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E. Geometry and Measurement · Geometry and Measurement: Circles, Triangles, Measurement Systems, Pythagoras and Volume

Geometry and Measurement: Circles, Triangles, Measurement Systems, Pythagoras and Volume: Unit Review

A1 A2 E1.1 E1.2 E1.3 E1.4 E1.5 E1.6About 40 minutes
Learning goals
  • I can convert units within and between measurement systems and tell the story of a measurement idea.
  • I can use circle and triangle properties to find unknown lengths and angles and to analyse designs.
  • I can predict how changing dimensions affects perimeter, area, surface area and volume, and find optimal shapes.
  • I can use the Pythagorean relationship and the volume relationships for prisms, pyramids, cylinders and cones.

Words to know

conversion factor
A fraction equal to 1 that changes units, like 100 cm / 1 m.
chord
A segment joining two points on a circle.
inscribed angle
An angle with its vertex on a circle; half the central angle on the same arc.
circumcentre
Where the perpendicular bisectors of a triangle's sides meet.
scale factor (k)
The number every length is multiplied by: areas change by k², volumes by k³.
hypotenuse
The longest side of a right triangle, opposite the right angle.
slant height
The distance from the apex of a cone or pyramid down its side to the base edge.

Let's review!

In this unit you researched the story of measurement and π, converted units within and between measurement systems, discovered circle properties (chords, tangents, central and inscribed angles, cyclic quadrilaterals) and triangle properties (exterior angles, the triangle inequality, and three triangle centres), and used them to create and analyse designs. You showed how changing dimensions changes perimeter, area, surface area and volume, found optimal shapes, used the Pythagorean relationship in real situations and composite shapes, and found the volumes of prisms, pyramids, cylinders and cones.

This review has a few questions from every lesson. Try each one, then press Show answer to check. If a part feels tricky, go back to that lesson for more practice.

length × karea × k²volume × k³k = 2: × 2, × 4, × 8k = 3: × 3, × 9, × 27k = 10: × 10, × 100, × 1000
The scale factor rule from the changing-dimensions lessons: lengths change by k, areas by k², and volumes by k³.
prismV = BhpyramidV = ⅓BhcylinderV = πr²hconeV = ⅓πr²h
The volume family from the two volume lessons. A pyramid holds ⅓ of the matching prism, and a cone holds ⅓ of the matching cylinder.

The Story of Measurement: Units, Systems and π

1

Why are agreed standard units important for trade and building?

2

A horse is 14.3 hands tall. Give its height in inches and in centimetres.

3

A regular hexagon is drawn inside a circle with a 20 cm diameter. What is the hexagon's perimeter, and what does it tell you about π?

Converting Units Within a Measurement System

1

Convert 3.6 m² to cm².

2

A box holds 4500 cm³. How many litres is that?

3

Convert 9 ft² to square inches.

4

Write 7 ft 11 in in inches.

Converting Between Measurement Systems, Including Units from Different Communities

1

Convert 6 ft 1 in to centimetres.

2

Convert −5 °C to °F.

3

Convert 2 acres to hectares (1 acre ≈ 0.4047 ha).

4

About how long are 5 linear arpents (1 arpent ≈ 58.5 m)?

Circle Properties: Chords and Tangents

1

A circle has radius 15 cm and a chord 24 cm long. How far is the chord from the centre?

2

P is 25 cm from the centre of a circle with radius 7 cm. Find the length of a tangent from P.

3

A chord is 12 cm from the centre of a circle with radius 13 cm. How long is the chord?

Circle Properties: Central and Inscribed Angles

1

A central angle is 96°. Find an inscribed angle on the same arc.

2

A cyclic quadrilateral has an angle of 47°. Find the opposite angle.

3

Triangle ABC is inscribed with AB a diameter and ∠A = 41°. Find ∠B.

Triangle Properties and the Centres of a Triangle

1

An exterior angle of a triangle is 140° and one remote interior angle is 65°. Find the other remote interior angle.

2

Can sides 6 cm, 8 cm and 15 cm form a triangle?

3

A right triangle has legs 9 cm and 12 cm. What is the radius of its circumscribed circle?

Creating and Analysing Designs with Circles and Triangles

1

For a regular decagon (10 sides) in a circle, find the central angle, the base angles of each isosceles triangle and the interior angle.

2

A regular hexagon is inscribed in a circle of radius 8 cm. Find its perimeter.

3

An equilateral triangle is inscribed in a circle of radius 12 cm. Find the area of the ring between this circle and the triangle's inscribed circle.

How Changing Dimensions Affects Perimeter, Circumference and Area

1

Every length of a shape is multiplied by 5. What happens to its area?

2

A circle has an area of 20 cm². Its radius is tripled. Find the new area.

3

Two similar shapes have areas in the ratio 49 : 1. What is the ratio of their lengths?

How Changing Dimensions Affects Surface Area and Volume

1

Every dimension of a box is multiplied by 4. What happens to its surface area and volume?

2

A cylinder's radius is halved and its height stays the same. What happens to its volume?

