Year 10 Extension Mathematics – Formula Sheet
MEASUREMENT
Perimeter: Add up all side lengths
𝑃(𝑆𝑞𝑢𝑎𝑟𝑒) = 4𝑙
𝑃(𝑅𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒) = 2𝑙 + 2𝑤
𝐶𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 2𝜋𝑟
𝑃(𝑆𝑒𝑐𝑡𝑜𝑟) =
𝑜𝑟
𝜃
2𝜋𝑟 + 2𝑟
360
𝜋𝐷
𝑜𝑟
𝜃
𝜋𝐷 + 𝐷
360
Area
𝐴(𝑆𝑞𝑢𝑎𝑟𝑒) = 𝑙 2
𝐴(𝑅𝑒𝑐𝑡𝑎𝑛𝑔𝑙𝑒) = 𝑙𝑤
𝐴(𝑃𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚) = 𝑏ℎ
1
𝐴(𝑇𝑟𝑖𝑎𝑛𝑔𝑙𝑒) = 2 𝑏ℎ or
𝐴(𝑇𝑟𝑎𝑝𝑒𝑧𝑖𝑢𝑚) =
𝑏ℎ
2
OR Using Heron’s Formula: 𝐴 = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) where 𝑠 = 𝑠𝑒𝑚𝑖𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟
1
(𝑎 + 𝑏)ℎ
2
𝑜𝑟
(𝑎 + 𝑏)ℎ
2
𝐴(𝐶𝑖𝑟𝑐𝑙𝑒) = 𝜋𝑟 2
𝐴(𝑆𝑒𝑐𝑡𝑜𝑟) =
𝜃
𝜋𝑟 2
360
Total Surface Area: Find the area of all the faces and add them together.
𝑇𝑆𝐴(𝐶𝑢𝑏𝑒) = 6𝑙 2
𝑇𝑆𝐴(𝑅𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑃𝑟𝑖𝑠𝑚) = 2𝑙𝑤 + 2𝑙ℎ + 2𝑤ℎ
𝑜𝑟
2(𝑙𝑤 + 𝑙ℎ + 𝑤ℎ)
𝑇𝑆𝐴(𝐶𝑦𝑙𝑖𝑛𝑑𝑒𝑟) = 2𝜋𝑟 2 + 2𝜋𝑟ℎ
𝑇𝑆𝐴(𝐶𝑜𝑛𝑒) = 𝜋𝑟 2 + 𝜋𝑟𝑠
𝑇𝑆𝐴(𝑆𝑝ℎ𝑒𝑟𝑒) = 4𝜋𝑟 2
Volume:
Volume of Prisms (and cylinder): Area of the base times the height
𝑉(𝑐𝑢𝑏𝑒) = 𝑙 3
𝑉(𝑅𝑒𝑐𝑡𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑃𝑟𝑖𝑠𝑚) = 𝑙𝑤ℎ
1
𝑉(𝑇𝑟𝑖𝑎𝑔𝑛𝑢𝑙𝑎𝑟 𝑃𝑟𝑖𝑠𝑚) = ( 𝑏ℎ) 𝐻
2
𝑉(𝐶𝑦𝑙𝑖𝑛𝑑𝑒𝑟) = 𝜋𝑟 2 ℎ
1
𝑉(𝑃𝑦𝑟𝑎𝑚𝑖𝑑) = 𝐴𝑜𝑓𝐵 × ℎ
3
𝑉(𝐶𝑜𝑛𝑒) =
1 2
𝜋𝑟 ℎ
3
4
𝑉(𝑆𝑝ℎ𝑒𝑟𝑒) = 𝜋𝑟 3
3
Trigonometry
sin 𝜃 =
𝑂
𝐻
cos 𝜃 =
𝐴
𝐻
tan 𝜃 =
𝑂
𝐴
𝑆𝑖𝑛𝑒 𝑅𝑢𝑙𝑒:
1
𝑎
𝑏
=
𝑆𝑖𝑛𝐴
𝑆𝑖𝑛𝐵
OR
𝑆𝑖𝑛𝐴
𝑆𝑖𝑛𝐵
=
𝑎
𝑏
𝐶𝑜𝑠𝑖𝑛𝑒 𝑅𝑢𝑙𝑒: 𝑐 2 = 𝑎2 + 𝑏 2 − 2𝑎𝑏 cos 𝐶
𝐴𝑟𝑒𝑎 = 2 𝑎𝑏𝑆𝑖𝑛𝐶
Exact Value Triangles:
Unit Circle:
𝑃 (𝑥, 𝑦) = (cos 𝜃, sin 𝜃)
√2
√3
60o
45o
𝜃
NUMBER AND ALGEBRA
Finance
Simple Interest:
𝑆𝐼 = 𝑃𝑟𝑡
Compound Interest:
𝐴 = 𝑃(1 + 𝑟)𝑡
