🔵Volume of Rectangular Prism
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 cm$^3$, m$^3$, in$^3$, ft$^3$).
| Measure | Dimensions | Unit | Everyday example |
|---|---|---|---|
| Length | 1-D | cm | how long a pencil is |
| Area | 2-D | cm$^2$ | the top of a table |
| Volume | 3-D | cm$^3$ | water a tank holds |
The Formula (and why it works)
Imagine filling a box completely with 1-cm cubes, leaving no gaps. The bottom layer is a rectangle holding $l \times w$ cubes — that is the area of the base. Stacking $h$ identical layers on top multiplies the count by $h$:
$$V = l \times w \times h$$
A second useful form is $V = (\text{area of base}) \times \text{height}$. Both give the same answer.
Special case — the cube. A cube has all edges equal, so if each edge is $a$ then $V = a \times a \times a = a^3$.
Where you meet volume in real life:
- How much water a fish tank or swimming pool holds
- How many small boxes fit inside a shipping carton
- How much soil fills a rectangular flower bed
Common mistakes to avoid:
- Writing the answer in square units (cm$^2$) instead of cubic units (cm$^3$)
- Adding the dimensions ($l+w+h$) instead of multiplying them
- Mixing units — always make $l$, $w$ and $h$ the same unit first
RECTANGULAR PRISM = a box. Fill it with 1-unit cubes:
+--------------+
/ /|
/ / |
+--------------+ | height (h)
| | |
| | +
| | /
| |/ width (w)
+--------------+
<-- length (l) -->
COUNTING IN LAYERS (l = 4, w = 2, h = 3):
Bottom layer = 4 x 2 = 8 cubes (this is the AREA of the base)
There are 3 such layers stacked up.
Total cubes = 8 x 3 = 24
So V = l x w x h = 4 x 2 x 3 = 24 cubic units (24 cm3)👀 See a worked example
Find the volume of a rectangular prism with length 6 cm, width 4 cm and height 5 cm.
- Write the formula: $V = l \times w \times h$
- Substitute: $V = 6 \times 4 \times 5$
- $V = 24 \times 5 = 120$
- Answer: $120$ cm$^3$
A fish tank is 10 inches long, 6 inches wide and 8 inches tall. Find its volume.
- $V = 10 \times 6 \times 8$
- $V = 60 \times 8 = 480$
- Answer: $480$ in$^3$
Find the volume of a cube whose edge is 7 cm.
- For a cube, $V = a^3 = a \times a \times a$
- $V = 7 \times 7 \times 7$
- $V = 49 \times 7 = 343$
- Answer: $343$ cm$^3$
Using the base, find the volume of a prism whose base area is 24 cm$^2$ and height is 5 cm.
- $V = (\text{area of base}) \times \text{height}$
- $V = 24 \times 5 = 120$
- Answer: $120$ cm$^3$
A box has volume 60 cm$^3$. Its length is 5 cm and width is 3 cm. Find its height.
- $V = l \times w \times h$
- $60 = 5 \times 3 \times h = 15 \times h$
- $h = 60 \div 15 = 4$
- Answer: $4$ cm
A room is 5 m long, 4 m wide and 3 m high. Find the volume of air it contains.
- $V = 5 \times 4 \times 3$
- $V = 20 \times 3 = 60$
- Answer: $60$ m$^3$
🤖 Vidi's Key Points
- Volume is the space inside a 3-D shape, measured in cubic units (cm$^3$, m$^3$, ...).
- Rectangular prism: $V = l \times w \times h$, or $V = (\text{base area}) \times \text{height}$.
- Cube: $V = a^3$ because all edges are equal.
- Multiply the three dimensions — never add them.
- Make all dimensions the same unit before you calculate.
- A missing dimension = volume $\div$ (product of the two known dimensions).
🟢Counting Cubic Units in Irregular 3D Shapes
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 $V = l \times w \times h$, then add them.
The golden rule: hidden cubes count too. A cube that you cannot see because another cube sits in front of it is still part of the solid, so it must be counted.
Tip: after counting, you can often check your answer with a different method — if both methods agree, you can be confident it is right.
IRREGULAR SOLID: count EVERY unit cube, including hidden ones. METHOD 1 - Layer by layer (add the cubes in each layer): Layer 1 (bottom) = 9 cubes Layer 2 = 4 cubes Layer 3 (top) = 1 cube Total = 9 + 4 + 1 = 14 cubes METHOD 2 - Height grid (top view; each number = column height): +---+---+---+ | 3 | 1 | 1 | +---+---+---+ | 2 | 1 | 1 | Total = 3+1+1 + 2+1+1 + 1+1+1 = 12 cubes +---+---+---+ | 1 | 1 | 1 | +---+---+---+ METHOD 3 - Break into boxes: Prism A (2x2x2)=8 + Prism B (3x1x2)=6 = 14
👀 See a worked example
A height grid (top view) shows these column heights: row 1 = [2, 1, 1], row 2 = [1, 1, 1], row 3 = [1, 1, 1]. Find the volume.
- Add every column height.
- $2+1+1 + 1+1+1 + 1+1+1$
- $= 2 + (8 \times 1) = 2 + 8 = 10$
- Answer: $10$ cubic units
A solid has 15 cubes in the bottom layer, 8 in the middle layer and 3 in the top layer. Find its volume.
