2-D MENSURATION

S.N
Name Figure Nomenclature Area Perimeter
o.

𝑙 → 𝑙𝑒𝑛𝑔𝑡ℎ 2𝑙 + 2𝑏 =
1. Rectangle 𝑙 × 𝑏 = 𝐼𝑏
𝑏 → 𝑏𝑟𝑒𝑎𝑑𝑡ℎ 2(𝑙 + 𝑏)

𝑎 → 𝑠𝑖𝑑𝑒 (i) 𝑎 × 𝑎 = 𝑎2 𝑎+𝑎+𝑎+𝑎

2. Square 𝑑 → 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑑2
𝑑 = 𝑎√2 (ii) = 4𝑎
2

𝑎, 𝑏 𝑎𝑛𝑑 𝑎𝑟𝑒 𝑡ℎ𝑟𝑒𝑒 𝑠𝑖𝑑𝑒𝑠 𝑜𝑓

𝑡𝑟𝑎𝑖𝑛𝑔𝑙𝑒 𝑎𝑛𝑑 𝑠 𝑡ℎ𝑒 1
𝑠𝑒𝑚𝑖𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟, 𝑤ℎ𝑒𝑟𝑒(i) ×𝑏×ℎ 𝑎+𝑏+𝑐
Triangle(Scal 2
3. 𝑎+𝑏+𝑐
(ii) √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐) = 2𝑠
ene) 𝑠=( )
2
b is the base and h is the (𝐻𝑒𝑟𝑜𝑛′ 𝑠 𝑓𝑜𝑟𝑚𝑢𝑙𝑎)
altitude of triangle
𝑎 → 𝑠𝑖𝑑𝑒 1
(iii) (i) × 𝑎 × ℎ 3𝑎
Equilateral ℎ → ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑟 𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒 2
4.
Triangle √3 (iv) (ii)
√3
𝑎2
ℎ→ 𝑎 4
2

𝑎 → 𝑒𝑞𝑢𝑎𝑙 𝑠𝑖𝑑𝑒𝑠
𝑏 → 𝑏𝑎𝑠𝑒 1
Isosceles (v) (i) × 𝑏 × ℎ
5. ℎ → ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑟 𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒 2
1 2𝑎 + 𝑏
Triangle √4𝑎2 − 𝑏 2 (vi) (ii) × 𝑏 × √4𝑎2 − 𝑏 2
4
ℎ=
2
𝑏 → 𝑏𝑎𝑠𝑒
𝑏+ℎ+𝑑
Right angled ℎ → 𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒/ℎ𝑒𝑖𝑔ℎ𝑡 1
6. 𝑑 → 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 (vii) ×𝑏×ℎ
Triangle 2
𝑑 = √𝑏 + ℎ
2 2

Isosceles
𝑎 → 𝑒𝑞𝑢𝑎𝑙 𝑠𝑖𝑑𝑒
right 1
7. 𝑑 → 𝑑𝑖𝑎𝑔𝑜𝑛𝑎𝑙 (viii) 𝑎2 2𝑎 + 𝑑
Angled 𝑑 = 𝑎√2
2
triangle
AC is the diagonal and
ℎ1 , ℎ2 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑎𝑙𝑡𝑖𝑡𝑢𝑑𝑒𝑠 𝑜𝑛
1 AB+BC+
8. Quadrilateral 𝐴𝐶 𝑓𝑟𝑜𝑚 𝑡ℎ𝑒 (ix) × 𝐴𝐶 × (ℎ1 + ℎ2 )
Verticals D and B
2 CD+AD
respectively
𝑎 𝑎𝑛𝑑 𝑏 𝑎𝑟𝑒 𝑠𝑖𝑑𝑒𝑠 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜
𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟.
Parallelogra
9. ℎ → 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛(x) 𝑎×ℎ 2(a + b)
m 𝑡ℎ𝑒 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙
𝑠𝑖𝑑𝑒
𝑎 → 𝑒𝑎𝑐ℎ 𝑒𝑞𝑢𝑎𝑙 𝑠𝑖𝑑𝑒
𝑜𝑓 𝑟ℎ𝑜𝑚𝑏𝑢𝑠
1
10. Rhombus (xi)
𝑑1 𝑎𝑛𝑑 𝑑2 are the diagonals × 𝑑1 × 𝑑2 4𝑎
2
𝑑1 → 𝐵𝐷
𝑑2 → 𝐴𝐶

