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MATH · GRADE 7Measurement

Volume of Right Prisms

Two students pick different faces as the base — and both get V = 24. How?

Grade 7
432V = 24V = 24?how can both be right…
In this lesson

What a prism is

A prism is a solid 3-D shape with two parallel congruent polygonal bases and rectangular lateral faces connecting them. The two bases are flat polygons of the same shape and size; the lateral faces are flat rectangles that join the bases edge to edge.

The prism takes its name from the shape of its base. A rectangular prism has rectangular bases; a triangular prism has triangular bases. Either way, the two bases face each other across the prism, and the lateral faces fill in the sides.

The volume of a prism or cylinder can be explained as a product of dimensions.

A right prism is one where the lateral faces are perpendicular to the bases. Every prism in this lesson is a right prism.

Three dimensions

A rectangular prism stretches in three independent directions. The curriculum names them length, width, and height. Each dimension acts on its own; stretching one changes the volume by that factor and leaves the other two unaffected.

For a rectangular prism, the base is a rectangle of area length × width, so the formula expands:

Vrect=ℓ×w×hV_{\text{rect}} = \ell \times w \times h

For a triangular prism, the base is a triangle, so:

Vtri=12×b×h△×hV_{\text{tri}} = \tfrac{1}{2} \times b \times h_{\triangle} \times h

where b is the triangle's base side, hₜ is the triangle's height (the perpendicular from the apex to that base side), and h is the prism's height (the perpendicular distance between the two triangular ends).

Layer-iteration: where V = Bh comes from

Imagine slicing the prism into thin horizontal layers, each one a copy of the base polygon. Each layer has volume equal to the base area B, multiplied by a thin slice of thickness. Stack h of them on top of each other and you get the whole prism.

That's where V = Bh comes from. The base sets one layer; the height counts how many layers stack on top of each other to fill the prism.

See it: stretch each dimension

The widget below has three sliders. Each one changes one dimension of the prism independently. The volume readout updates live. Try the two preset shapes: a rectangular base, then a triangular base. The "Show layers" toggle (rectangular only) draws the layer separators on the front face so you can count the stacking.

Try it

Stretch any dimension. The volume updates as the product of all three.

base area B = 12

V = 4 × 3 × 2 = 24

volume V (cubic units)

Each slider stretches one dimension. The volume is the product of all three.

Notice as you work the widget:

  • Each slider changes only its dimension. The volume scales by exactly that factor. Doubling the height doubles the volume; doubling the length doubles the volume; doubling both quadruples it.
  • The base equation rewrites itself. When you slide on the rectangular preset, the formula shows V = ℓ × w × h. Switch to triangular and it becomes V = ½ × b × hₜ × h.
  • The layer toggle reveals the structure. Layers are exactly the rows of the front face. Count them: the number of layers equals the prism height.

Any face can be the base

A rectangular prism doesn't really have a "natural" base. Any of the three pairs of faces could serve as the two parallel bases. Volume comes out the same regardless of which pair you pick, because the product B × h is symmetric in the three dimensions.

Base on the bottom

B = 4 × 3 = 12; h = 2; V = 24

Base on the front

B = 4 × 2 = 8; h = 3; V = 24

Base on the side

B = 3 × 2 = 6; h = 4; V = 24

Same prism. Three choices of base. V = 24 in every case.

The same 4 × 3 × 2 prism is shown three times. In the first panel, the 4 × 3 face is the base and the height is 2. In the second, the 4 × 2 face is the base and the height is 3. In the third, the 3 × 2 face is the base and the height is 4. All three give V = 24. The choice of base is a choice of perspective, not a choice of formula.

Any face of a rectangular prism can be interpreted as the base.

For a triangular prism, only the triangular ends serve as the bases, because the rectangular lateral faces aren't congruent polygons of the right shape. For rectangular prisms, the symmetry holds.

Units must match

Volume is measured in cubic units: cubic centimetres (cm³), cubic metres (m³), cubic kilometres (km³), and so on. The unit on the volume follows directly from the unit on the dimensions.

If length, width, and height are all in centimetres, the volume is in cm³. If they're in metres, the volume is in m³. If the dimensions are given in mixed units, say 30 cm × 50 cm × 2 m, you have to convert them to a single unit before multiplying. Otherwise the product is meaningless.

