Investigation 6: Attributes of Geometric Solids

New Concept

Geometric solids are three-dimensional objects that take up space. We can describe them by their key attributes:

• Face: a flat surface of a polyhedron
• Edge: a line where two faces meet
• Vertex: a point where three or more edges meet

What’s next

Soon, you'll practice identifying and drawing various solids and solve problems by counting their faces, edges, and vertices.

Property

Geometric solids have length, width, and height; they take up space. A solid with all flat, polygon faces is a polyhedron. Polyhedrons do not have any curved surfaces.

Examples

A cube is a polyhedron because all its faces are flat squares.
A sphere is not a polyhedron because its entire surface is curved.
A pyramid is a polyhedron since it is built from a flat base and flat triangular faces.

Explanation

Think of it this way: if you can build a shape using only flat paper polygons like squares and triangles, it's a polyhedron! Your toy building blocks are polyhedrons. But if a shape has any curves, like a basketball or a can of soda, it is a non-polyhedron. It's all about flat faces versus cool curves!

Property

Face: a flat surface of a polyhedron. Edge: a line where two faces meet. Vertex: a point where three or more edges meet.

Examples

A cube has $6$ flat faces (the squares), $12$ edges (the lines), and $8$ vertices (the corners).
A triangular prism has $5$ faces ($2$ triangles and $3$ rectangles), $9$ edges, and $6$ vertices.
A square-based pyramid has $5$ faces ($1$ square and $4$ triangles), $8$ edges, and $5$ vertices.

Explanation

Imagine a cardboard box! The flat sides you can draw on are the faces. The lines where the sides fold and meet are the edges—it is like the skeleton of the shape. And the sharp corners where you can poke your finger? Those are the vertices! Every cool polyhedron is just a collection of these simple parts working together.

Property

The sum of the areas of a polyhedron's faces is called the surface area of the solid.

Examples

For a cube with each edge being $4$ cm, one face has an area of $4 \cdot 4 = 16 \text{ cm}^2$.
Since a cube has $6$ identical faces, the total surface area is $6 \cdot 16 \text{ cm}^2 = 96 \text{ cm}^2$.
For a box that is $5 \times 3 \times 2$ inches, the surface area is $2(5 \cdot 3) + 2(5 \cdot 2) + 2(3 \cdot 2) = 62 \text{ in}^2$.

Explanation

Ever wonder how much wrapping paper you need for a gift? You are actually calculating its surface area! It is the total space covering the outside of a 3D object. To find it, you just find the area of each individual face—like the front, back, top, bottom, and sides—and then add them all up. It is like gift-wrapping with math!