Geometry and measures - Maths

Pythagoras’ theorem

a² + b² = c²

How to answer an Pythagoras’ theorem question

1. Label the longest side of the triangle ‘c’

2. Label the other two sides ‘a’ and ‘b’

3. Write out the formula for Pythagoras’ Theorem

4. Substitute the values ‘a’, ‘b' and ‘c’.

5. Rearrange the formula and solve.

A triangle when answering a Pythagoras” Theorem question should…

• Have the formula being short² + short² = long²

• Be a right angled triangle

• Have the lengths of two known sides

• The length of the third side missing

Hypotenuse

The longest side of a right-angled triangle

Adjacent

The side located where the right angle is and the side needed to be worked out.

Opposite

The side opposite to the angle used for working out/to be worked out.

sin (x) =

Opposite / Hypotenuse

cos (x) =

Adjacent / Hypotenuse

tan (x) =

Opposite / Adjacent

How to answer a trigonometry question

1. Label the hypotenuse first - it is the longest side

2. Label the side adjacent to the angle/side you want to work out.

3. Label the side opposite to the angle/side you want to work out.

4. Remember the formula for the triangles

   • if (x) is at the bottom of the fraction:

     • Divide the known side by sin/cos/tan (x)

   • if (x) is at the top of the fraction:

     • Multiply the known side with sin/cos/tan (x)

   • If two of the sides are known:

     • Find out the sin/cos/tan inverse (___^-1)of the fraction.

5. Write all figures of the calculator display then round your answer.

A triangle when answering a Trigonometry question should…

• Include either:

  • One known side and angle

  • Two known sides and a missing angle

• Be a right angled triangle.

Exact values for sin(45) and cos(45)

1/√2 (not √2/2)

Area of a triangle

½ b × h

Area of a quadrilateral

b × h

Area of a trapezium

½ (a + b) h

How to find the area of a complex shape

• Split it into parts where you can find the area of those parts

• If a shape looks cut out:

  • Take away the area of the cutted out part away from the entire area.

How to find the area of shapes

• Make sure the lengths are in the same unit of measure.

• Give the answer in square (unit of measure).

• If the unit is not present next to the space provided to write your answer:

  • Write the unit next to your answer.

To answer questions with unfamiliar problems…

• Write out the formula before substituting values.

• Write lengths or angles you work out neatly on the diagram given.

• Draw your own sketch if that helps.

• Use algebra for unknown variables.

• If it’s tricky, use some numbers instead of variables.

Area conversions

1cm² = 10²mm² = 100mm²

1m² = 100²cm² = 10,000cm²

1km² = 1000²m² = 1,000,000m²

Volume conversions

1cm³ = 10³mm³ = 100mm³

1m³ = 100³cm³ = 10,000cm³

1km³ = 1000³m³ = 1,000,000m³

Unit of area/volume conversions

• The multiplier for an area conversion is the length multiplier squared

• The multiplier for a volume conversion is the length of the multiplier cubed

Prism

A 3D shape which has a constant cross-section

Surface area

The area of all of the faces combined in a 3D shape

Circumference of a circle

2πr or πd

Area of a cylinder

πr²h

Surface area of a cylinder

2πr² × 2πrh

Sector

A fraction of a circle cut out along the circumference with the two straight sides being the radius of the circle it was derived from.

Formula including sectors

Sector = x/360

Area of sector = x/360 × πr²

Arc length = x/360 × 2πr

Volume of a cone

⅓ πr²h

Volume of a sphere

4/3πr³

Volume of a pyramid

⅓ × Area of base × vertical height

Curved surface area of a cone

πrl

Total surface area of a cone

πrl + πr²

^{(l means the slant height)}

Surface area of a sphere

4πr²

Elevation

A 2D drawing of a 3D shape from a different direction.

Command word: Construct

Draw accurately with a pencil, ruler, compasses or protractor. You shouldn’t rub out any construction lines.

75 - Exterior angle

An angle found outside the area of a polygon.

75 - Interior angle

An angle found inside the area of a polygon.

75 - Sum of exterior angles

360°

75 - Formula for the sum of the interior angles

180° × (number of sides - 2)

75 - What to avoid when answering a polygon question

Do not try to draw the polygon if there is no diagram given in a polygon question.

75 - Features of a regular polygon

• All sides and angles are equal

• Each exterior angle is (360° / number of sides)

75 - What to remember when answering a polygon question.

• It is easier to work with exterior angles.

• To find the number of sides use the formula 360° / Exterior angle.

• For some questions, use the ratio of the size of the exterior angle and the interior angle if information is given

• To find out the value of the exterior angle, do 180°/ Total value of ratio.