# Right Solid: Definition, Area & Volume

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

We will define a right solid and look at some types of shapes that fall into this category. After looking at the volume and surface area of a right solid and the formulas involved in finding these values, we will do a couple of applications practicing what we've learned.

## Right Solid

Would you be willing to live in a teepee for a week to raise money for your favorite charity? Suppose you just signed up to do exactly this! Let's take a look at this teepee that will be your home for the next 7 days!

An actual teepee is shaped a little differently from this illustration, but for the purpose of our lesson, let's pretend it looks like this cone, and has a pole down its center (teepee poles outline its perimeter). Do you notice how the pole creates a right angle with the floor of the teepee?

A right solid is a three-dimensional object in which either the sides of the object, or its axis, is perpendicular to its base. The base of a right solid is the bottom side of the solid, and the axis is a straight line that divides the solid into symmetrical halves. The top of a right solid is either a point, or a shape that is exactly the same as its base. If the top is a point, that point is called the apex, and it lies directly above the center of the base. As we said, otherwise, the top is the same shape as the base and not a point. In this case, the shape is directly in line with the base and not shifted to either side.

Can you see why this is a right solid? Huh, do you think it sounds fancier to say you're going to live in a teepee or a right solid for a week?

Examples of right solids include pyramids, cylinders, prisms and cones.

## Volume of a Right Solid

Okay, back to your living arrangements! Now that you've signed up, you're curious as to how much room this new living space will actually provide. Technically speaking, what we want to find is the volume of the teepee, because the volume of a three-dimensional object is how much space is inside the object.

You find out that the radius of the floor of the teepee is 8 feet, and the length of the center pole is 11 feet.

In the case of the teepee, we have a nice formula for the volume of a cone as follows:

Volume of a cone = (1/3) π r2 h, where r is the radius of the base and h is the height of the cone.

Great! We actually have these exact measurements! We have that the radius is 8 feet, and the height is 11 feet, so we plug r = 8 and h = 11 into the formula and find the volume.

We get that the volume of the teepee is approximately 737.23 cubic feet. That's not huge, but it's only a week! You'll manage, and it's for a great cause!

When it comes to the volume of a right solid, we will find that it is always proportional to the area of the base multiplied by the height. Let's take a look at some different formulas for volumes of various right solids. The area of the base times the height is always involved.

It's nice that we have these formulas! They make finding the volume of a right solid quite easy! It's just a matter of finding measurements and plugging them into the formulas, just like we did with the teepee.

## Surface Areas of Right Solids

Another aspect of right solids that we are interested in is the surface area. The surface area of a right solid is the amount of area that the total surface of the solid takes up. Thankfully, to find the surface area of right solids, we also have formulas.

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