to power of two.
18² = 128 + c²
324 = 128 + c²
Subtract 128 to both sides
324 - 128 = c²
196 = c²
c = √196
c = 14 cm
Surface area is:
A = 2 ( a ∙ b + a ∙ c + b ∙ c )
A = 2 ( 8 ∙ 8 + 8 ∙ 14 + 8 ∙ 14 )
A = 2 ( 64 + 112 + 112 )
A = 2 ∙ 288
A = 576 cm²
Volume is:
V = a b c
V = 8 ∙ 8 ∙ 14
V = 896 cm³
To calculate the surface area of a cuboid, you need to find the areas of all of its six sides and then add them together. The formula to find the surface area of a cuboid is:
Surface Area = 2 * (length * width + length * height + width * height)
Given that the base of the cuboid is a square with a side length of 8 cm and the longest diagonal is 18 cm, we can determine the length (l), width (w), and height (h) of the cuboid using these measurements.
Since the diagonal can be considered as the hypotenuse of a right-angled triangle formed by the height, the base, and the diagonal itself, we can use the Pythagorean theorem to find the height.
In a right-angled triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b):
c^2 = a^2 + b^2
In this case, the base of the triangle is 8 cm, and the hypotenuse (diagonal) is 18 cm. We can set up the equation as follows:
18^2 = 8^2 + b^2
324 = 64 + b^2
b^2 = 324 - 64
b^2 = 260
Taking the square root of both sides, we get:
b = √260
After finding the value of b, we can substitute it into the surface area formula to calculate the surface area of the cuboid.
Next, to determine the volume of the cuboid, we can use the formula:
Volume = length * width * height
Substituting the values we have found for length, width, and height into this formula will give us the solution.
Let's solve it step by step:
1. Calculate the height (h) using the Pythagorean theorem:
b = √260
2. Calculate the surface area:
Surface Area = 2 * (length * width + length * height + width * height)
3. Calculate the volume:
Volume = length * width * height
By following these steps, you will be able to calculate the surface area and volume of the given cuboid.