Neel’s Math Review

N E E L E N D R A N A T H 1

CONTENT

Cuboid

Cube

Cylinder

Cone

Pyramid

Sphere

N E E L E N D R A N A T H 2

𝐹𝑎𝑐𝑒 𝐴𝑟𝑒𝑎𝑠, 𝑥 = 𝑙 × 𝑏; 𝑦 = 𝑏 × ℎ; 𝑧 = 𝑙 × ℎ

𝐿𝑆𝐴 = 2 𝑙 + 𝑏 ℎ

𝑇𝑆𝐴 = 2 𝑥 + 𝑦 + 𝑧 = 2 𝑙𝑏 + 𝑏ℎ + 𝑙ℎ

𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑙 × 𝑏 × ℎ = 𝑥𝑦𝑧 = 𝑥ℎ = 𝑦𝑙 = 𝑧𝑏

𝐵𝑜𝑑𝑦 𝐷𝑖𝑎𝑔𝑜𝑛𝑎𝑙 = 𝑙2 + 𝑏2 + ℎ2

N E E L E N D R A N A T H 3

CUBOID

LSA – Lateral Surface Area (Area of four walls)

TSA – Total Surface Area = LSA + Top & Bottom Surfaces

𝐹𝑎𝑐𝑒 𝐴𝑟𝑒𝑎𝑠, 𝑥 = 𝑎2; 𝑦 = 𝑎2; 𝑧 = 𝑎2

𝐿𝑆𝐴 = 4𝑎2

𝑇𝑆𝐴 = 6𝑎2

𝑉𝑜𝑙𝑢𝑚𝑒 = 𝑎3

𝐵𝑜𝑑𝑦 𝐷𝑖𝑎𝑔𝑜𝑛𝑎𝑙 = 3𝑎

N E E L E N D R A N A T H 4

CUBE

LSA – Lateral Surface Area (Area of four walls)

TSA – Total Surface Area = LSA + Top & Bottom Surfaces

𝐹𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 𝜋𝑟2

𝐶𝑆𝐴 = 2𝜋𝑟ℎ

𝑇𝑆𝐴 = 2𝜋𝑟(𝑟 + ℎ)

𝑉𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟2ℎ

N E E L E N D R A N A T H 5

CYLINDER

CSA – Curved Surface Area

TSA – Total Surface Area = CSA + Top & Bottom Surfaces

𝐹𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 𝜋𝑟2

𝑆𝑙𝑎𝑛𝑡 𝐻𝑒𝑖𝑔ℎ𝑡, 𝑙 = 𝑟2 + ℎ2

𝐶𝑆𝐴 = 𝜋𝑟𝑙

𝑇𝑆𝐴 = 𝜋𝑟(𝑙 + 𝑟)

𝑉𝑜𝑙𝑢𝑚𝑒 =1

3𝜋𝑟2ℎ

N E E L E N D R A N A T H 6

CONE

CSA – Curved Surface Area

TSA – Total Surface Area = CSA + Bottom Surface

𝐹𝑎𝑐𝑒 𝐴𝑟𝑒𝑎𝑠, 𝑡𝑜𝑝 = 𝜋𝑟2 𝑏𝑜𝑡𝑡𝑜𝑚 = 𝜋𝑅2

𝑆𝑙𝑎𝑛𝑡 𝐻𝑒𝑖𝑔ℎ𝑡, 𝑙 = (𝑅 − 𝑟)2+ℎ2

𝐶𝑆𝐴 = 𝜋(𝑅 + 𝑟)𝑙

𝑇𝑆𝐴 = 𝜋𝑙 𝑅 + 𝑟 + 𝜋𝑟2 + 𝜋𝑅2

𝑉𝑜𝑙𝑢𝑚𝑒 =1

3𝜋ℎ(𝑅2 + 𝑟2 + 𝑅𝑟)

𝐻𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑂𝑟𝑖𝑔𝑖𝑛𝑎𝑙 𝐶𝑜𝑛𝑒, 𝐻 =𝑅ℎ

𝑅−𝑟

N E E L E N D R A N A T H 7

CONE - FRUSTUM

CSA – Curved Surface Area

TSA – Total Surface Area = CSA + Top & Bottom Surface

𝐿𝑆𝐴 =1

2× 𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 𝑜𝑓 𝑏𝑎𝑠𝑒 × 𝑆𝑙𝑎𝑛𝑡 𝐻𝑒𝑖𝑔ℎ𝑡

𝑇𝑆𝐴 = 𝐿𝑆𝐴 + 𝐴𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒

𝑉𝑜𝑙𝑢𝑚𝑒 =1

3𝐴𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡

N E E L E N D R A N A T H 8

PYRAMID

LSA – Lateral Surface Area

TSA – Total Surface Area = CSA + Bottom Surface

*Base area will depend upon type of polygon at the base

𝐿𝑆𝐴 =1

2× (𝑃1 + 𝑃2) × 𝑆𝑙𝑎𝑛𝑡 𝐻𝑒𝑖𝑔ℎ𝑡

𝑇𝑆𝐴 = 𝐿𝑆𝐴 + 𝐴1 + 𝐴2

𝑉𝑜𝑙𝑢𝑚𝑒 =1

3× ℎ(𝐴1 + 𝐴2 + 𝐴1𝐴2

N E E L E N D R A N A T H 9

PYRAMID – FRUSTUM

P1 & P2 = Perimeter of Top & Bottom Surfaces

A1 & A2 = Area of Top & Bottom Surfaces

LSA – Lateral Surface Area

TSA – Total Surface Area = CSA + Bottom Surface

*Top & bottom areas will depend upon type of polygon at the base

Sphere

𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐴𝑟𝑒𝑎 = 𝐶𝑆𝐴 = 𝑇𝑆𝐴 = 4𝜋𝑟2

𝑉𝑜𝑙𝑢𝑚𝑒 =4

3𝜋𝑟3

Hemisphere

𝐶𝑆𝐴 = 2𝜋𝑟2

𝑇𝑆𝐴 = 2𝜋𝑟2 + 𝜋𝑟2 = 3𝜋𝑟2

𝑉𝑜𝑙𝑢𝑚𝑒 =2

3𝜋𝑟3

N E E L E N D R A N A T H 10

SPHERE & HEMISPHERE

CSA – Curved Surface Area

TSA – Total Surface Area = CSA + Flat Surface