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GM5-4: Find the perimeters and areas of circles and composite shapes and the volumes of prisms, including cylinders.

Elaboration on this Achievement Objective

This means that students should find the ratio of the diameter of a circle to its perimeter (π) by measuring several circles and looking for the relationship.

They should understand that the formula for the circumference of a circle can be expressed in two ways:
C = 2πr (2 times pi times the radius), or
C = πD (pi times the diameter)

 

They should apply the formula to find the circumference of any circle, for example the circumference of a bicycle tyre. Students should find the formula for area of a circle by cutting several circles into increasingly small sectors and reassembling the sectors to approximate rectangles, A = πr2 (pi times the radius squared). They should apply this formula to finding the area of any circle.

Students will find the perimeters of composite shapes by adding the lengths of the sides together. At Level Five composite shapes are made up of common polygons (for example rectangles, triangles, semicircles). Students will find the area of composite shapes by calculating the areas of the parts and adding them together. Students will find the volumes of prisms by multiplying the area of their cross section by their length, for example for a cylinder multiply the area of the circle by the length of the cylinder.

 

Teaching resources for this Achievement Objective
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A4 containers

Level Five
Geometry and Measurement
Problem solving activities
Describe the relationship between the surface area and volume of a container
Devise and use problem solving strategies to explore situations mathematically (be systematic, draw a diagram, use a model)
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A4 Cylinders

Level Five
Geometry and Measurement
Rich learning activities
The purpose of this activity is for students to create open cylinders and calculate their volume. In doing so students consider the relative effects of increasing and decreasing height and radius on the volume.
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Area of a circle

Level Five
Geometry and Measurement
Rich learning activities
The purpose of this multi-level task is to engage students in an investigation of the area of circles.
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Around We Go

Level Five
Geometry and Measurement
Units of Work
The focus for this unit is the measurement of circles and the relationships to be found within them. Students investigate a range of circles, each with a different radius, diameter and circumference and assemble this information in their search for relationships.
  • Measure the circumference of circles
  • State the relationship between the circumference and diameter of a circle
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Bill's badge

Level Five
Geometry and Measurement
Problem solving activities
Construct two intersecting circles where the radius of the second circle is on the circumference of the first.
Use Pythagoras’ theorem to find the area of a rhombus
Devise and use problem solving strategies to explore situations mathematically (be systematic, draw a diagram, use a model)
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Circle Links

Level Four
Level Five
Geometry and Measurement
Figure It Out activities
This is a level 4 and 5 measurement strand activity from the Figure It Out series.A PDF of the student activity is included.

find approximate value of pi from measurement

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Colossal Kiwifruit

Level Five
Geometry and Measurement
Figure It Out activities
This is a level 5 measurement strand activity from the Figure It Out series. A PDF of the student activity is included.

find the circumference and area of a circle

find the area of a cylinder

fidnt he volume of a cuboid

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Dan's badge

Level Five
Geometry and Measurement
Problem solving activities

Use area formulas of circles and squares to solve problems

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Fences and Posts

Level Five
Integrated
Units of Work
This measurement unit concerns Pick’s Rule which applies to finding areas of polygons where all the vertices are lattice points, (in this application, the nails on geoboards). The unit centres on finding linear patterns.
  • find areas of shapes
  • find simple two-variable linear patterns relating to areas
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G5.30: Finding the area of a regular polygon

Level Five
Geometry and Measurement
Student e-Ako
This interactive learning module supports students to explore rules for the angle properties of intersecting and parallel lines.
  • Showing that a polygon is composed of rectangles and triangles.
  • Showing that a non-right angled triangle is composed of two right-angled triangles.
  • Given the length of one side, finding the area of a regular polygon.
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Gumboot Games

Level Five
Integrated
Figure It Out activities
This is a level 5 measurement strand activity from the Figure It Out series. A PDF of the student activity is included.

order distances

use rates to solve problems involving measuring units

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Karen's tiles

Level Five
Geometry and Measurement
Problem solving activities
investigate how perimeter changes as area changes
see the relation between area and perimeter for similar shapes
pose questions for mathematical exploration
prove or refute mathematical conjectures
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Marketing Tricks

Level Five
Geometry and Measurement
Rich learning activities
The purpose of this multi-level task is to encourage students to use algebraic techniques when solving a measurement problem.
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On the Right Track

Level Five
Geometry and Measurement
Figure It Out activities
This is a level 5 measurement strand activity from the Figure It Out series. A PDF of the student activity is included.

find the perimeter (cicumference) of a circle

find the area of a circle

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Penny's box

Level Five
Geometry and Measurement
Problem solving activities
Find the volume and surface area of rectangular prisms
Devise and use problem solving strategies to explore situations mathematically (be systematic, draw a diagram).
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Peter's second string

Level Five
Geometry and Measurement
Problem solving activities
determine the maximum area of a rectangle with a given perimeter
determine the range of areas of a rectangle with a given perimeter
pose questions for mathematical exploration
prove or refute mathematical conjectures
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Playdough balls

Level Five
Integrated
Problem solving activities
compare the volume of a sphere and a cylinder by either measuring or applying a formula.
devise and use problem solving strategies to explore situations mathematically (be systematic, make a model).
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Puck's girdle

Level Five
Geometry and Measurement
Problem solving activities
Find the circumference of a circle.
Understand the relationship between changes in the circumference of a circle and changes in the radius.
Devise and use problem solving strategies to explore situations mathematically (be systematic, make a model).
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Round the Bend

Level Five
Geometry and Measurement
Figure It Out activities
This is a level 5 measurement strand activity from the Figure It Out series. A PDF of the student activity is included.

calculate circumference of a sphere

explore the radius and circumference relationship

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Round the Track

Level Five
Integrated
Units of Work
This unit is about making calculations in the real life situations of athletics. The main task is to construct a scale plan of an athletics track.
  • Draw a plan to scale, of an object based on a rectangle and two semi-circles.
  • Understand the relation of length on the plan to actual length.
  • Find lengths and areas use these for costing purposes.
  • Be able to link speed, distance and time (given two find the third).
  • Estimate distances and300
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Scale Factors for Areas and Volumes

Level Five
Integrated
Units of Work
This unit of work provides students with the opportunity to discover the effect of a change of length by a given factor on area and volume.
  • use scale factors to investigate areas being enlarged
  • use scale factors to investigate volumes being enlarged
  • solve real life context problems involving scale factors
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Tennis ball tubes

Level Five
Geometry and Measurement
Problem solving activities
Use the formula for the circumference and diameter of a sphere to solve a problem
Devise and use problem solving strategies to explore situations mathematically (be systematic, make a model)
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Transforming shapes

Level Five
Geometry and Measurement
This unit applies geometry and measurement in the creation of bags, door stops and other fabric items.
  • Use translations, reflections and rotations to create friezes and Escher type tessellations.
  • Create tivaevae and other symmetric patterns and describe the lines of symmetry and rotational symmetry.
  • Apply knowledge of circles (perimeters, area and surface area) to create nets for cylinders.