Perimeter, Area and Volume of Regular Shapes

Camborne School of Mines

Perimeter, Area and Volume of Regular Shapes

University of Exeter

Perimeter of Regular Polygons Perimeter means the total length of all sides, or distance around the edge of a polygon.

For a polygon with straight sides this is the sum of all sides.

Eg. triangle

rectangle

parallelogram

trapezium

8

11

9

5

5

6

6

7

7

4

4

4

11

5 + 5 + 4 = 14cm 6 + 6 + 11 + 11 = 34cm All dimensions given in cm (not drawn to scale)

8

8 + 8 + 7 + = 30cm

3

4 + 4 +9 + 3 = 20cm

For polygons with curved sides the perimeter is known as the circumference and is given by the formula

Circumference = 2r for a circle and

2 a? + b? for an ellipse 2

Where

is a mathematical constant with the value of 3.142 (correct to 3 decimal places)

r is the radius of the circle (distance from centre to circumference)

a is the major radius of an ellipse ................

b is the minor radius of an ellipse .._.._.._.._.._

Eg.

a = 6cm

b = 4cm

Circumference = 2r

2 x 3.142 x 5

31.42cm

circumference = 2 x 3.142 36 + 16

2

radius = 5

= 32.04cm

Area of Regular Polygons The area of a polygon is the space it occupies in a single plane.

For squares, rectangles and parallelograms the area is given by

Area = base x height

Eg.

12

4

ht = 7

12

12

8

12 x 12 = 144cm?

12 x 4 = 48cm?

8 x 7 = 56cm?

Height is defined as the perpendicular distance between the pair of parallel sides

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All dimensions given in cm (not drawn to scale)

For Triangles

area = ? x base x height

University of Exeter

Where height is distance from apex to meet base at right angle

Area = ? x 12 x 3 = 18cm?

3 cm

For Trapeziums

12 cm

area = ? sum of parallel sides x height

8 cm

7 cm

Area = ? x (8 + 14) x 7 = 77cm?

For Circles

14 cm

area = r?

r = 5

Area = r? = 3.142 x 5? = 78.54cm?

For a sector of a Circle

r = 5 60?

area = area of circle x sector angle 360

Area of sector = r? x 60 = 13.1cm? 360

For Ellipse

b = 5

a = 5

area = ab

Area = 3.142 x 10 x 5 = 157cm?

Complex shapes for which there are no formulas should be divided into simple shapes. The area of each is then calculated and added together to determine the overall area.

A1 A5 A3

A2

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Area = A1 + A2 + A3 + A4 - A5

A4

CSM1027 Maths 1A Foundation

Camborne School of Mines

University of Exeter

Volume of Regular Shapes

Volume is the amount of space in 3 dimensions occupied by a shape.

Prism A prism is any shape where the cross-sectional area is constant.

For any prism:

Volume

= area of base x height

Rectangular Prism

h

b I Shaded area is the base

area of base =

volume

=

length x breadth length x breadth x height

eg. calculate the volume of a block with a square base of side 6cm and a height of 10cm

volume

= 1 x b x h = 6 x 6 x 10 = 360cm?

Triangular Prism

h1

h2

b

Shaded area is the base

Circular Prism

h r Shaded area is the base

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area of base =

volume

=

? x base x height ? x base x h1 x h2

eg. determine the volume of a component 16cm long with a triangular cross-section which has a base of 4cm and perpendicular height of 5cm

area of base =

volume

=

=

? x 4 x 5 ? x 4 x 5 x 16 160cm?

area of base =

volume

=

r? r? x height

eg. calculate the volume of a cylinder with a radius of 5cm and a height of 4cm.

volume

= r? x height = 3.142 x 5? x 4 = 314.2cm?

CSM1027 Maths 1A Foundation

Camborne School of Mines

University of Exeter

The volume of certain non-prismatic shapes can be determined by using the correct formula.

Sphere r

volume of a sphere =

4 r3 3

eg. determine the volume of a spherical component with the radius of 7cm.

Pyramid and cone

volume

= 4 x 3.142 x 7? = 1436.76cm? 3

h b

1

h r

volume

Pyramid volume

Cone volume

= 1 x base area x height 3

= 1x1xbxh 3

= 1 x r? x h 3

eg. calculate the volume of a cone with base radius of 6cm and perpendicular height of 10cm

Volume

= 1 x 3.142 x 6? x 10 = 376.00cm?

3

Volumes of irregular shapes can be determined by calculation if the mass and density of the

material from which it is known or by displacement.

Calculation of volume using density and mass.

eg. density of substance from which an irregular object is made is 8500kg/m?. if it has a mass of 425kg, calculate its volume.

Volume

=

mass

density

= 425 = 0.05m? 8500

Measurement of volume using displacement

500cc 2nd reading

300cc

1st reading

volume

= 2nd reading ? 1st reading = 500 ? 300 = 200cc

Measuring cylinder

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Perimeter, Area and Volume of Regular Shapes Worksheet 1

Calculate the area of the following shapes

1.

2.

15 cm

9.5 cm

5 cm

3.

15 cm

4.

1.3 cm

7.8 cm

4.5 cm 6 cm

1.3 cm

3 cm

5.

3.5 cm

12 cm 12 cm

47.5 cm

2.5 cm

24 cm

5 cm

6. A water tank is a cuboid with a base of 1.2m by 0.8m. How deep is the water when the tank contains 0.384m? of water?

7. A classroom is 5m x 6m x 3m. Health regulations require that each student must have a minimum of 5m? of air. How many students can occupy the room?

Calculate the volume of the following shapes. All dimensions in cm.

8. 2 2

9.

2

10.

12

8

15

2 6

Internal r = 0.75 External r = 1.00

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