Mathematics Year 6: Volume of a cylinder and of solids made of parts
Mathematics, Year 6 (Portugal): A cylinder is a stack of equal circles, so its volume is the area of the base times the height; then add the parts of a bigger solid.
Circles stacked up
A cylinder is like a cuboid, but its base is a circle. Stack identical thin circles up to the height h and you get a cylinder. So, as with the cuboid, V = area of the base × height. The area of a circle of radius r is π × r², therefore V = π × r² × h. In calculations use π ≈ 3.14, or the π key of a calculator.
One rule for cuboids and cylinders
Both solids are stacks of identical layers, so both follow the same rule: volume = area of the base × height. That is why you can vary the height and see the volume grow in proportion. Many real objects are not a single cuboid or cylinder but are made of several. Then you split the solid into parts you know, find each volume in the same unit, and add them (or subtract a part that is missing).
A trophy base made of two parts
A trophy is a slab 12 cm by 12 cm by 2 cm with a cylinder on top, radius 3 cm and height 5 cm. Slab: 12 × 12 × 2 = 288 cm³. Cylinder: base area 3.14 × 3² = 3.14 × 9 = 28.26 cm², then 28.26 × 5 = 141.3 cm³. Total: 288 + 141.3 = 429.3 cm³. Both parts are in cm³, so they can be added. Rounding, it is a little over 0.4 l of material.
The radius, not the diameter
The formula asks for the radius r, but a can or a pipe is usually described by its diameter. A can with diameter 8 cm and height 10 cm has radius 4 cm: V = 3.14 × 4² × 10 = 502.4 cm³. Put 8 where r should be and you get 3.14 × 8² × 10 = 2009.6 cm³, four times too much, because r is squared. Halve the diameter first, then square.
Cans, pipes and tanks
Cylinders are everywhere: tins of food, candles, water pipes, round tanks and silos. The can of the previous card has a volume of 502.4 cm³, which is about 0.5 dm³, so it holds around half a litre. Builders and engineers split complicated shapes into cuboids and cylinders to estimate how much water, concrete or air they hold, and sculptors do the same to plan how much material a piece needs.
Test what you learned
A cylinder has a base of radius r and a height h. Which expression gives its volume?
- a) π × r² × h, the area of the base times the height
- b) π × r × h
- c) π × r²
- d) 2 × π × r × h
Correct answer: a) π × r² × h, the area of the base times the height — Yes: π × r² is the area of the circular base, and it is repeated over the height h, as in a cuboid.
A pedestal is a cuboid 10 cm by 10 cm by 3 cm with a cylinder on top (radius 2 cm, height 8 cm). Using π ≈ 3.14, what is the volume of the whole pedestal?
- a) 100.48 cm³
- b) 350.24 cm³
- c) 400.48 cm³
Correct answer: c) 400.48 cm³ — Yes: cuboid 10 × 10 × 3 = 300, cylinder 3.14 × 2² × 8 = 100.48, and 300 + 100.48 = 400.48 cm³.
A can has a diameter of 6 cm and a height of 10 cm. Using π ≈ 3.14, what is its volume?
- a) 1130.4 cm³
- b) 282.6 cm³
- c) 94.2 cm³
Correct answer: b) 282.6 cm³ — Yes: r = 6 ÷ 2 = 3 cm, base area 3.14 × 3² = 28.26 cm², and 28.26 × 10 = 282.6 cm³.
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