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Mensuration

The document provides formulas for calculating the area, perimeter, and diagonal of various geometric shapes including squares, rectangles, triangles, circles, and three-dimensional figures like cubes and cylinders. Each shape is detailed with its respective mathematical expressions for these properties. Additionally, it includes specific cases for different types of triangles and paths within rectangles.

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ajithselvaraj32
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5 views3 pages

Mensuration

The document provides formulas for calculating the area, perimeter, and diagonal of various geometric shapes including squares, rectangles, triangles, circles, and three-dimensional figures like cubes and cylinders. Each shape is detailed with its respective mathematical expressions for these properties. Additionally, it includes specific cases for different types of triangles and paths within rectangles.

Uploaded by

ajithselvaraj32
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Mensuration

Figure Area Perimeter Diagonal

Square a2 4a √2𝑎

Rectangle 𝑙 ×𝑏 2 ( (𝑙 × 𝑏 ) √𝑙 2 × 𝑏 2

Rectangle of 2𝑥(𝑙+𝑏−2𝑥) 4 ( 𝑙 + 𝑏 − 2𝑥) -


path inside

Rectangle of 2𝑥 ( 𝑙 + 𝑏 + 2𝑥) 4 ( 𝑙 + 𝑏 + 2𝑥) -


path outside

Rectangle of 𝑥(𝑙+𝑏−𝑥) 2 ( 𝑙 + 𝑏 − 2𝑥) -


path midway

𝒍𝒆𝒏𝒈𝒉𝒕 𝒐𝒇 𝒐𝒏𝒆 𝒅𝒊𝒂𝒈𝒐𝒏𝒂𝒍 𝒂𝒓𝒆𝒂 = 𝟐√𝒔(𝒔 − 𝒂 )(𝒔 − 𝒃)(𝒔 − 𝒅)

Parallelogram
𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡 2 (𝑎 + 𝑏 ) 𝑑12 + 𝑑 22 = 2 ( 𝑎2 + 𝑏2 )

Rhombus 1 4𝑎 (𝑜𝑟) 4𝑎2 = 𝑑12 + 𝑑22


𝑑 𝑑
2 1 2
Trapezium 1 𝑎+𝑏+𝑐+𝑑 𝑑12 + 𝑑22 = 𝑐 2 + 𝑑2 + 2𝑎𝑏
(𝑎+𝑏)×ℎ
2

Triangle 1 𝑠𝑒𝑚𝑖 𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 𝒊𝒏𝒄𝒊𝒓𝒄𝒍𝒆 𝑐𝑖𝑟𝑐𝑢𝑚𝑟𝑎𝑑𝑖𝑢𝑠


𝑏𝑎𝑠𝑒 𝑎+𝑏+𝑐 𝑎𝑟𝑒𝑎 𝑜𝑓 ∆ 𝑎𝑏𝑐
2
× ℎ𝑒𝑖𝑔ℎ𝑡 𝑠= 𝑅=
2 𝑠𝑒𝑚𝑖 𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟 4 × 𝑎𝑟𝑒𝑎 𝑜𝑓 ∆

Isosceles 1 2𝑎 + 𝑏 √4𝑎 2 − 𝑏2
×𝑏 ℎ𝑒𝑖𝑔ℎ𝑡 =
triangle 2 2
√4𝑎2 − 𝑏2
×
2
Equilateral √3 2 𝑠 = 3𝑎 ℎ𝑒𝑖𝑔ℎ𝑡 𝑖𝑛𝑟𝑎𝑑𝑖𝑢𝑠 𝑐𝑖𝑟𝑐𝑢𝑚𝑟𝑎𝑑𝑖𝑢𝑠
triangle 𝑎 𝑎 ℎ 𝑎
4 √3𝑎 = 𝑜𝑟 =
= 3 √3
2 2√3
√𝟑 𝒂
𝑷𝟏 + 𝑷𝟐 + 𝑷𝟑 = 𝑶𝑹 𝒂 = ( 𝑷𝟏 + 𝑷𝟐 + 𝑷𝟑 )
𝒂 √𝟑
Right angle 1 𝑃+𝐵+𝐻 𝑖𝑛𝑟𝑎𝑑𝑖𝑢𝑠 𝑐𝑖𝑟𝑐𝑙𝑒
𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟
triangle 2 𝑃+𝐵−𝐻
× 𝑏𝑎𝑠𝑒 =
2

CIRCLE 𝜋𝑑 2 2𝜋𝑟 (𝑜𝑟 )𝜋𝑑


𝜋𝑟 2 (𝑜𝑟)
4
Semi circle
𝜋𝑟 2 𝑐𝑖𝑟𝑐𝑢𝑚𝑓𝑒𝑟𝑒𝑛𝑐𝑒 = 𝜋𝑟 + 𝑑 (𝑜𝑟) 𝜋𝑟 + 2𝑟
2

Sector of 𝜃 𝐴𝐶 + 𝐴𝐵 + 2 𝑟 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑠𝑒𝑐𝑡𝑜𝑟 = 𝐴𝐶𝐴𝐵


𝜋𝑟 2 ×
circle 360 (𝑜𝑟) 𝜃
= 2𝜋𝑟
𝑙 + 2𝑟 360

Cube 4𝑎2 𝑡𝑜𝑡𝑎𝑙 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑜𝑓 𝑐𝑢𝑏𝑒 = √3𝑎


= 6𝑎2
𝑓𝑎𝑐𝑒 𝑜𝑓 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑜𝑓 𝑐𝑢𝑏𝑒 = √2𝑎

𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑐𝑢𝑏𝑒 = 𝑎3

Cuboid 𝑣𝑜𝑙𝑢𝑚𝑒 𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑐𝑢𝑏𝑜𝑖𝑑


=𝑙 ×𝑏 ×ℎ = 2(𝑙+𝑏)×ℎ = 2 ( 𝑙𝑏 + 𝑏ℎ + 𝑙ℎ )
𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑜𝑓 𝑐𝑢𝑏𝑜𝑖𝑑𝑑 = √𝑙 2 + 𝑏2 + ℎ 2

Cylinder 𝑣𝑜𝑙𝑢𝑚𝑒 = 𝜋𝑟 2 ℎ 𝑐𝑢𝑟𝑣𝑒𝑑 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑡𝑜𝑡𝑎𝑙 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎


= 2𝜋𝑟ℎ = 2𝜋𝑟ℎ + 2𝜋𝑟 2 ℎ
𝑜𝑟
2𝜋𝑟 (𝑟 + ℎ)

Cone 𝑣𝑜𝑙𝑢𝑚𝑒 = 𝑐𝑢𝑟𝑣𝑒𝑑 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎 𝑡𝑜𝑡𝑎𝑙 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎


1
𝜋𝑟 2 ℎ = 𝜋 𝑟𝑙 = 𝜋𝑟𝑙 + 𝜋𝑟 2
3
𝑜𝑟
𝜋𝑟 (𝑟 + 𝑙)
𝑠𝑙𝑎𝑛𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 = 𝐿
= √𝑅2 + ℎ 2

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