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 𝑎 𝑎𝑛𝑑 𝑏 𝑎𝑟𝑒 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑠𝑖𝑑𝑒 𝑡𝑜 𝑒𝑎𝑐ℎ 𝑜𝑡ℎ𝑒𝑟
𝑎𝑛𝑑 ℎ 𝑖𝑠 𝑡ℎ𝑒 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑎+𝑏 𝐴𝐵 + 𝐵𝐶 + 𝐶𝐷
11. Trapezium ( )×ℎ
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 2 + 𝐴𝐷
𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑠𝑖𝑑𝑒

Regular 3√3
12. (xii)
𝑎 → 𝑒𝑎𝑐ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑙 𝑠𝑖𝑑𝑒 𝑎2 6𝑎
Hexagon 2

Regular
13. 𝑎 → 𝑒𝑎𝑐ℎ 𝑜𝑓 𝑒𝑞𝑢𝑎𝑙 (xiii)
𝑠𝑖𝑑𝑒 2𝑎2 (1 + √2) 8𝑎
Octagon

2𝜋𝑟
𝑟 → 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 (𝑐𝑎𝑙𝑙𝑒𝑑 𝑎𝑠 )
14. Circle 22 (xiv) 𝜋𝑟 2
𝜋= = 3.1416(𝑎𝑝𝑝𝑟𝑜𝑥) Circumfere
7
nce)

1
15. Semicircle (xv)
𝑟 → 𝑟𝑎𝑑𝑖𝑢𝑠𝑜𝑓 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 𝜋𝑟 2 𝜋𝑟 + 2𝑟
2

1 1
16. Quadrant 𝑟 → 𝑟𝑎𝑑𝑖𝑢𝑠 (xvi) 𝜋𝑟 2 𝜋𝑟 + 2𝑟
4 2

Ring or (outer)→
circular path 𝑅 → 𝑜𝑢𝑡𝑒𝑟 𝑟𝑎𝑑𝑖𝑢𝑠 2𝜋𝑅
17. (xvii) 𝜋(𝑅2 − 𝑟 2 )
(shaded 𝑟 → 𝑜𝑚𝑚𝑒𝑟 𝑟𝑎𝑑𝑖𝑢𝑠 (Inner)→
region) 2𝜋𝑟

𝑂 → 𝑐𝑒𝑛𝑡𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒

𝑟 → 𝑟𝑎𝑑𝑖𝑢𝑠 𝜃
Sector of a 𝑙 → 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 (xviii)
𝑎𝑟𝑒 (i) 𝜋𝑟 2 ( )
18. 360° 𝑙 + 2𝑟
circle 𝜃 → 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑒𝑐𝑡𝑜𝑟
(xix)
1
(ii) 𝑟 × 𝑙
𝜃 2
𝑙 = 2𝜋𝑟 ( )
360°

(xx)
𝜃 → 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑠𝑒𝑐𝑡𝑜𝑟 Area of segment 𝜋𝜃
Segment of a 𝑟 → 𝑟𝑎𝑑𝑖𝑢𝑠 2𝑟 [ +
19. ACB(minor segment) 360°
circle 𝐴𝐵 → 𝑐ℎ𝑜𝑟𝑑 𝜋𝜃 𝑠𝑖𝑛𝜃
𝜃
𝑠𝑖𝑛 ( )]
(xxi)
𝐴𝐶𝐵 → 𝑎𝑟𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 =𝑟 2 ( − ) 2
360° 2

Pathways
𝑙 → 𝑙𝑒𝑛𝑔𝑡ℎ 2(𝑙 + 𝑏)
running
𝑏 → 𝑏𝑟𝑒𝑎𝑑𝑡ℎ − 4𝑤
20. across the 𝑤
(xxii) (𝑙 + 𝑏 − 𝑤)𝑤
= 2[𝑙 + 𝑏
middle of a → 𝑤𝑖𝑑𝑡ℎ 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑎𝑡ℎ (𝑟𝑜𝑎𝑑) − 2𝑤]
rectangle
(inner)→
𝑙 → 𝑙𝑒𝑛𝑡ℎ
Outer 2(𝑙 + 𝑏)
21. 𝑏 → 𝑏𝑟𝑒𝑎𝑑𝑡ℎ (𝑙 + 𝑏 + 2𝑤)2𝑤
Pathways 𝑤 → 𝑤𝑖𝑑𝑡ℎ𝑛𝑒𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑎𝑡ℎ (outer)→
(𝑙 + 𝑏 + 4𝑤)

