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Surface area
The total area of all the faces (or curved surfaces) on the outside of a 3-dimensional solid.
Volume
The amount of space a 3-dimensional solid occupies, measured in cubic units such as cm$^3$.
Although this article focuses on surface area, it helps to remember the big picture:
Area and surface area use the same units (square units), but they refer to different kinds of objects: 2D area is for flat shapes, 3D surface area is the total of all outer surfaces.
A 3D solid may have flat faces, curved faces, or both.
Polyhedron
A 3D solid that has only plane (flat) faces.
A powerful strategy is to identify the faces of a solid and match each face to a 2D shape you already know how to find the area of.
If you can draw a quick sketch and label what each face is (rectangle, triangle, circle), the surface area method often becomes "add the areas of the parts".
A pyramid has a polygon base, and the other faces are triangles meeting at a point called the apex.
Apex
The point where the triangular faces of a pyramid (or the curved surface of a cone) meet.
For many pyramids you will see two heights:
Vertical eight
The perpendicular distance from the apex straight down to the base plane.
Slant height
For a pyramid, the distance from the midpoint of a base edge to the apex, measured along a triangular face.
The slant height is used to find the area of each triangular face because each face is a triangle whose height is the slant height.
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