A frustum of a pyramid is 16 cm square at the bottom 6cm square at the top and 12cm high find the volume of the frustum
There are lots of resources online to explain and derive the following formula.
V = 12/3 (16+6+√(16*6)) = 88+16√6
To find the volume of the frustum, we can use the formula for the volume of a frustum of a pyramid, which is:
V = (1/3) * h * (A1 + A2 + √(A1 * A2))
where V is the volume, h is the height, A1 is the area of the larger base, and A2 is the area of the smaller base.
In this case, the height (h) of the frustum is given as 12 cm. The area of the larger base (A1) is 16 cm², and the area of the smaller base (A2) is 6 cm².
Now we can substitute these values into the formula and calculate the volume:
V = (1/3) * 12 cm * (16 cm² + 6 cm² + √(16 cm² * 6 cm²))
V = (1/3) * 12 cm * (22 cm² + √(96 cm⁴))
Next, let's calculate the area of the product under the square root:
√(96 cm⁴) = √((16 cm² * 6 cm²)²) = 16 cm² * 6 cm² = 96 cm⁴
Now we can substitute this value back to the formula:
V = (1/3) * 12 cm * (22 cm² + √(96 cm⁴))
V = (1/3) * 12 cm * (22 cm² + 96 cm²)
V = (1/3) * 12 cm * 118 cm²
V = 4 cm * 118 cm²
V = 472 cm³
Therefore, the volume of the frustum is 472 cm³.