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IntroductionTopic dealing with the use of formulae to calculate Perimeters, Areas and Volumes of plain shapes and solid ones (prisms).

Plane:A plane is a flat surface (think tabletop) that extends forever in all directions.It is a two-dimensional figure.Three non-collinear points determine a plane.So far, all of the geometry we’ve done in these lessons took place in a plane.But objects in the real world are three-dimensional, so we will have to leave the plane and talk about objects like spheres, boxes, cones, and cylinders.

Solid: Geometric figure in three dimensionsSurface Area: Total area of all the surfaces of a solid shape or prism.Volume: This is the space occupied by a solid shape or prism.

© iTutor. 2000-2013. All Rights Reserved


Areas of geometrical shapes

RECTANGLE

l

w l w

a

a SQURE a a

TRIANGLE

b

h 1/2 b h

h

b

PARALLELOGRAM b h

ShapeDiagram Area



ShapeDiagram

Area

CIRCLE r2r

TRAPEZIUM ½(a +b)h

a

hb



AREA The perimeter

of a shape is a measure of distance around the outside.

The area of a shape is a measure of the surface/space contained within its perimeter.

Area is measured in units2

Units of distance

mm

cm

m

km

inches

feet

yards

miles

1 cm1 cm2

1 cm

1 cm

Metric

Imperial

Units of area

mm2

cm2

m2

km2

inches2

feet2

yards2

miles2

Metric

Imperial



Area of a rectangleExamples

To Find the area of a rectangle simply multiply the 2 dimensions together. Area = l x w (or w x l)

Find the area of each rectangular shape below.

100 m

50 m

120 m

40 m

1 2

34

5

8½ cm

5½ cm

90 feet

50 feet

210 cm

90 cm

5 000 m2

4500 ft2

4 800 m2

46.75 cm2

18 900 cm2 © iTutor. 2000-2013. All Rights Reserved


Area of a Triangle

rectangle area = 2 + 2 triangle area = ½ rectangle area

base

height

Area of a triangle = ½ base x height

The area of a triangle = ½ the area of the surrounding rectangle/parallelogram



Area of a Triangle Example Find the area of the following

triangles.

8 cm 10 cm

14 cm

12 cm9 cm 16

cm

Area = ½ b x h

3.2 m

4.5 m

7 mm

5 mm

Area = ½ x 8 x 9 = 36

cm2

Area = ½ x 10 x 1 = 60 cm2

Area = ½ x 14 x 16 = 112 cm2

Area = ½ x 3.2 x 4.5 = 7.2 m2

Area = ½ x 7 x 5 =

17.5 mm2© iTutor. 2000-2013. All Rights Reserved


The Area of a Trapezium

Area = (½ the sum of the parallel sides) x (the perpendicular height)

A = ½(a + b)h

a

b

h½ah

½bh

Area = ½ah + ½bh = ½h(a + b)

= ½(a + b)h

Find the area of each trapezium

1

8 cm

12 cm

9 cm

25 cm

7 cm

6 cm

3

5 cm

3.9 cm

7.1 cm

Area = ½ (8 + 12) x 9

= ½ x 20 x 9

= 90 cm2

Area = ½(7 + 5) x 6

= ½ x 12 x 6

= 36 cm2

Area = ½(3.9 + 7.1) x 5

= ½ x 11 x 5

= 27.5 cm2© iTutor. 2000-2013. All Rights Reserved


32 Sectors

Transform

Remember C = 2πr

?

?

As the number of sectors , the transformed shape becomes more and more like a rectangle. What will the

dimensions eventually become?

½C

r

πr

A = πr x r = πr2

The Area of a Circle



A = r2

A = x 82

A = 201.1 cm2

A = r2

A = x 102

A = 314 cm2

Find the area of the following circles. A = r2

8 cm

1

10 cm

2

Examples



Three Dimensional GeometryThree-dimensional figures, or solids, can be made up of flat or curved surfaces. Each flat surface is called a face. An edge is the segment that is the intersection of two faces. A vertex is the point that is the intersection of three or more faces.



