Lesson 12
Edge Lengths and Volumes
Let’s explore the relationship between volume and edge lengths of cubes.
Problem 1
 What is the volume of a cube with a side length of
 4 centimeters?
 \(\sqrt[3]{11}\) feet?
 \(s\) units?
 What is the side length of a cube with a volume of
 1,000 cubic centimeters?
 23 cubic inches?
 \(v\) cubic units?
Problem 2
Write an equivalent expression that doesn’t use a cube root symbol.
 \(\sqrt[3]{1}\)
 \(\sqrt[3]{216}\)
 \(\sqrt[3]{8000}\)
 \(\sqrt[3]{\frac{1}{64}}\)
 \(\sqrt[3]{\frac{27}{125}}\)
 \(\sqrt[3]{0.027}\)
 \(\sqrt[3]{0.000125}\)
Problem 3
Find the distance between each pair of points. If you get stuck, try plotting the points on graph paper.
 \(X=(5,0)\) and \(Y=(\text4,0)\)

\(K=(\text21,\text29)\) and \(L=(0,0)\)
Problem 4
Here is a 15by8 rectangle divided into triangles. Is the shaded triangle a right triangle? Explain or show your reasoning.
Problem 5
Here is an equilateral triangle. The length of each side is 2 units. A height is drawn. In an equilateral triangle, the height divides the opposite side into two pieces of equal length.
 Find the exact height.
 Find the area of the equilateral triangle.
 (Challenge) Using \(x\) for the length of each side in an equilateral triangle, express its area in terms of \(x\).