Common geometric formulae
1-D (perimeter)
● Circle {circumference}: 2π × 𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠)
2-D (area)
2
𝑙 (𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑) × 𝑛 (𝑛𝑜. 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠)
● General formula for regular polygons: 180˚
4 × 𝑡𝑎𝑛 ( 𝑛
)
3-sided
1
● Triangle, with 2 known ⟂ lengths: 2
× 𝑏 (𝑏𝑎𝑠𝑒) × ℎ (ℎ𝑒𝑖𝑔ℎ𝑡)
1
● Triangle, with 1 known angle & without known ⟂ lengths: 2
× 𝑎 × 𝑏 × 𝑠𝑖𝑛 𝐶
● Triangle, with 3 known side lengths & without known angles:
𝑎+𝑏+𝑐
𝑠 (𝑠𝑒𝑚𝑖 𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟) × (𝑠 − 𝑎) × (𝑠 − 𝑏) × (𝑠 − 𝑐) where 𝑠 = 2
4-sided
2
● Square: 𝑠 (𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑)
● Rectangle: 𝑏 (𝑏𝑎𝑠𝑒) × ℎ (ℎ𝑒𝑖𝑔ℎ𝑡)
● Parallelogram: 𝑏 (𝑏𝑎𝑠𝑒) × ℎ (ℎ𝑒𝑖𝑔ℎ𝑡)
(𝑎 [𝑡𝑜𝑝 𝑙𝑒𝑛𝑔𝑡ℎ] + 𝑏 [𝑏𝑎𝑠𝑒 𝑙𝑒𝑛𝑔𝑡ℎ]) × ℎ (ℎ𝑒𝑖𝑔ℎ𝑡)
● Trapezium, with 2 known ∥ lengths: 2
● Trapezium, with 2 known opposite angles & without any ∥ lengths:
2 θ
(𝑠 [𝑠𝑒𝑚𝑖 𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟] − 𝑎) × (𝑠 − 𝑏) × (𝑠 − 𝑐) × (𝑠 − 𝑑) − 𝑎𝑏𝑐𝑑 × 𝑐𝑜𝑠 2
𝑎+𝑏+𝑐+𝑑
where 𝑠 = 2
, θ = θ1 + θ2
1
● Rhombus: 2
× 𝑑1 (𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 1) × 𝑑2 (𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 2)
1
● Kite: 2
× 𝑑1 (𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 1) × 𝑑2 (𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 2)
Circular
2
● Circle: π × 𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑)
● Ellipse: π × 𝑎 (𝑟𝑎𝑑𝑖𝑢𝑠, 𝑎𝑡 𝑔𝑟𝑒𝑎𝑡𝑒𝑠𝑡) × 𝑏 (𝑟𝑎𝑑𝑖𝑢𝑠, 𝑎𝑡 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡)
3-D (surface area)
4-sided (as base)
2
● Cube: 6𝑠 (𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑)
, ● Cuboid: 2 × (𝑙𝑤 + 𝑤ℎ + 𝑙ℎ) where 𝑙 = 𝑙𝑒𝑛𝑔𝑡ℎ, 𝑤 = 𝑤𝑖𝑑𝑡ℎ, ℎ = ℎ𝑒𝑖𝑔ℎ𝑡
Circular (as base)
● Cylinder: 2π𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑏𝑎𝑠𝑒) × (𝑟 + ℎ [ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟])
● Cone: π𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑏𝑎𝑠𝑒) × (𝑟 + 𝑙 [𝑠𝑙𝑎𝑛𝑡 ℎ𝑒𝑖𝑔ℎ𝑡])
2
● Sphere: 4π𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑)
2
● Hemisphere: 3π𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑏𝑎𝑠𝑒 ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑)
3-D (volume)
● General formula for pyramids with regular-sided bases:
2 180˚
𝑛 (𝑛𝑜. 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠) × ℎ (ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑝𝑦𝑟𝑎𝑚𝑖𝑑) × 𝑙 (𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑) × 𝑐𝑜𝑡 ( 𝑛
)
12
4-sided (as base)
3
