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Useful Formulas

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What is a problem? A problem = a fact + a judgment. That is a simple formula that tells us something about the way the world works. Maths is full of formulas, and that can intimidate some people if they don’t understand them or can’t remember the right one to use.

However, formulas should be our friends as they help us to do complex calculations accurately and repeatably in a consistent and straightforward way. The following is a list of the most useful ones I’ve come across while teaching Maths to a variety of students at a variety of ages and at a variety of stages in their education.

Averages

  • The mean is found by adding up all the values and dividing the total by how many there are, eg the mean of the numbers 1-10 is 5.5, as 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55, and 55 ÷ 10 = 5.5.
  • The mode is the most common value (or values), eg the mode of 1, 2, 2, 3, 4, 5 is 2.
  • The median of an odd number of values sorted by size is the one in the middle, eg the median of the numbers 1-5 is 3. The median of an even number of values is the mean of the two numbers in the middle, eg the median of the numbers 1-10 is 5.5, as 5 and 6 are the numbers in the middle, and 11 ÷ 2 = 5.5.
  • The range is the highest value minus the lowest, eg the range of the numbers 1-10 = 10 – 1 = 9.

Sample Questions

  1. What is the mean of the numbers 6, 5, 8, 2 and 4?
  2. What is the mode of the numbers 1, 5, 5, 12 and 3?
  3. What is the median of the numbers 2, 8, 9, 6 and 5?
  4. What is the median of the numbers 1, 13, 4, 6, 8 and 20?
  5. What is the range of the numbers 15, 2, 3, 8 and 4?

Circle Theorems

  • Alternate segment theorem
    The angle between a tangent and a chord is equal to the angle subtended by the same chord in the alternate segment.
  • Angles at the centre and at the circumference
    The angle formed at the centre of the circle by lines starting from two points on the circumference is double the angle formed on the other side of the circle by lines originating from the same points.
  • Angles in the same segment
    If you create two triangles between the ends of the chord and the side of the circle, the angles in the same segment are equal.
  • Angles in a semicircle
    If you draw a triangle using the diameter and any point on the circumference, the angle at that point is 90°.
  • Perpendicular chord bisection
    The perpendicular from the centre of a circle to a chord will always bisect the chord (divide it into two equal parts).
  • Tangent of a circle
    a) The angle between a tangent and radius is 90°.
    b) Tangents that meet at the same point outside the circle are equal in length.
  • Cyclic quadrilateral
    The opposite angles in a cyclic quadrilateral add up to 180°.
  • Angles subtended on the same arc
    Angles formed from two points on the circumference of a circle are equal to other angles, in the same arc, formed from those two points.
  • Angle in a semi-circle
    Angles created by drawing lines from the ends of the diameter of a circle to the circumference form a right angle.

Geometry

  • Angles around a point add up to 360º
  • Angles on a straight line add up to 180º
  • Vertically opposite angles are equal, ie the two pairs of angles opposite each other when two straight lines bisect (or cross) each other
  • Alternate angles are equal, ie the angles under the arms of a ‘Z’ formed by a line (or ‘transversal’) bisecting two parallel lines
  • Corresponding angles are equal, ie the angles under the arms of an ‘F’ formed by a line (or ‘transversal’) bisecting two parallel lines
  • Complementary angles add up to 90º
  • Any straight line can be drawn using y = mx + c, where m is the gradient and c is the point where the line crosses the y-axis (the ‘y-intercept’)
  • The gradient of a straight line is shown by δy/δx (ie the difference in the y-values divided by the difference in the x-values of any two points on the line, usually found by drawing a triangle underneath it)

Polygons

  • Number of diagonals in a polygon = (n-3)(n÷2) where n is the number of sides
  • The sum of the internal angles of a polygon = (n-2)180º, where n is the number of sides
  • Any internal angle of a regular polygon = (n-2)180º ÷ n, where n is the number of sides

Sample Questions

  1. How many diagonals are there in a square?
  2. How many diagonals are there in an octagon?
  3. What is the sum of the internal angles of a pentagon?
  4. What is the sum of the internal angles of a hexagon?
  5. What are the internal angles of a regular decagon?
  6. What are the internal angles of a regular heptagon?

Probability

  • The probability of something happening = the number of ways of getting what you want divided by the number of possible outcomes, eg the chance of getting heads if you flip a coin is 1/2 (or 0.5 or 50%).
  • The probability of a sequence of events = the probability of all the events multiplied together, eg the chance of getting two heads in a row if you flip a coin is 1/2 x 1/2 = 1/4 (or 0.25 or 25%).

Sample Questions

  1. What is the probability of rolling a six?
  2. If there are two red and three black balls in a bag, what is the probability of taking out a red one?
  3. What is the probability of rolling a number greater than 3?
  4. What is the probability of tossing three heads in a row?
  5. If the chance of hitting treble 20 is 1/10, what is the probability of scoring 180 with three darts?

