Which of the following solids have five faces, eight edges, and five vertices?
The diagram shows five different 3D shapes. We have a sphere, what looks like a cube, a cylinder, a square-based pyramid, and a cone. Now, although these 3D shapes are just drawings, we can tell what they look like all the way around because we can see dotted lines on some of them. This shows us what each shape is like at the back and underneath. Now, we’re asked to find which of the solids have five faces, eight edges, and five vertices. We know that a face is a flat surface on a shape. An edge is like a line where two faces meet. And the word vertices is the plural of vertex. A vertex is a corner where edges meet. So let’s look at the features one by one.
To begin with, which of the following solids have five faces? A sphere, which is a ball shape, has a curved surface that goes all the way around. This shape doesn’t have five faces. A cube is made of several square faces. But are there five of them? Let’s count. One, that’s the one at the top, two, that’s the one on the bottom. And then, we have four more faces as we look all the way around. One on the right, one on the front, one hidden face on the left, and a final hidden face on the back. A cube has six faces. So this isn’t the shape we’re looking for either.
How many faces does a cylinder have? One circular face on the top. And a dotted line shows us that there’s another face on the bottom. Around the outside is a curved surface. A cylinder doesn’t have five faces. What about a square-based pyramid? We know the square that forms the base of the pyramid is a flat face. And then, we can also include all the triangles that form the sides of the pyramid, one on the right, one at the front. And then, the dotted lines show us that there’s one on the left and one more at the back of the shape. A square-based pyramid does have five faces. It looks like this could be the shape we’re looking for.
Let’s quickly check our final shape, the cone. A cone has a flat circular face on the bottom. And all around the outside is a curved surface. It doesn’t have five faces. It looks like the solid shape that we’re looking for is a square-based pyramid. But does it contain eight edges and five vertices? Let’s check. When we’re looking at a diagram of a 3D shape like this, we can tell the number of edges by looking at the lines that make up the drawing. And we need to also include the dotted lines. These represent the edges at the back of the shape that we can’t see.
So let’s count the edges. There are four edges that form the square at the base of the pyramid, one, two, and then two round the back, four edges altogether. But there are also four more edges that take our pyramid up to a point. So we already have four edges, five, six, seven, and one around the back. Our square-based pyramid does have eight edges. Finally, does it have five vertices? Remember that a vertex is a corner where edges meet. We can see that our square-based pyramid has four vertices or four corners around the base, one, two, three, and one around the back. That’s four altogether. And then, of course, there’s a vertex at the top, where each of the edges meet. A square-based pyramid has five vertices altogether.
And so out of the five solids, the one that has five faces, eight edges, and five vertices is the square-based pyramid.