A solid granite pillar is crafted as a right prism whose base is an equilateral triangle with a side length of 12 cm. The total height of the pillar is 20 cm. A central cylindrical hole with a diameter of 6 cm is drilled completely through the pillar from the top surface to the bottom surface. Determine the volume of the remaining granite block. (Use π ≈ 3.1416 and √3 ≈ 1.732)

The correct answer is 681.6 cm³.

To find the volume of the remaining granite block, we calculate the total volume of the triangular prism and subtract the volume of the central cylindrical hole drilled completely through it.

Step 1: Calculate the volume of the triangular prism

The base of the prism is an equilateral triangle with side length a = 12 cm. The area of an equilateral triangle is given by the formula:

$A_{\text{base}}=\frac{\sqrt{3}}{4}\times a^2$

Substituting $\sqrt{3}\approx 1.732$ and a = 12 cm:

$A_{\text{base}}=\frac{1.732}{4}\times {12}^2=0.433\times 144=62.352\text{cm}{2}^{}$

Given the total height of the pillar h = 20 cm, the volume of the triangular prism is:

$V_{\text{prism}}=A_{\text{base}}\times h=62.352\times 20=1247.04\text{cm}{3}^{}$

Step 2: Calculate the volume of the cylindrical hole

The diameter of the cylinder is d = 6 cm, which gives a radius of:

$r=\frac{6}{2}=3\text{cm}$

The volume of the cylindrical hole extending through the total height h = 20 cm is:

$V_{\text{cylinder}}=\pi \times r^2\times h$

Using $\pi \approx 3.1416$:

$V_{\text{cylinder}}=3.1416\times 3^2\times 20=3.1416\times 9\times 20=565.488\text{cm}{3}^{}$

Step 3: Calculate the volume of the remaining granite block

Subtracting the volume of the cylindrical hole from the total volume of the prism:

$V_{\text{remaining}}=V_{\text{prism}}-V_{\text{cylinder}}$

$V_{\text{remaining}}=1247.04-565.488=681.552\text{cm}{3}^{}$

Rounding off to one decimal place gives 681.6 cm³.