The fix when the units mismatch: pick one unit (usually the most natural for the answer) and convert the other dimensions to match. 30 cm × 50 cm × 2 m becomes 30 cm × 50 cm × 200 cm = 300 000 cm³, or equivalently 0.30 m × 0.50 m × 2 m = 0.30 m³. Same volume, two valid ways of writing it.

Where it shows up in real life

A grain bin on an Alberta farm is a tall cylinder sitting on a square or rectangular concrete pad, but plenty of farm storage is genuinely prismatic. A grain hopper at the loading end of a combine is a rectangular prism with a triangular prism funneling down to the auger. The grain volume is the sum of the two prism volumes, each computed by its own V = Bh formula.

A shed holding hay, tools, or animal feed is usually a rectangular prism with a triangular prism stacked on top to form the peaked roof. The total interior volume is the rectangular box's volume plus the triangular prism's volume.

A swimming pool with a constant rectangular cross-section is a rectangular prism. The volume in litres tells you how much water it holds; a pool 4 m wide, 8 m long, and 1.5 m deep is a 4 × 8 × 1.5 = 48 m³ prism, holding about 48 000 litres.

In each case, the volume in real units (cubic metres, litres, bushels) determines a real quantity (water, grain, hay) that someone has to measure, store, or pay for. V = Bh is exactly the calculation a farmer or a builder runs.

Worksheet

These aren't graded. Get them right, get them wrong. The goal is fluency with V = Bh in both directions, plus an instinct for keeping units straight.

Practice · Not graded

MA.7.MEA.2

Practice the idea

01 / 08

A rectangular prism has length 3, width 4, and height 5. What is its volume?

Multiple choice: volume of a rectangular prism with dimensions 3 by 4 by 5.
Show common mistakes

Student says

“Treats the base as the bottom face only. 'The prism is sitting on the ground, so the bottom is the base.'”

What it reveals

Hasn't generalized 'base' beyond 'the side it's resting on.' For a rectangular prism, ANY pair of parallel faces can serve as the bases; the choice doesn't change the volume.

Targeted response

Use the PrismOrientationStrip above. The same 4 × 3 × 2 prism is shown three times with three different faces highlighted as the base. The volume comes out to 24 in every case. The 'base' is a label that helps you apply the formula; it's not a physical orientation. Once you internalize that, you'll notice you can pick whichever face makes the arithmetic easiest.

Student says

“For a triangular prism, computes B = b × hₜ (forgets the one-half) or uses the prism's height as the triangle's height.”

What it reveals

Two common slips. The triangle area formula is ½ × b × hₜ, not b × hₜ; and the triangle's height (apex-to-base distance) is different from the prism's height (distance between the two triangular ends).

Targeted response

The formula for triangular prism volume is V = ½ × b × hₜ × h. Two heights show up because there are two perpendiculars in the problem: one inside the triangle (apex to its base) and one along the prism's length (between the two triangular ends). Use the PrismStretcher above and switch to the triangular preset. The slider labels make both heights explicit.

Student says

“Multiplies dimensions in mixed units without converting. 'V = 30 cm × 0.5 m × 20 cm = 300.'”

What it reveals

Treats numbers as values to multiply without checking the units. The product 30 × 0.5 × 20 is arithmetically 300, but the result isn't a volume in any meaningful unit; it's mixed cm³ and cm² · m.

Targeted response

Always convert all dimensions to the same unit before multiplying. Either all in cm (30 × 50 × 20 = 30 000 cm³) or all in m (0.30 × 0.50 × 0.20 = 0.030 m³). Both express the same real volume, just written differently.

Going further

Next up: volume of a right cylinder. The same V = Bh formula applies, but the base is a circle instead of a polygon, so B = πr² and the formula becomes V = πr²h. The lateral surface of a cylinder is curved, not flat, so it's a "surface" rather than a collection of "faces." The layer-iteration logic is identical: a stack of h circular disks of area πr².

In Grade 8, the same V = Bh idea extends to pyramids and cones with a one-third factor (V = ⅓Bh), because those shapes taper to a point rather than maintaining a constant cross-section. The Grade 7 right-prism case is the foundation; the tapered cases add one factor on top.