Maths / Trig volume and area

Trigonometry, Volume and Area

Trigonometry

Right-angled triangles can be studied in maths to help us solve real-life problems.

E xample:

How far is the ship from crashing on the rocks?

Hypotenuse = Longest side

Opposite = Side you hit when you fire a gun from the

Adjacent = Remaining side (next to )

= = =

SOH – CAH – TOA (Sock - A - Toa)

Examples: find the unknown sides

1. = 35^o

opposite – x

adjacent = 15

As we have 0 and A: it must be TOA ie. Tangent

tan =

tan35^o =

15 tan35^o = x

10.5 = x

2. = 35^o

opposite = x

hypotenuse = 15

We have O and H ; therefore we use sine

Sin35^o =

= sin35^o

x = 15 sin35^o

x = 8.6

Find the unknown angle:

Example 1.

Adjacent = 7

Hypotenuse = 13

We have A and H : therefore we use cosine

Cos =

= cos^-1

= 57.4^o

Example 2.

Adjacent = 7

Opposite = 10

We have A and O; therefore we use tangent

tan =

tan^-1

= 55^o

Practice Questions

Calculate the unknown

Answers

1. Cos =

= cos¹

= 53.1^o

2. = Cos 37^o

X = 10 cos 37^o

= 7.99

3. Sin 20^o =

= sin20^o

X = 5 sin20^o

= 1.71

4. Opposite = 10

Adjacent = 15

We have O and A ; therefore we use tangent

tan =

tan^-1

= 33.7^o

5. Opposite = 7

Adjacent = 9

We have O and A ; therefore we use tangent

tan =

tan^-1

= 37.97^o

6. = 67^o

Opposite = x

Adjacent = 5

We have O and A ; therefore we use tangent

= tan 67^o

x 5 tan 67^o

= 11.8

Area

Let’s review the rules for some basic shapes

1

s

. Square

Area = s x s (side x side)

2 . Rectangle L

Area = L x W w

3. Parallelogram

Area = b x h (perpendicular height)

4. Trapezium

Area = h

5. Triangle

Area = bh

6. Circle

Area =

Practice Questions

Calculate the area of the following shapes:

1. Square 5cm

2. Rectangle 15mm

3cm

3. Circle

4 . Trapezium 50mm

10cm

5. Parallelogram

6. Triangle

Answers

1. Area of square = S x S

= 5 x 5

= 25 cm²

2. Area of rectangle = L x W

= 3 x 1.5 (make sure you use W= 1.5cm not 15mm)

= 4.5 cm²

3. Area of circle =

= (radius = diameter) ( = 3.14159)

= 12.57 cm²

4. Area of Trapezium = x h

= x 4

= 30 cm²

5. Area of Parallelogram = b x h

= 12 x 3.9

= 46.8 cm²

6. Area of Triangle = bh

= x 6 x 3

= 9cm²

Volume

The volume of an 3D object represents how much space it occupies. The capacity refers to how much solid, liquid or gas it could hold.

1 . Cylinder

Volume =

= area of circle

x height

2. Triangular Prism

Volume = bh x L

= area of triangle

x length

3

H

. Rectangular Prism

V

W

olume = L x W x H

= area of rectangle L

x height

4 . Cube

Volume = S² x H

= area of square x Height H

S

Practice Questions

Find the volume

1.

2.

3.

4.2cm

4.2cm

4.2 cm

4 .

2cm

3 cm

10.2cm

Answers

1. Volume of triangular prism = bh x L

= x 3 x 4 x 8

= 48cm³

2. Volume of cylinder =

= x 20

= 6283.10 cm³

3. Volume of cube = S x S x S

= 4.2 x 4.2 x 4.2

= 74.1 cm³

4. Volume of rectangular prism = L x W x H

= 10.2 x 3 x 2

= 61.2 cm³