compounded annually
𝑟
𝑛
𝐴 = 𝑃(1 + )𝑛𝑡
Growth/Decay:
compounded more frequently than annually
𝑉 = 𝑃0 × (1 ± 𝑟)𝑛
Index Laws
𝑎𝑚 × 𝑎𝑛 = 𝑎𝑚+𝑛
𝑎𝑚 ÷ 𝑎𝑛 = 𝑎𝑚−𝑛
(𝑎𝑚 )𝑛 = 𝑎𝑚×𝑛
𝑎−𝑚 = 𝑎𝑚
1
𝑎0 = 1
𝑚
𝑛
( √𝑎)𝑛 = 𝑎𝑚
Logarithms
log 𝑎 𝑎 = 1
log 𝑎 𝑝 + log 𝑎 𝑞 = log 𝑎 𝑝𝑞
log 𝑎 𝑝 − log 𝑎 𝑞 = log 𝑎
𝑁 log 𝑎 𝑝 = log 𝑎 𝑝𝑁
𝑝
𝑞
Linear Algebra
𝑦 = 𝑚𝑥 + 𝑐
𝑚=
𝐷𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = √(𝑦2 − 𝑦1 )2 + (𝑥2 − 𝑥1 )2
(𝑦2 − 𝑦1 )
(𝑥2 − 𝑥1 )
𝑥 + 𝑥1 𝑦2 + 𝑦1
,
)
2
2
𝑀𝑖𝑑𝑝𝑜𝑖𝑛𝑡 = (𝑥̅ , 𝑦̅) OR ( 2
Quadratic Algebra
General Form: 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
𝑦 = (𝑥 ± 𝑎)(𝑥 ± 𝑏)
Root Form:
Turning Point Form:
𝑥=
Quadratic Equation:
𝑦 = (𝑥 ± ℎ)2 ± 𝑣
Discriminant = 𝑏 2 − 4𝑎𝑐
−𝑏
Turning Point: 𝑥 = 2𝑎
Exponential Relationships
𝑦 = 𝑘𝑎 𝑥
where 𝑘 = 𝑦 − intercept and 𝑎 = growth rate
Hyperbolas
𝑦=
1
𝑥
Circle Graphs
𝑦 = (𝑥 ± ℎ)2 + (𝑦 ± 𝑣)2 = 𝑟 2
Polynomials
Remainder Theorem: When 𝑃(𝑥) is divided by (𝑥 − 𝑎) the remainder is 𝑃(𝑎)
Factor Theorem: When 𝑃(𝑥) is divided by (𝑥 − 𝑎) the remainder of 𝑃(𝑎) = 0
STATISTICS AND PROBABILITY
Statistics
𝑚𝑒𝑎𝑛 𝑥 =
∑𝑥
𝑛
𝑟𝑎𝑛𝑔𝑒 = max −𝑚𝑖𝑛
𝑛+1
𝐼𝑄𝑅 = 𝑄3 − 𝑄1
2
𝑂𝑢𝑡𝑙𝑖𝑒𝑟 = 𝑄1 − (1.5 × 𝐼𝑄𝑅) < 𝑥 < 𝑄3 + (1.5 × 𝐼𝑄𝑅)
𝑚𝑒𝑑𝑖𝑎𝑛 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 =
Probability
∪ = Union = ‘Or’
∩ = Intersection = ‘And’
Addition Rule: Pr(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
Conditional Probability: P(𝐴 | 𝐵) =
Independent Events:
𝑃(𝐴 | 𝐵) = 𝑃(𝐴)
𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) × 𝑃(𝐵)
𝑃(𝐴 ∩ 𝐵)
𝑃(𝐵)
or P(𝐵 | 𝐴) =
𝑃(𝐵 ∩ 𝐴)
𝑃(𝐴)
−𝑏±√𝑏2 −4𝑎𝑐
2𝑎