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Squares pegs = square holes. Triangular pegs = triangular holes. But where does a sphere go? In this lesson, review volumes of common shapes while contrasting a sphere and a cylinder – after all, they both go into the circular hole… right?

Square Peg in a Square Hole

Let’s take a few minutes to review the volumes of shapes. These get quite important in calculus problems because we often use calculus to do things like optimize areas, optimize volumes – you know, how much shampoo should you put in a shampoo bottle to make the most money? – that kind of thing.

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I also like to call this the square peg in square hole problem. And this is because of how I classify shapes.

2D Shapes

Formula for finding the area of triangles
2D Triangle Area

Let’s start with 2D shapes. So, now instead of looking at volume, let’s look at the area.

So, if you have a square with a side length s, then the area of the square is s2. The perimeter, that is the length that borders the shape, is 4 * s. Now, a square is just a fancy type of rectangle. A rectangle has some height, h, and some width, w, so the area of a rectangle is wh – the width times the height.

You can find the length of this line that surrounds the rectangle, or the perimeter, as 2 * w + 2 * h because we’re adding up w + h + w + h when we go around the rectangle.Now, the second of our square peg in square hole area is the triangle. So, let’s take a look at two different types of triangles.

We’ve got this guy here, and we’ve got this guy here. Now both of these have a base that’s the width of the bottom part of the triangle, and it’s always going to be along some side. We’re going to call that b.

Now the triangles also have some height. It’s really important that the height be measured as being perpendicular to the base. So, for some triangles, such as this one here, we need to measure the height actually outside of the base because we need this height to be straight up from the base, to be perpendicular from the base.

In these cases, the area is just 1/2 bh. Now, I know this is all a review, so I won’t explain why.Now, the last of my three types of shapes in my square peg, square hole is the circle. We usually classify a circle by some radius, r, measured from the center of the circle straight out to the edge of the circle.

So, this radius is the same no matter where along the circle you measure it. The area of a circle like this is pi r2.

3D Shapes

Spheres are 3D shapes that do not have the same dimensions from top to bottom
How to find the volume of a cylinder
Cylinder Volume

Volume of 3D Shapes

So, what does this mean in terms of finding the volume? Well, for anything that has the same shape on the top and the bottom and all the way through, like the can, the cube and the prism, the volume is just the height times the area of the base.

So, here we’ve got our can. We’ve got height. The base area is just pi r2.

So, there’s my base. So, the volume is h(pi r2). Say now I’ve got this box, this trunk of sorts, with a base that is a square. So, now my base area is s2 because that’s the area of a square, and my volume is going to be the base area, s2 times h. So my volume is h(s2). So, you can do this for anything where the cross-section is the same from the top to the bottom. Say your base area is an E.

It looks like a big E. Same thing. Base area times the height will give you the volume.

So, what about things that change? Let’s look at some of those. For example, a sphere: the volume of a sphere is 4/3 pi r3. A hemisphere is just half a sphere, and so the volume of a hemisphere is 1/2 (4/3 pi r3), which is also 2/3 pi r3.Pyramids and cones are treated very similarly to one another.

So, each one has a height, and the volume is the h/3 (area of the base). You can remember this for both of those. So, the area of the base of a cone, obviously, is the area of this circle, and for a pyramid it’s going to be, in this case, a triangle, sometimes a square, but that’s going to be the area of the base.

How to find the volume of pyramids and cones
Pyramid Cone Volume

Lesson Summary

So, let’s recap. When you’re trying to find the volume of something just remember if it’s a square peg in a square hole.

If the base cross-section is the same as the cross-section throughout the entire volume then your volume is the base area, or one of those cross-sections, times the height. If it’s not the same, then you need to remember one of the other rules. For example, for cones, your volume is h/3 (base area). For circles, just remember the equation for the volume of a sphere: 4/3 pi r3.

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