● Cube: 𝑠 (𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ ~ 𝑐𝑢𝑏𝑒𝑑)
● Cuboid: 𝑙 (𝑙𝑒𝑛𝑔𝑡ℎ) × 𝑤 (𝑤𝑖𝑑𝑡ℎ) × ℎ (ℎ𝑒𝑖𝑔ℎ𝑡)
ℎ 2 2
● Pyramidal square frustum: 3
(𝑎 + 𝑎𝑏 + 𝑏 )
where ℎ = ℎ𝑒𝑖𝑔ℎ𝑡, 𝑎 = 𝑏𝑎𝑠𝑒 𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ, 𝑏 = 𝑡𝑜𝑝 𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ
Circular (as base)
2
● Cylinder: π × 𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑏𝑎𝑠𝑒 ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑) × ℎ (ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟)
1 2
● Cone: 3
π × 𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑏𝑎𝑠𝑒 ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑) × ℎ (ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑐𝑜𝑛𝑒)
4 3
● Sphere: 3
π × 𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 ~ 𝑐𝑢𝑏𝑒𝑑)
2 3
● Hemisphere: 3
π × 𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 ~ 𝑐𝑢𝑏𝑒𝑑)
ℎ1𝐵1−ℎ2𝐵2
● Conical frustum: 3
where ℎ1 = ⟂ 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑎𝑝𝑒𝑥 𝑡𝑜 𝑏𝑎𝑠𝑒 𝑝𝑙𝑎𝑛𝑒, 𝐵1 = 𝑏𝑎𝑠𝑒 𝑎𝑟𝑒𝑎
ℎ2 = ⟂ 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑎𝑝𝑒𝑥 𝑡𝑜 𝑡𝑜𝑝 𝑝𝑙𝑎𝑛𝑒, 𝐵2 = 𝑡𝑜𝑝 𝑎𝑟𝑒𝑎
Probability
For 2-set Venn diagrams
● 𝑃(𝐴') = 1 − 𝑃(𝐴)
● 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
● 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) × 𝑃(𝐵|𝐴)
𝑃(𝐵|𝐴)×𝑃(𝐴)
● 𝑃(𝐴|𝐵) = 𝑃(𝐵|𝐴)×𝑃(𝐴)+𝑃(𝐵|𝐴')×𝑃(𝐴')
● For independent events A & B:
○ 𝑃(𝐴|𝐵) = 𝑃(𝐴)
1-D (perimeter)
● Circle {circumference}: 2π × 𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠)
2-D (area)
2
𝑙 (𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑) × 𝑛 (𝑛𝑜. 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠)
● General formula for regular polygons: 180˚
4 × 𝑡𝑎𝑛 ( 𝑛
)
3-sided
1
● Triangle, with 2 known ⟂ lengths: 2
× 𝑏 (𝑏𝑎𝑠𝑒) × ℎ (ℎ𝑒𝑖𝑔ℎ𝑡)
1
● Triangle, with 1 known angle & without known ⟂ lengths: 2
× 𝑎 × 𝑏 × 𝑠𝑖𝑛 𝐶
● Triangle, with 3 known side lengths & without known angles:
𝑎+𝑏+𝑐
𝑠 (𝑠𝑒𝑚𝑖 𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟) × (𝑠 − 𝑎) × (𝑠 − 𝑏) × (𝑠 − 𝑐) where 𝑠 = 2
4-sided
2
● Square: 𝑠 (𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑)
● Rectangle: 𝑏 (𝑏𝑎𝑠𝑒) × ℎ (ℎ𝑒𝑖𝑔ℎ𝑡)
● Parallelogram: 𝑏 (𝑏𝑎𝑠𝑒) × ℎ (ℎ𝑒𝑖𝑔ℎ𝑡)
(𝑎 [𝑡𝑜𝑝 𝑙𝑒𝑛𝑔𝑡ℎ] + 𝑏 [𝑏𝑎𝑠𝑒 𝑙𝑒𝑛𝑔𝑡ℎ]) × ℎ (ℎ𝑒𝑖𝑔ℎ𝑡)
● Trapezium, with 2 known ∥ lengths: 2
● Trapezium, with 2 known opposite angles & without any ∥ lengths:
2 θ
(𝑠 [𝑠𝑒𝑚𝑖 𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟] − 𝑎) × (𝑠 − 𝑏) × (𝑠 − 𝑐) × (𝑠 − 𝑑) − 𝑎𝑏𝑐𝑑 × 𝑐𝑜𝑠 2
𝑎+𝑏+𝑐+𝑑
where 𝑠 = 2
, θ = θ1 + θ2
1
● Rhombus: 2
