Compute the exact area or length as directed. Square roots do not need to be simplified, but arithmetic does.

What is the surface area of a cylinder with diameter 4 and height 4? \( 16\pi + 4\pi + 4\pi = 24\pi \)

A prism with a rectangle base has dimensions 2 by 3 by 4. What is the surface area of this prism? \( 2 \cdot 3 + 2 \cdot 3 + 2 \cdot 4 + 2 \cdot 4 + 3 \cdot 4 + 3 \cdot 4 = 52 \)

A hemisphere has a "base" area of \(16 \pi\) units^{2}. What is the surface area of the hemisphere (which does not include a base area)? \( r = 4; \text{surface area} = 32\pi \)

What is the surface area of a tetrahedron with edge length of 3? \( 6 \sqrt{\frac{27}{4}} = 6 \sqrt{6.75} = 9\sqrt{3} \)

What is the surface area of a cone with radius of 2 and height of 6? \( 4\pi + 2\pi\sqrt{40} = 4\pi + 4\pi\sqrt{10} \)

What is the surface area of a square pyramid with base lengths of 4 and slant heights of 6? (Note: slant height is not edge length) \( 16 + 4 \cdot 12 = 64 \)

What is the surface area of a square pyramid with base lengths of 4 and lateral edge lengths of 6? \( 16 + 8\sqrt{32} = 16 + 32 \sqrt{2} \)

What is the surface area of a square pyramid with base lengths of 4 and pyramid height of 6? \( 16 + 8 \sqrt{40} = 16 + 16\sqrt{10}\)

What is the surface area of a sphere inscribed in a cylinder with height of 5? \( 25\pi \)

What is the surface area of a sphere inscribed in a cylinder with circumference of 6? \( r = \dfrac{3}{\pi}; A = \dfrac{36}{\pi} \)

Challenge problems

Anne made a two-layer cake for her sister's wedding, which she plans to ice. Each of the two layers is a cylinder. The bottom layer has a diameter of 12", while the top layer has a diameter of 8". Each layer is exactly 7" tall. Anne is going to ice only the visible surfaces. What is the surface area of the cake that Anne will ice?
\( \text{ lateral area of small cake } = 56 \pi \\
\text{ lateral area of big cake } = 84 \pi \\
\text{ area of top } = 36 \pi \\
\text{ total } = 176 \pi \)

Consider a cylinder with diameter 4 and height 4. Further, this cylinder has a string that spirals around the lateral side exactly once, from one base to the other. What is the length of this string? \( \sqrt{16 + 16\pi^2} = 4\sqrt{1 + \pi^2} \)