Question

a cone has a height of 16 centimeters and a radius of 12 centimeters. what is the exact lateral and surface area of the cone? type the correct answer in each box. use numerals instead of words. the lateral area is square centimeters. the total surface area is square centimeters.

Explanation:

Step1: Find the slant height \(l\)

Use the Pythagorean theorem \(l=\sqrt{h^{2}+r^{2}}\), where \(h = 16\) and \(r=12\). So \(l=\sqrt{16^{2}+12^{2}}=\sqrt{256 + 144}=\sqrt{400}=20\).

Step2: Calculate the lateral - area \(L\)

The formula for the lateral area of a cone is \(L=\pi rl\). Substitute \(r = 12\) and \(l = 20\) into the formula, we get \(L=\pi\times12\times20 = 240\pi\).

Step3: Calculate the base - area \(B\)

The formula for the base - area of a cone (a circle) is \(B=\pi r^{2}\). Substitute \(r = 12\), so \(B=\pi\times12^{2}=144\pi\).

Step4: Calculate the total surface area \(S\)

The formula for the total surface area of a cone is \(S = L + B\). Substitute \(L = 240\pi\) and \(B=144\pi\), we get \(S=240\pi+144\pi=384\pi\).

Answer:

The lateral area is \(240\pi\) square centimeters. The total surface area is \(384\pi\) square centimeters.