DWG NO. 11.2 — Lesson 2 of 5

Volume of Prisms and Cylinders

Unit 11: Surface Area and Volume · ~15–30 min

Objective

Find the volume of a prism or cylinder by multiplying the area of its base by its height.

Volume as stacked layers

Volume measures how much space a solid fills, in cubic units. For a prism or a cylinder, picture the base as a flat region and imagine stacking infinitely many razor–thin copies of it on top of each other until you reach the height h. Each layer contributes an area B (the base area), so the total volume is just that area repeated h times.

This works because a prism or cylinder has the same cross–section at every height — slice it anywhere parallel to the base and you get a congruent copy of the base.
It does not matter whether the base is a triangle, a hexagon, or a circle — the "stack the base" logic is identical, which is why one formula covers every prism.

Volume formulas

Let B be the area of the base and h be the height (the perpendicular distance between the two bases).

Solid	Volume
Prism	V = Bh
Cylinder	V = πr²h (since B = πr²)

Notice the cylinder formula is nothing new — it's just V = Bh with the circle's area formula substituted in for B.

Stacking the base area B through height h fills the whole prism — the same idea works for a cylinder's circular base.

Worked Example 1 · Triangular prism

ProblemA prism's triangular base has legs 5 in and 12 in meeting at a right angle (area = ½(5)(12)). The prism's height (length) is 20 in. Find its volume.

1.Base area: B = ½(5)(12) = 30 in².

2.Volume: V = Bh = 30(20) = 600 in³.

Volume = 600 in³

Worked Example 2 · Cylinder

ProblemA cylindrical water tank has radius 4 ft and height 10 ft. Find its volume in terms of π, then to the nearest tenth.

1.Base area: B = πr² = π(4)² = 16π.

2.Volume: V = Bh = 16π(10) = 160π ≈ 502.7 ft³.

Volume = 160π ft³ ≈ 502.7 ft³

Guided practice

1.Find the volume of a rectangular prism with length 9 m, width 4 m, and height 6 m.
Show answer
B = 9(4) = 36 m². V = Bh = 36(6) = 216 m³.
2.A cylinder has diameter 8 cm and height 15 cm. Find its volume in terms of π.
Show answer
Radius r = 4 cm. V = πr²h = π(4)²(15) = π(16)(15) = 240π cm³.
3.A prism's hexagonal base has an area of 41.6 ft², and the prism is 7 ft tall. Find its volume to the nearest tenth.
Show answer
V = Bh = 41.6(7) = 291.2 ft³.
4.Two prisms have equal base areas, but Prism Y is three times as tall as Prism X. How do their volumes compare?
Show answer
Volume is directly proportional to height when the base area is fixed (V = Bh), so Prism Y's volume is exactly three times Prism X's volume.
5.Explain why V = Bh works for a cylinder even though a circle isn't a polygon.
Show answer
The "stack the base" reasoning behind V = Bh only requires that every cross–section parallel to the base be a congruent copy of the base — that's true for a cylinder just as it is for a prism, so the same formula applies with B = πr².

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