Rectangles

  • Perimeter of a rectangle = 2(l + w), where l = length and w = width
    Note that this is the same formula for the perimeter of an L-shape, too.
  • Area of a rectangle = lw, where l = length and w = width

Sample Questions

  1. What is the perimeter of a rectangle measuring 5 x 7 cm?
  2. What is the perimeter of a rectangle measuring 13 x 4 cm?
  3. What is the perimeter of an L-shape measuring 12 x 8 m overall with a 4 x 2 m piece missing?
  4. What is the area of a rectangle measuring 3 x 12 cm?
  5. What is the area of a rectangle measuring 6 x 8 cm?

Trapeziums

  • Area of a trapezium = lw, where l = average length and w = width (in other words, you have to add both lengths together and divide by two to find the average length)

Sample Questions

  1. What is the area of a trapezium with a height of 7 cm and horizontal sides of 4 and 8 cm?
  2. What is the area of a trapezium with a height of 10 inches and horizontal sides of 5 and 4 inches?
  3. What is the area of a trapezium with a height of 5 cm and horizontal sides of 2 and 4 cm?
  4. What is the area of a trapezium with a height of 12 m and horizontal sides of 6 and 9 m?
  5. What is the area of a trapezium with a height of 10 cm and horizontal sides of 4 and 5 cm?

Triangles (Trigonometry)

  • Area of a triangle = ½bh, where b = base and h = height
  • Angles in a triangle add up to 180º
  • Pythagoras’s Theorem: in a right-angled triangle, a² + b² = c², ie the area of a square on the hypotenuse (or longest side) is equal to the sum of the areas of squares on the other two sides

 

 

 

 

 

 

Circles

  • Circumference of a circle = 2πr, where r = radius
  • Area of a circle = πr², where r = radius (π = 3.14 to two decimal places and is sometimes given as 22/7)

Sample Questions

  1. What is the area of a triangle with a height of 5 cm and a base of 3 cm?
  2. What is the remaining internal angle of a triangle if the others are 20º and 30º?
  3. What is the length of the remaining side of a right-angled triangle if the others are 4 cm and 3 cm?
  4. What is the circumference of a circle with a radius of 5cm, assuming π = 3.14?
  5. What is the area of a circle with a diameter of 4 m, assuming π = 22/7?

Spheres

  • Surface area of a sphere = 4πr², where r = radius
  • Volume of a sphere = 4/3πr³, where r = radius

Sample Questions

  1. What is the surface area of a sphere with a radius of 15 cm, assuming π = 3.14?
  2. What is the surface area of a sphere with a diameter of 7 cm, assuming π = 3.14?
  3. What is the volume of a sphere with a radius of 1 cm, assuming π = 22/7?
  4. What is the volume of a sphere with a radius of 5 cm, assuming π = 3.14?
  5. What is the volume of a sphere with a diameter of 5 m, assuming π = 22/7?

Cuboids

  • Surface area of a cuboid = 2(lw + lh + wh), where l is length, w is width and h is height
  • Volume of a cuboid = lwh, where l is length, w is width and h is height

Sample Questions

  1. What is the surface area of a cuboid measuring 3 x 4 x 5 cm?
  2. What is the surface area of a cuboid measuring 6 x 2 x 12 cm?
  3. What is the volume of a cuboid measuring 8 x 5 x 6 cm
  4. What is the volume of a cuboid measuring 2 x 15 x 7 cm
  5. What is the volume of a cuboid measuring 18 x 9 x 2 cm

Number Sequences

  • The nth term of any arithmetic sequence (with regular intervals) = xn ± k, where x is the interval (or difference) between the values, n is the value’s place in the sequence and k is a constant that is added or subtracted to make sure the sequence starts at the right number, eg the formula for 5, 8, 11, 14…etc is 3n + 2
    • If you know where the term is in the sequence, you can find it by plugging in n and solving the equation, eg the fourth term of the sequence above = 3n + 2 = 3 x 4 + 2 = 14
    • If you know what the term is, you can find its position in the sequence by adding the number as the result of the formula and solving for n, eg if the term in the sequence above = 14, then 3n + 2 = 14, 3n = 12 and n = 4.
  • The sum of n consecutive numbers is n(n + 1)/2, eg the sum of the numbers 1-10 is 10(10 + 1)/2 = 110/2 = 55

Sample Questions

  1. What is the formula for the nth term of 2, 4, 6, 8…?
  2. What is the formula for the nth term of -5, -3, -1, 1…?
  3. What is the seventh term of the sequence 4, 8, 12, 16…?
  4. What position is 66 in the sequence 3, 6, 9, 12…?
  5. What is the sum of the numbers 12 to 18?

Other

  • Speed = distance ÷ time
  • Time = distance ÷ speed
  • Distance = speed x time
  • Profit = sales – cost of goods sold
  • Profit margin = profit ÷ sales
  • Mark-up = profit ÷ cost of goods sold

Sample Questions

  1. What is the speed of a vehicle that travels 20 miles in two hours?
  2. How long does it take to travel 210 km at 70 km/h?
  3. What profit does a butcher make on three steaks he buys at £2 and sells for £2.50?
  4. What’s his profit margin on those steaks?
  5. What’s his mark-up on those steaks? 

     

     

     

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