Volume and surface area: cylinders: Year 9: Planning tool

Expected level of development

Australian Curriculum Mathematics V9: AC9M9M01

Numeracy Progression: Understanding units of measurement: P9

At this level, students will calculate the surface area, volume and capacity of cylinders and solve related problems.

Using nets, students will see that the shapes that make up the surface area of a cylinder are two circles and a rectangle. They should know that the rule for the surface area of a cylinder is: A = 2πrh + 2πr².

Building on students’ previous knowledge of the volume of prisms (Volume = Base Area × Height, where the base area is the same cross-sectional area throughout the height), the formula for the volume of a cylinder can be derived. Students will see that the base area is πr² and therefore the rule for the volume of a cylinder will be V = πr²h.

The use of concrete materials, such as play dough, can help students to visualise the cross-section of different objects, such as a cylinder. The use of nets demonstrates that area is a 'flat' or two-dimensional (2D) quantity and so to find surface area students need to find the area of each surface of the shape. Turning the nets into a 3D solid will show students that volume is the amount of space occupied by an object.

Identifying, understanding and using the correct units of measurement for volume and area allows students to communicate with specificity and context. Students consolidate their understanding of using and providing units to show that their calculations, formulas and comparisons are correct for different contexts.

Teaching and learning summary:

Revise knowledge of circles, such as circumference and area.
Connect knowledge of circles and nets to find the surface area of a cylinder.
Connect knowledge of the volume of a prism to finding the volume of a cylinder.
Expand knowledge of the connection between volume and capacity.
Use knowledge of circles and cylinders to solve problems.
Connect knowledge of circles and cylinders with knowledge of other shapes and objects to find the area, surface area and volume of composite shapes and objects.
Provide opportunities for students to use these rules in solving surface area and volume-related problems.
Ensure students identify, understand and use correct units of measurement for volume and area.

Students:

draw the net of a cylinder
calculate the surface area of a cylinder from a net
calculate the volume of a cylinder
solve simple problems involving the surface area and volume of a cylinder
combine knowledge of circles and cylinders with knowledge of other shapes and objects to find the area, surface area and volume of composite shapes and objects.

Some students may:

confuse radius and diameter.
not appreciate that π is a number.
use the wrong measurement.
have difficulty in leaving answers in terms of π, preferring rounded decimals.
have difficulty in visualising the net of a cylinder.
confuse the various formulas for the area of a circle and the volume of a cylinder.
confuse the rules for surface area and volume.
be confused by the word ‘base’. They should be able to recognise this does not always refer to the ‘bottom’ of the object, but the face by which the prism or pyramid is named.

The Learning from home activities are designed to be used flexibly by teachers, parents and carers, as well as the students themselves. They can be used in a number of ways including to consolidate and extend learning done at school or for home schooling.

Learning intention

We are expanding knowledge of circles to three-dimensional objects such as cylinders.
We are learning how to calculate the surface area and volume of a cylinder.
We are learning how to apply our knowledge of circles and cylinders to practical situations.

Why are we learning about this?

A knowledge of area and volume is useful in many situations, such as wrapping presents and filling containers.

What to do

Total surface area and volume of a cylinder

Open this GeoGebra exercise: Total surface area and volume of cylinder (with net shown).
Follow the activity prompts that will lead you through finding the surface area and volume of a cylinder.
Use the interactive net to vary the radius and height of the cylinder. Create at least three cylinders of different sizes.
Draw or take a screen capture of your results.

Finding the volume and surface area of a cylinder

Open the resource Efficient cutting. Using a sheet of A4 paper, your challenge is to find those dimensions that allow you to make a cylinder with the greatest volume.
Work through the task, creating a cylinder with these dimensions. Next, work out the surface area of your cylinder.
Explain how you worked it out.

Success criteria

I can draw the net of a cylinder.
I can calculate the surface area of a cylinder from a net.
I can calculate the volume of a cylinder.
I can solve simple problems involving the surface area and volume of a cylinder.
I can combine my knowledge of circles and cylinders with my knowledge of other shapes and objects to find the area, surface area and volume of composite shapes and objects.

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