📏 Volume

What is Volume?
Volume is the amount of space inside a 3-dimensional (solid) object. Length tells you how long something is, area measures a flat surface, and volume measures how much a solid can hold.
Volume is always measured in cubic units little cubes that are exactly 1 unit long, 1 unit wide and 1 unit tall (written cm³, m³, in³, ft³).
MeasureDimensionsUnitEveryday exampleLength1-Dcmhow long a pencil isArea2-Dcm²the top of a tableVolume3-Dcm³water a tank holdsThe Formula (and why it works)
Imagine filling a box completely with 1-cm cubes, leaving no gaps. The bottom layer is a rectangle holding l × w cubes that is the area of the base. Stacking h identical layers on top multiplies the count by h:
V = l × w × h
A second useful form is V = (area of base) × height. Both give the same answer.
Special case the cube. A cube has all edges equal, so if each edge is a…

Not every solid is a perfect box. Some shapes are made by stacking unit cubes in an uneven way (think of steps, an L-shape or a staircase of blocks). To find their volume we simply count every unit cube including the ones hidden behind or underneath others.
There are three reliable ways to count, and they always give the same answer. Pick whichever suits the picture.
Method 1 Layer by layer.
• Count the cubes in the bottom layer.
• Count the cubes in each layer above it.
• Add all the layers together.

Method 2 Height grid (top view). A top-view grid shows how many cubes are stacked in each column. Add all the column heights to get the total number of cubes.
Method 3 Break it into boxes. Split the irregular shape into two or three rectangular prisms, find the volume of each with V = l × w × h, then add them.
The golden rule: hidden cubes count too. A cube that you…

Many real problems give you the volume and ask for a length you cannot see the depth of a pool, the height of a carton, the width of a tank. The same formula handles every case once you know how to rearrange it.
The four problem types
TypeWhat is givenWhat to do1. Find the volumel, w, hMultiply: V = l × w × h2. Find a missing dimensionV and two of l, w, hDivide: V ÷ (product of the two known)3. Compare volumestwo solidsFind both volumes, then compare4. Multi-step / capacitymixedMay need unit change (1 L =1000 cm³)Rearranging the formula. Starting from V = l × w × h:
• height = V ÷ (l × w)
• width = V ÷ (l × h)
• length = V ÷ (w × h)

In words: the missing dimension equals the volume divided by the product of the two dimensions you do know.
A 4-step method that always works:
• Read the problem and note the units.
• Write V = l × w × h and fill in what you know.
•…