Academic Skills Advice
Using Formulae Refresher Sheet
Substitution:
Substitution just means replacing the letters in a formula with the given values.
Examples:
Area of rectangle = 𝑙 × 𝑤 (length x width)

Find the area of a rectangle with length 5cm and width 7cm.
Area = 𝑙 × 𝑤
Area = 5 x 7
Area = 35cm2.
(replace 𝑙 with 5 and 𝑤 with 7)
Area of a circle = π r2

Find the area of a circle with radius (r) 4cm. π is approximately 3.14.
Area = π r2 (which means π x r x r).
Area = 3.14 x 4 x 4 (replace π with 3.14 and both r’s with 4).
Area = 50.24cm2.
Rearranging:
Rearranging (sometimes called transposing) a formula involves ‘reversing’ or ‘undoing’ the
formula in order to change the subject.
Examples:

Make 𝑥 the subject of the formula.
𝑦=𝑥−3
𝑦+3=𝑥
𝑥 =𝑦+3

(at the moment 𝑦 is the subject).
(add 3 to each side to leave 𝑥 on it’s own (note: -3 +3 = 0))
(now 𝑥 is the subject of the formula)
Make ‘a’ the subject of the formula.
v = u + at
v – u = at
v–u=a
t
a=v–u
t
© H Jackson 2008 / Academic Skills
(v is the subject of the formula)
(subtract ‘u’ from each side)
(divide each side by ‘t’ to make ‘a’ the subject)
(we can re-write it with ‘a’ on the LHS)
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Some Useful Formulae:
Area of triangle = ½ base x height
Circumference of circle = π d
(where d = diameter)
Area of circle = π r2
(where r = radius).
Volume of cylinder V = π r2h
(where r = radius, h = height).
Volume of sphere V = 4/3 π r3
(where r = radius).
F = ma
(where F = force, m = mass, a = acceleration).
v = u + at
(where u = initial speed, v = final speed,
a = acceleration, t = time, s = distance travelled).
v2 = u2 + 2as
S = ut + ½ at2
Kinetic Energy E = ½ mv2
The period T of a pendulum = 2𝜋 √
(where m = mass and v = speed).
𝑙
𝑔
(where l = length of pendulum
and g = acceleration due to gravity (9.8m/s2)).
Practice Questions: (Using π = 3.14)
1. Find the force when mass is 20kg and acceleration is 15m/s2.
2. Find the circumference of a circle with a diameter of 12cm.
3. Find the kinetic energy of a 7kg object travelling at 15m/s.
4. A taxi driver charges a fixed minimum price of £3 and then 40p per mile. Write the
relationship (formula) between the charge (C) and the number of miles covered (m).
5. Rearrange the formula for volume of a cylinder to make ‘r’ the subject.
6. The formula for the cost of hiring a car can be written as c = d x r, when d = number of
days and r = rate per day. Work out:
a) The cost for 5 days at £60 per day.
b) The cost for 15 days at £70 per day.
7. Find the area of a circle with a diameter of 12cm.
8. Find the volume of a sphere with a radius of 2cm.
© H Jackson 2008 / Academic Skills
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9. If the circumference of a circle is 18.84cm, what is the radius? (rearrange the formula
first.)
10. Write the formula to find the area of this worktop:
a
b
x
y
11. Find the volume of a round tower with radius 6m and height 40m.
12. Find the area of a triangle with base 8cm and height 11cm.
Challenge Questions.
13. A car starts at rest and accelerates at 7m/s2 for 5 seconds. What is its final speed?
14. How far does the car in question 13 travel?
15. A stone is dropped from a cliff 160m high. Calculate its final speed before it hits the
ground at the bottom.
16. If a person is sat on a ledge ¾ of the way up the cliff (in question 15) what speed will
the stone be travelling at as it passes the person?
17. If a stone takes 5 seconds to hit the bottom of a well – how deep is the well to the
nearest whole number.
18. A second well is tested and the stone takes 3 seconds to hit the bottom – how much
deeper is the 1st well than the 2nd?
19. Rearrange the formula for Kinetic Energy to make ‘m’ the subject.
20. Find the period of a grandfather clock with a pendulum of length 1.5m.
© H Jackson 2008 / Academic Skills
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