# Lesson 12

Edge Lengths and Volumes

Let’s explore the relationship between volume and edge lengths of cubes.

### Problem 1

1. What is the volume of a cube with a side length of
1. 4 centimeters?
2. $$\sqrt{11}$$ feet?
3. $$s$$ units?
2. What is the side length of a cube with a volume of
1. 1,000 cubic centimeters?
2. 23 cubic inches?
3. $$v$$ cubic units?

### Problem 2

Write an equivalent expression that doesn’t use a cube root symbol.

1. $$\sqrt{1}$$
2. $$\sqrt{216}$$
3. $$\sqrt{8000}$$
4. $$\sqrt{\frac{1}{64}}$$
5. $$\sqrt{\frac{27}{125}}$$
6. $$\sqrt{0.027}$$
7. $$\sqrt{0.000125}$$

### Problem 3

Find the distance between each pair of points. If you get stuck, try plotting the points on graph paper.

1. $$X=(5,0)$$ and $$Y=(\text-4,0)$$
2. $$K=(\text-21,\text-29)$$ and $$L=(0,0)$$

(From Unit 8, Lesson 11.)

### Problem 4

Here is a 15-by-8 rectangle divided into triangles. Is the shaded triangle a right triangle? Explain or show your reasoning.

(From Unit 8, Lesson 9.)

### 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.

3. (Challenge) Using $$x$$ for the length of each side in an equilateral triangle, express its area in terms of $$x$$.