MODULE 2 Space, Shape and Orientation: SUMMARY (Levels 2 ...

MATHEMATICAL LITERACY LEVELS 2,3 AND 4 P a g e | 1 APRIL 2020 T SWANEPOEL/ts

MODULE 2

Space, Shape and Orientation:

SUMMARY

(Levels 2, 3 and 4)

VOCABULARY

Perimeter

The distance around a flat (two-dimensional) shape, measured in units

such as mm, cm, m and km

Diagonal

A line segment joining two points that are not adjacent to each other

Area

The measurement of the size of a surface that is covered by the shape

and measured in units such as mm2, cm2, m2 and km2

Angle

The space (usually measured in degrees) between two intersecting lines

Diameter

The line that passes through the centre of a circle from one side to the other

Radius

A line segment from the centre of a circle to the circle boundary.

The radius is always equal to half the diameter.

Circumference

The distance around a circle. The perimeter of a circle is called it’s circumference,

which is measured in units such as km, m, cm and mm.

Volume

The space occupied by a shape. When dealing with liquids, the volume of a

container can also be called the capacity of the container. Capacity is measured in

units such as mm3, cm3, l and kl.


Point of intersection

Where lines or surfaces meet

Perpendicular

When lines intersect at an angle of 90ᵒ

Parallel

When lines are an even distance from each other at any given position.

Parallel lines never intersect (cross each other or meet).

TWO-DIMENSIONAL SHAPES

Rectangle

A shape with four sides at 90ᵒ angles to each other.

The opposite sides are the same length to form two breadths and

two lengths. The opposites sides are parallel.

Perimeter:

2 x length + 2 x breadth

(2 x l) + (2 x b)

Area:

length x breadth

l x b


Square

A shape with four equal sides at 90ᵒ angles to each other.

The opposite sides are parallel.

Perimeter:

4 x side length

4 x s

Area:

side length x side length

s x s

Triangle

A basic shape with three corners and three sides.

Perimeter:

Add the three side lengths

Area:

½base x perpendicular height

½b x h


Circle

A simple shape consisting of joined points that are at an equal distance

from the centre.

Perimeter:

(called circumference)

2 x pi x radius

2 x л x r

Area:

pi x radius x radius

л x r2

Semicircle

Half a circle. The straight line is the diameter of the circle.

Perimeter:

Diameter + pi x radius

d + л x r

Area:

2

radiusradiuspi

2

2r


THREE-DIMENSIONAL SHAPES

Rectangular prism

An object that has six faces that are all rectangles

Surface area:

Add the areas of the 6 sides

or

2lb + 2lh + 2bh

Volume:

Area of base x height

or

length x breadth x height

Cube

A special rectangular prism with six square faces

Surface area:

6 x area of a side

or

6 x side length x side length


or

6(side length)2

Volume:


or

side length x side length x side length

or

(side length)3

Cylinder

Sphere

A basic geometrical shape with a circular base

Surface area:

2 x area of base + height x circumference of base

or

2лr2 + 2лrh

Volume:


or

лr2h


Sphere

A three-dimensional object shaped like a ball.

Every point on the surface is the same distance from the centre.

Surface area:

4лr2

Volume:

3

4лr3

Cone

A shape with a circle at the bottom and sides that narrow to a point.

Surface area:

pi x radius x side + pi x radius x radius

or

лrs + лr2

Volume:

3

1 x pi x radius x radius x height

or

3

1лr2h


Triangular prism

An object with two triangular bases and three rectangular sides

Surface area:

2 x area of base + length x base + 2 x side x length

or

bh + bl + 2sl

Volume:


or

½bhl


The Theorem of Pythagoras

For triangles where one of the angles is 90ᵒ, the theorem of Pythagoras can be used

to calculate the length of the hypotenuse.

The theorem of Pythagoras states that the square of the hypotenuse equals the sum

of the squares of the two opposite sides.

Manipulation of Pythagoras leads to the calculation of the lengths of the right-angled

sides.

XZ2 (XZ is the hypotenuse) = XY2 + YZ2

Therefore XZ = )( 22 YZXY

XY2 = XZ2 – YZ2

Therefore XY = )( 22 YZXZ

YZ = )( 22 XYXZ


SUMMATIVE ASSESSMENT

1. Determine the perimeter of the shape below.

2. Use the diagram below to answer the following questions.

2.1 Determine the area of the shape.

2.2 Calculate the perimeter of the shape.


3. Calculate the area of the red section of the diagram below, if

the diameter of the circle below is 8,2 cm.