× 𝑑1 (𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 1) × 𝑑2 (𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 2)
1
● Kite: 2
× 𝑑1 (𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 1) × 𝑑2 (𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑙𝑒𝑛𝑔𝑡ℎ 2)
Circular
2
● Circle: π × 𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑)
● Ellipse: π × 𝑎 (𝑟𝑎𝑑𝑖𝑢𝑠, 𝑎𝑡 𝑔𝑟𝑒𝑎𝑡𝑒𝑠𝑡) × 𝑏 (𝑟𝑎𝑑𝑖𝑢𝑠, 𝑎𝑡 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡)
3-D (surface area)
4-sided (as base)
2
● Cube: 6𝑠 (𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑)
, ● Cuboid: 2 × (𝑙𝑤 + 𝑤ℎ + 𝑙ℎ) where 𝑙 = 𝑙𝑒𝑛𝑔𝑡ℎ, 𝑤 = 𝑤𝑖𝑑𝑡ℎ, ℎ = ℎ𝑒𝑖𝑔ℎ𝑡
Circular (as base)
● Cylinder: 2π𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑏𝑎𝑠𝑒) × (𝑟 + ℎ [ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟])
● Cone: π𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑏𝑎𝑠𝑒) × (𝑟 + 𝑙 [𝑠𝑙𝑎𝑛𝑡 ℎ𝑒𝑖𝑔ℎ𝑡])
2
● Sphere: 4π𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑)
2
● Hemisphere: 3π𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑏𝑎𝑠𝑒 ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑)
3-D (volume)
● General formula for pyramids with regular-sided bases:
2 180˚
𝑛 (𝑛𝑜. 𝑜𝑓 𝑠𝑖𝑑𝑒𝑠) × ℎ (ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑝𝑦𝑟𝑎𝑚𝑖𝑑) × 𝑙 (𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑) × 𝑐𝑜𝑡 ( 𝑛
)
12
4-sided (as base)
3
● Cube: 𝑠 (𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ ~ 𝑐𝑢𝑏𝑒𝑑)
● Cuboid: 𝑙 (𝑙𝑒𝑛𝑔𝑡ℎ) × 𝑤 (𝑤𝑖𝑑𝑡ℎ) × ℎ (ℎ𝑒𝑖𝑔ℎ𝑡)
ℎ 2 2
● Pyramidal square frustum: 3
(𝑎 + 𝑎𝑏 + 𝑏 )
where ℎ = ℎ𝑒𝑖𝑔ℎ𝑡, 𝑎 = 𝑏𝑎𝑠𝑒 𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ, 𝑏 = 𝑡𝑜𝑝 𝑠𝑖𝑑𝑒 𝑙𝑒𝑛𝑔𝑡ℎ
Circular (as base)
2
● Cylinder: π × 𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑏𝑎𝑠𝑒 ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑) × ℎ (ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟)
1 2
● Cone: 3
π × 𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑏𝑎𝑠𝑒 ~ 𝑠𝑞𝑢𝑎𝑟𝑒𝑑) × ℎ (ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑐𝑜𝑛𝑒)
4 3
● Sphere: 3
π × 𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 ~ 𝑐𝑢𝑏𝑒𝑑)
2 3
● Hemisphere: 3
π × 𝑟 (𝑟𝑎𝑑𝑖𝑢𝑠 ~ 𝑐𝑢𝑏𝑒𝑑)
ℎ1𝐵1−ℎ2𝐵2
● Conical frustum: 3
where ℎ1 = ⟂ 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑎𝑝𝑒𝑥 𝑡𝑜 𝑏𝑎𝑠𝑒 𝑝𝑙𝑎𝑛𝑒, 𝐵1 = 𝑏𝑎𝑠𝑒 𝑎𝑟𝑒𝑎
ℎ2 = ⟂ 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑎𝑝𝑒𝑥 𝑡𝑜 𝑡𝑜𝑝 𝑝𝑙𝑎𝑛𝑒, 𝐵2 = 𝑡𝑜𝑝 𝑎𝑟𝑒𝑎
Probability
For 2-set Venn diagrams
● 𝑃(𝐴') = 1 − 𝑃(𝐴)
● 𝑃(𝐴 ∪ 𝐵) = 𝑃(𝐴) + 𝑃(𝐵) − 𝑃(𝐴 ∩ 𝐵)
● 𝑃(𝐴 ∩ 𝐵) = 𝑃(𝐴) × 𝑃(𝐵|𝐴)
𝑃(𝐵|𝐴)×𝑃(𝐴)
● 𝑃(𝐴|𝐵) = 𝑃(𝐵|𝐴)×𝑃(𝐴)+𝑃(𝐵|𝐴')×𝑃(𝐴')
● For independent events A & B:
○ 𝑃(𝐴|𝐵) = 𝑃(𝐴)