3

A 1 : 50 model tank holds 0.2 L. How much does the real tank hold?

Finding the Best Dimensions: Maximum Area and Minimum Surface Area

1

A rectangle has a perimeter of 52 m. What is its greatest possible area?

2

A rectangle has an area of 144 m². What is its least possible perimeter?

3

A pen along a wall uses 90 m of fence on three sides. Find the best dimensions and area.

The Pythagorean Relationship in Real-Life Situations

1

A 13 m ladder has its foot 5 m from a wall. How high does it reach?

2

A 32-inch TV has a 16 : 9 screen. Find its width and height to one decimal.

3

Find the space diagonal of a box 12 cm × 16 cm × 15 cm.

4

Is a triangle with sides 8, 15, 17 a right triangle?

Using the Pythagorean Relationship with Composite Shapes

1

An isosceles trapezoid has parallel sides 7 m and 19 m and slanted sides of 10 m. Find its area.

2

A rhombus has diagonals 18 cm and 24 cm. Find its perimeter and area.

3

A cone has radius 7 cm and slant height 25 cm. Find its height.

The Volume of Prisms and Pyramids

1

A square pyramid has a base of 12 m and a height of 10 m. Find its volume.

2

A triangular prism has a triangular base with base 5 cm and height 4 cm, and a length of 20 cm. Find its volume.

3

A pyramid has a volume of 72 cm³ and a base area of 24 cm². Find its height.

The Volume of Cylinders and Cones

1

A cylinder has a diameter of 10 cm and a height of 20 cm. Find its volume in litres, to two decimals.

2

Find the volume of a cone with r = 9 cm and h = 10 cm.

3

A cone has r = 5 cm and slant height 13 cm. Find its volume.

Just for fun

Geometry review riddle: what do a pizza, a salt pile and a stop sign have in common?

They are all great places to practise this unit: area, cone volume, and a regular octagon with 135° angles!

How to use this review

  • Work through one section at a time and write your steps, not just the answers.
  • Put a check mark beside each question you got right on your own. Put a star beside any you needed help with.
  • If you starred two or more questions in a section, go back to that lesson and redo its "More practice" questions.
  • Finish with the quiz. For every quiz mistake, read the explanation and find the lesson it came from.
  • Mistakes are useful: they show exactly what to practise next.
Check your understanding

Quiz

1.

Convert 2 m² to cm².

2.

Convert 68 °F to °C.

3.

A circle has radius 10 cm. A chord is 6 cm from the centre. How long is the chord?

4.

An inscribed angle stands on the same arc as a central angle of 150°. The inscribed angle is:

5.

Which point of a triangle is the centre of its circumscribed circle?

6.

A regular octagon is drawn in a circle. Its central angle is:

7.

Every dimension of a cylinder is doubled. Its volume is multiplied by:

8.

You have 32 m of fence for a rectangle (4 sides). The greatest possible area is:

9.

A 2.5 m ladder reaches 2.4 m up a wall. How far is its foot from the wall?

10.

A cylinder and a cone have the same radius and height. The cylinder holds 600 mL. The cone holds:

For parents and teachers

How to use the review results:

  • Unsure about units, standards, or the story of π → revisit The Story of Measurement: Units, Systems and π.
  • Area and volume units converted with × 100 instead of × 10 000, or mixed units read as decimals → revisit Converting Units Within a Measurement System.
  • Errors converting between metric and imperial, or with temperature → revisit Converting Between Measurement Systems, Including Units from Different Communities.
  • Forgetting to halve a chord, or using the wrong hypotenuse with tangents → revisit Circle Properties: Chords and Tangents.
  • Doubling instead of halving inscribed angles, or cyclic quadrilateral errors → revisit Circle Properties: Central and Inscribed Angles.
  • Exterior angle, triangle inequality, or mixing up circumcentre and incentre → revisit Triangle Properties and the Centres of a Triangle.
  • Trouble with central angles of regular polygons or design areas → revisit Creating and Analysing Designs with Circles and Triangles.
  • Using k instead of k² for area → revisit How Changing Dimensions Affects Perimeter, Circumference and Area.
  • Using k² instead of k³ for volume, or radius vs height in cylinders → revisit How Changing Dimensions Affects Surface Area and Volume.
  • Fence and box problems (forgetting the constraint, or the wall) → revisit Finding the Best Dimensions: Maximum Area and Minimum Surface Area.
  • Adding instead of subtracting squares, or two-step (3D) diagonals → revisit The Pythagorean Relationship in Real-Life Situations.
  • Missing heights in trapezoids, rhombuses, cones and pyramids → revisit Using the Pythagorean Relationship with Composite Shapes.
  • Using a side instead of base area, or the slant height instead of the height → revisit The Volume of Prisms and Pyramids.
  • Using the diameter in πr²h, or forgetting the ⅓ for cones → revisit The Volume of Cylinders and Cones.

Keep practice short and frequent. Ask your teen to explain one answer out loud each day: explaining and making connections are the Grade 9 math processes (A1 and A2).