4. John wants to build a fireplace in the shape below.

He wants to cover the whole surface with fireproof tiles.

The dimensions of a tile are 8 cm by 8 cm.

4.1 Calculate exactly how many tiles are needed to cover the surface.

Add an additional 5% tiles for possible breakage


5.

.

The diagram shows a concrete tank on Mr Mudau’s farm.

The height of the tank is 85 cm and the width of the wall of the tank is 20 cm.

5.1 Calculate the outside diameter of the tank.

5.2 Calculate the volume of the inside of the tank.

5.3 Calculate th volume of the wall of the tank.

5.4 Mr Mudau wants to paint the outside of the tank.

Determine the surface area of the outside of the tank.

(formula: outside area = 2лrh)

5.5 One litre of paint is needed to paint an area of 28 000 cm2.

What will it cost to paint the outside area, if 1 litre of paint costs R49,95?

6.


A brick has a length of 20 cm, a breadth of 8 cm and a height of 8 cm.

There are two circular holes in the brick, each with a diameter of 3 cm.

Calculate the volume of the clay that the brick is made of.

7. Calculate the quantity of sheet metal needed to manufacture 50 cylindrical jugs,

each with a diameter of 20 cm and a height of 30 cm. The jugs are open at the

top without lids. The metal sheets can only be bought in sheets of 1 m2.

8. The area of a semi-circle is 5 024 cm2.

8.1 Calculate the length of the straight lined side.

8.2 What is the circumference of the semi-circle?

Give your answer to the nearest centimetre.

9.

9.1 The area of a netball court is 465,125 m2. If the breadth of the court is 15,25 m,

what is the length?

9.2 A netball court is divided into thirds. What will the area of a third be?

Give your answer to the nearest square metre.

10.

10.1 The volume of a cylindrical jug is 2 000 cm3.

What is the diameter of the jug if the height is 200 mm?


11. Calculate the height of a rectangular prism with a base area of 120 cm2 and

capacity of 48 l. (1 l = 1 000 cm3)

12 If the volume of a cube is 64 cm3, what is the length of its side?

13. The area of a circle is 706,5 cm2.

13.1 What is the radius of the circle?

13.2 If the area is doubled, what would the radius be?

14.

Given:

Area of a circle = лr2

Volume of a cylinder = area of base x height

1 m3 = 1 kl

If the cement wall of the dam is 35 cm thick, calculate the volume of the water in

the dam if the dam is 80% full. Give the answer in kilolitres.

15.


The height of the triangle above is 8 cm and the area is 1 800 mm2.

Calculate the base length.

16.

The diameter of a swimming area, including the pool and a paved area of

1,5 m wide around the pool, is 15 m.

Formulae: Area of circle = л x r2

Circumference of circle = 2 x л x r

Volume of a cylinder = л x r2 x h

Use л = 3,14

16.1 Determine the area of the paving.

16.2 Determine the circumference of the pool.

16.3 Determine the volume of the water in the pool if the depth is 1,2 m and the

pool is 85% full.

16.4 Calculate the number of bricks required for the paving, if 52 bricks cover an

area of one square metre.

Make provision for breakages and cutting by adding an extra 10%.


17

17.1 Calculate the area of the concrete slab below.

17.2 The concrete is mixed by combining 1 unit of cement with 4 units of sand and

8 units of gravel. To produce 0,25 cubic metre of concrete, 2 bags of cement

are needed. How many bags of cement are required to lay this slab if it is 5 cm

thick?

17.3 How many bags of sand are required?

17.4 For every two bags of cement, 37 l of water is used. How much water will be

used to lay the slab?

18 The area af the mirror with a semi-circular shape is 502 400 mm2.

18.1 Calculate the length of the straight lined side.

18.2 What length of frame is needed for the mirror? Give the answer to the nearest

centimetre.


19 Calculate the length of AC.

20. Calculate the length of BC.

21. The diagram below shows a kite, ABCD. The diagonals cut at right angles and

intersect at O. Calculate the length of the diagonal AC.

(All lengths are given in centimetres.)


[Some extracts were made from the text book: Mathematical Literacy HANDS-ON

TRAINING by Cecile Bruwer & Salome Voges]

MODULE 2 Space, Shape and Orientation: SUMMARY (Levels 2 ...

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