Solids
273. The Solid's Shape · Volume, Surface Area, and Space
The cube's geometry remains consistent as parameters change.
A cube is a solid with six equal square faces. For edge a: Volume = a³ and Total surface area = 6a², with 12 equal edges and 8 vertices — its geometry stays consistent as the edge changes.
What this lesson covers
Try to break it
Drag P along the edge to trace one dimension. A solid is anything that takes up space with definite length, width, and height — the cube has all three. That's why volume needs three measurements (l × b × h) and surface area needs the total wrapping over all six faces.
How you build it
Build a cube and explore its geometry.
- Draw the FRONT face ABCD as a rectangle.
- Draw the BACK face EFGH the same size, offset up and to the right.
- Depth edge 1: join front corner A to back corner E.
- Depth edge 2: join front corner B to back corner F.
- Depth edge 3: join front corner C to back corner G.
- Depth edge 4: join front corner D to back corner H. Done — a cube has 8 vertices, 12 edges, and 6 square faces.
The proof, step by step
Prove that the object is a solid with three dimensions.
- A solid occupies space and has a definite shape.
- Surface area is the sum of the areas of all faces.
- Volume is the space occupied by the solid.
- Capacity of a container equals its internal volume.
Worked example
A cubical water tank has an edge of 2 m. What is its volume and total surface area?
Volume = edge³ = 2³ = 8 m³. Total Surface Area = 6 × edge² = 6 × 2² = 24 m².
- Volume = 8 m³, TSA = 24 m² — correct
- Volume = 6 m³, TSA = 24 m²
- Volume = 8 m³, TSA = 12 m²
- Volume = 12 m³, TSA = 24 m²