- Layer method: add the layers.
- $15 + 8 + 3 = 26$
- Answer: $26$ cubic units
An L-shaped solid is made of two boxes: box A is $2 \times 2 \times 2$ and box B is $3 \times 1 \times 2$. Find the total volume.
- Box A: $V = 2 \times 2 \times 2 = 8$
- Box B: $V = 3 \times 1 \times 2 = 6$
- Total $= 8 + 6 = 14$
- Answer: $14$ cubic units
A staircase of cubes has 6 cubes on the bottom step, 4 on the next and 1 on top. How many unit cubes in all?
- $6 + 4 + 1 = 11$
- Answer: $11$ cubic units
A solid is built from a $3 \times 3$ bottom layer with a single $1 \times 1$ tower of 2 extra cubes on one corner. Find the volume.
- Bottom layer: $3 \times 3 = 9$ cubes
- Extra tower: $2$ cubes
- Total $= 9 + 2 = 11$
- Answer: $11$ cubic units
🤖 Vidi's Key Points
- For an irregular solid, count every unit cube — hidden cubes included.
- Layer method: add the cubes in each layer from bottom to top.
- Height-grid method: add all the column heights in the top view.
- Decompose method: split into boxes, find each volume, then add.
- Check your total with a second method whenever you can.
🟣Volume Word Problems with Missing Dimensions
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
| Type | What is given | What to do |
|---|---|---|
| 1. Find the volume | $l$, $w$, $h$ | Multiply: $V = l \times w \times h$ |
| 2. Find a missing dimension | $V$ and two of $l, w, h$ | Divide: $V \div (\text{product of the two known})$ |
| 3. Compare volumes | two solids | Find both volumes, then compare |
| 4. Multi-step / capacity | mixed | May need unit change ($1$ L $=1000$ cm$^3$) |
Rearranging the formula. Starting from $V = l \times w \times h$:
- height $= V \div (l \times w)$
- width $= V \div (l \times h)$
- length $= V \div (w \times h)$
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 $V = l \times w \times h$ and fill in what you know.
- Solve — multiply to find a volume, or divide to find a missing dimension.
- Check the unit (cubic for volume, single for a length) and that the answer is sensible.
Capacity link. Liquids are measured in litres, but volume of a container is in cm$^3$. Remember $1$ litre $= 1000$ cm$^3$, so a tank of $5000$ cm$^3$ holds $5$ litres.
STRATEGY FOR VOLUME WORD PROBLEMS:
1. Write the formula: V = l x w x h
2. Put in the numbers you KNOW.
3. If a dimension is MISSING, divide:
missing = V / (product of the two known dimensions)
4. Check the unit is cubic (cm3, m3, ...).
MISSING-DIMENSION EXAMPLE: V = 120, l = 5, w = 4, find h
120 = 5 x 4 x h
120 = 20 x h
h = 120 / 20 = 6
CAPACITY LINK: 1 litre = 1000 cm3 (so a 2 L jug holds 2000 cm3)👀 See a worked example
A shipping container is 12 m long, 8 m wide and 6 m tall. Find its volume.
- $V = 12 \times 8 \times 6$
- $V = 96 \times 6 = 576$
- Answer: $576$ m$^3$
A rectangular prism has volume 240 cm$^3$. Its length is 10 cm and width is 6 cm. Find its height.
- height $= V \div (l \times w)$
- $h = 240 \div (10 \times 6) = 240 \div 60$
- $h = 4$
- Answer: $4$ cm
Box A is $8 \times 5 \times 4$ cm and Box B is $6 \times 6 \times 5$ cm. Which box has the greater volume?
- Box A: $V = 8 \times 5 \times 4 = 160$ cm$^3$
- Box B: $V = 6 \times 6 \times 5 = 180$ cm$^3$
- $180 > 160$
- Answer: Box B
A rectangular pool is 20 m long and 10 m wide. Its volume of water is 600 m$^3$. How deep is the pool?
- depth $= V \div (l \times w)$
- $= 600 \div (20 \times 10) = 600 \div 200$
- $= 3$
- Answer: $3$ m deep
A juice carton holds 2 litres. Its base is 10 cm by 10 cm. Find its height. (Use $1$ L $=1000$ cm$^3$.)
- $2$ L $= 2000$ cm$^3$
- height $= V \div (l \times w) = 2000 \div (10 \times 10)$
- $= 2000 \div 100 = 20$
- Answer: $20$ cm
How many 2-cm cubes fit inside a box that is 8 cm by 6 cm by 4 cm?
- Box volume $= 8 \times 6 \times 4 = 192$ cm$^3$
- Each small cube $= 2 \times 2 \times 2 = 8$ cm$^3$
- Number of cubes $= 192 \div 8 = 24$
- Answer: $24$ cubes
🤖 Vidi's Key Points
- Always start by writing $V = l \times w \times h$.
- Missing dimension = volume $\div$ (product of the two known dimensions).
- To compare solids, work out both volumes first, then compare.
- Capacity: $1$ litre $= 1000$ cm$^3$ — convert before dividing.
- Check your unit: a volume is cubic; a single length is not.
- A quick check: multiply your answer back and see if you get the original volume.
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