BoxesA box (also called a right parallelepiped) is just what the name box suggests. One is shown to the right.A box has six rectangular faces, twelve edges, and eight vertices.A box has a length, width, and height (or base, height, and depth).These three dimensions are marked in the figure.

LW

H

The volume of a three-dimensional object measures the amount of “space” the object takes up.Volume can be thought of as a capacity and units for volume include cubic centimeters (cm3) cubic yards, and gallons.The surface area of a three-dimensional object is, as the name suggests, the area of its surface.



Volume and Surface Area of a BoxThe volume of a box is found by multiplying its three dimensions together:

LW

H

V L W H

Example

Find the volume and surface area of the box shown.The volume is

The surface area is

The surface area of a box is found by adding the areas of its six rectangular faces. Since we already know how to find the area of a rectangle, no formula is necessary.

8 5 4 40 4 160

8 5 8 5 5 4 5 4 8 4 8 4

40 40 20 20 32 32

184

85

4



Cube A cube is a box with three equal dimensions (length = width = height). Since a cube is a box, the same formulas for volume and surface area hold. If s denotes the length of an edge of a cube, then its volume is s3 and its surface area is 6s2.

A cube is a prism with six square faces. Other prisms and pyramids are named for the shape of their bases.



Prisms

A prism is a three-dimensional solid with two congruent bases that lie in parallel planes, one directly above the other, and with edges connecting the corresponding vertices of the bases.The bases can be any shape and the name of the prism is based on the name of the bases.For example, the prism shown at right is a triangular prism.The volume of a prism is found by multiplying the area of its base by its height.The surface area of a prism is found by adding the areas of all of its polygonal faces including its bases.



Solution

(a)T. S. A. = Area of the 2 triangles + Area of rectangle 1 + Area of rectangle 2

+ Area of rectangle 3 = 2 (½ x 6 x 8) + (6 x 5.5) + (10 x 5.5) + (8 x 5.5)

= (2 x 24) + 33 + 55 + 44 = 48 + 33 + 55 + 44

Therefore T. S. A. = 180cm²

(b) V = Base area x height = 24 x 5.5 V = 132cm³

6cm

8cm5.5cm

10cm

A triangular prism has a base in form of a right-angled triangle, with sides 6cm, 8cm and 10cm. If the height of the prism is 5.5cm, sketch the prism and calculate, (a) its total surface area, (b) its volume.

Example


CylindersA cylinder is a prism in which the bases are circles.The volume of a cylinder is the area of its base times its height:

The surface area of a cylinder is:

h

r2V r h

22 2A r rh

8cm

3cm

Find the surface area of the cylinder.

Surface Area = 2 x x 3(3 + 8)

= 6 x 11

= 66

= 207 cm2

Example



Pyramids

A pyramid is a three-dimensional solid with one polygonal base and with line segments connecting the vertices of the base to a single point somewhere above the base.

There are different kinds of pyramids depending on what shape the base is. To the right is a rectangular pyramid.

To find the volume of a pyramid, multiply one-third the area of its base by its height.

To find the surface area of a pyramid, add the areas of all of its faces.



Find the volume of the following prisms.

9 m2

V = 9 x 5 = 45 m3

8 cm2 7 mm2

5 m 4 cm10 mm

20 mm2

10 mm

30 m2

2½ m 40 cm2

3 ¼ cm

V = 8 x 4 = 32 cm3

V = 7 x 10 = 70 mm3

V = 20 x 10 = 200 mm3

V = 30 x 2½ = 75 m3

V = 40 x 3¼ = 130 cm3

1 2 3

4 5 6


ConesA cone is like a pyramid but with a circular base instead of a polygonal base.The volume of a cone is one-third the area of its base times its height:

The surface area of a cone is:

h

r

21

3V r h

2 2 2A r r r h



SpheresSphere is the mathematical word for “ball.” It is the set of all points in space a fixed distance from a given point called the center of the sphere.A sphere has a radius and diameter, just like a circle does.The volume of a sphere is:

The surface area of a sphere is:

r

34

3V r

24A r



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