Distance Between Two Points on a Solid
Finding the distance between two points on a solid isn't too hard as long as we're familiar with right triangles and the Pythagorean Theorem. Right triangles are triangles that contain a right angle (90degree angle). The two shorter sides of a right triangle are called the legs of the triangle, and the longest side is called the hypotenuse.
In a right triangle, if the shorter sides have lengths a and b, and the hypotenuse has length c, then the Pythagorean Theorem relates the lengths of these sides together using the following formula:
Now, let's look at why it is important to be familiar with these concepts when dealing with two points on a solid.
Notice that the piece of wood creates the right triangle ACM, where AM is the new piece of wood.
We know the length of the side CM is onehalf the length of CG, because M is the midpoint of CG. Therefore, CM = 2ft. Also notice that the side AC is the hypotenuse of the right triangle ACD, where AD = 7ft. and CD = 4ft., so we can find AC using the Pythagorean Theorem:
We get that AC = √65 ft..
Now we have the lengths of two of the sides of the right triangle ACM, so we can use the Pythagorean Theorem to find the length of our piece of wood, AM.
Awesome, we have that the piece of wood will be √69 ≈ 8.3 ft.. Captain Cute is going to be so happy!
Another Example
As we just saw, finding the distance between two points on a solid is just a matter of using right triangles and the Pythagorean Theorem. That is, to find the distance between two points on a solid, we follow these steps:
 Create a right triangle with one of its sides connecting the two points you want to find the distance between.
 Use properties of the solid, other right triangles, and the Pythagorean Theorem to ultimately find the distance between your two points.
For example, suppose that after seeing how much fun Captain Cute has been having on his jungle gym, you decide to make another one. This one is a wood structure in the shape of a cylindrical solid with height 3 ft. and radius 2 ft. Once again, you want to add another piece of wood from the edge of the bottom base to the edge of the top base, and you need to know how long that piece of wood will be.

Do you see the right triangle? The piece of wood creates a right triangle such that the height of the cylinder is one leg, the diameter (2 × radius) of the base is another leg, and your new piece of wood, call it w, is the hypotenuse.
We know the height is 3 feet, and since the radius is 2 ft., the diameter is 2 × 2 = 4 ft., so we have the lengths of the legs of the triangle. We can use the Pythagorean Theorem to find the length of the piece of wood!
We see that the piece of wood will be 5 ft. long. This is kind of fun…like a puzzle of sorts!
Lesson Summary
A solid is a threedimensional object. We can find the distance between points on a solid using right triangles, or triangles that have a right angle, and the Pythagorean Theorem. In a right triangle, we call the two shorter sides legs, and we call the longest side the hypotenuse. The Pythagorean Theorem states that if the legs have lengths a and b, and the hypotenuse has length c, we have the following:
To find the distance between two points on a solid, we use the following steps:
 Create a right triangle with one of its sides connecting the two points you want to find the distance between.
 Use properties of the solid, other right triangles, and the Pythagorean Theorem to ultimately find the distance between your two points.
Not only is this useful in everyday applications, as we've seen, it is also pretty fun to try to fit all the pieces of information together to find a solution. Huh! Who knew math could be so entertaining?