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 (𝑜𝑢𝑡𝑒𝑟)
𝑙 → 𝑙𝑒𝑛𝑔𝑡ℎ
→ (𝑙 + 𝑏)
22. Inner path 𝑏 → 𝑏𝑟𝑒𝑎𝑑𝑡ℎ (𝑙 + 𝑏 − 2𝑤)2𝑤
𝑤 → 𝑤𝑖𝑑𝑡ℎ𝑛𝑒𝑠𝑠 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑎𝑡ℎ (inner)→
(𝑙 + 𝑏 − 4𝑤)

Quadrilateral:
 The sum of opposite sides of a quadrilateral circumscribed about a circle is constant..
 Lines joining the mid points of the adjacent sides of a quadrilateral form a parallelogram.
 Area of triangle is half of the area of a parallelogram which lie on the same base and between the same parallel
lines.
 A parallelogram inscribed in a circle is a rectangle.
 A parallelogram that is circumscribed about a circle is a rhombus.

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 3-D MENSURATION

Curved/Lateral
S.No. Name Figure Nomenclature Volume Total surface area
surface area
1. Cuboid 𝑙 → 𝑙𝑒𝑛𝑔𝑡ℎ 𝑙𝑏ℎ 2(𝑙 + 𝑏)ℎ 2(𝑙𝑏 + 𝑏ℎ + ℎ𝑙)
𝑏 → 𝑏𝑟𝑒𝑎𝑑𝑡ℎ
ℎ → ℎ𝑒𝑖𝑔ℎ𝑡
2. Cube 𝑎 𝑎3 4𝑎2 6𝑎2
→ 𝑒𝑑𝑔𝑒/𝑠𝑖𝑑𝑒

3. Right circular 𝑟 → 𝑟𝑎𝑑𝑖𝑢𝑠 𝑜𝑓 𝜋𝑟 2 ℎ 2𝜋𝑟ℎ 2𝜋𝑟(ℎ + 𝑟)

cylinder 𝑏𝑎𝑠𝑒
ℎ → ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓
𝑡ℎ𝑒 𝑐𝑦𝑙𝑖𝑛𝑑𝑒𝑟
4. Right circular 𝑟 → 𝑟𝑎𝑑𝑖𝑢𝑠 1 2 𝜋𝑟𝑙 𝜋𝑟(𝑙 + 𝑟)
𝜋𝑟 ℎ
cone ℎ → ℎ𝑒𝑖𝑔ℎ𝑡 3
𝑙 → 𝑠𝑙𝑎𝑛𝑡
ℎ𝑒𝑖𝑔ℎ𝑡
𝑙 = √𝑟 2 + ℎ 2
5. Right Area of base Perimeter of base × Lateral surface area + 2
triangular × height height (area of base)
prism
6. Right pyramid 1 1 𝑙𝑎𝑡𝑒𝑟𝑎𝑙 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎
× 𝑎𝑟𝑒𝑎 𝑜𝑓 × 𝑝𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟
3 2 + 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒
𝑡ℎ𝑒 𝑏𝑎𝑠𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑏𝑎𝑠𝑒
× ℎ𝑒𝑖𝑔ℎ𝑡 × 𝑠𝑙𝑎𝑛𝑡 ℎ𝑒𝑖𝑔ℎ𝑡

7. Sphere 𝑟 → 𝑟𝑎𝑑𝑖𝑢𝑠 4 3 4𝜋𝑟 2

𝜋𝑟
3

8. Hemisphere 𝑟 → 𝑟𝑎𝑑𝑖𝑢𝑠 2 3 2𝜋𝑟 2 3𝜋𝑟 2

𝜋𝑟
3
9. Spherical shell 𝑟 → 𝑖𝑛𝑛𝑒𝑟 4 4𝜋[𝑅2 + 𝑟 2 ]
𝜋[𝑅3 − 𝑟 3 ]
𝑟𝑎𝑑𝑖𝑢𝑠 3
𝑅 → 𝑜𝑢𝑡𝑒𝑟
𝑟𝑎𝑑𝑖𝑢𝑠
10. Frustum of a 𝜋 𝜋(𝑟 + 𝑅)𝑙 𝑙𝑡𝑒𝑟𝑎𝑙 𝑠𝑢𝑟𝑓𝑎𝑐𝑒 𝑎𝑟𝑒𝑎
ℎ(𝑟 2 + 𝑅𝑟
cone 3 𝜋[𝑅2 + 𝑟 2 ]
+ 𝑅2 )

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