Mensuration Basic Fundas - Vikas Saini

• Sphere

Volume (V) = (4/3) x π x (radius)^3
Surface area (SA) = 4 x π x (radius)^2

1. The radius of a sphere is 14.7 cm, what is its volume?
Solution :-
Volume = (4/3) x (22/7) x (14.7)^3 = 13311.144 cu. Cm

2. The volume of a sphere is 457.34π cu. cm. What is its radius?
Solution :-
(4/3) x π x (radius)^3 = 457.34 π
Radius^3 = 457.34 x 3 / 4

3. If the surface area of a sphere is 616 sq. cm, find its radius.
Solution :- 4 π radius^2 = 616

4. The volume of a sphere is 7241.15 cu. cm, find its surface area.
Solution :-
(4/3) x π x (radius)^3 = 7241.15
SA = 4 π (radius)^2 = 1810.29 sq cm.

Hemi sphere

Volume (V) = (2/3) x π x (radius)^3
Surface area (SA) = 2 x π x (radius)^2

1. The diameter of a hemisphere is 30 cm, find its volume. (π = 3.14)
Solution :-
Radius = 30/2 = 15 cm.
V = (2/3) x π x (radius)^3
V = 7065 cu cm.

2. Find the total surface area of a hemisphere whose volume is 452.16 cu. cm. (π = 3.14).
Solution :-
452.16 = (2/3) x π x (radius)^3
CSA =2 x π x (radius)^2 = 339.12 sq cm.
Cube
Volume = side^3
Surface area = 6 x side^2

3. The volume of a cube is 46656 cu. cm. What is its total surface area?
Solution :-
V = Side^3
Side = (46656)^(1/3)
Side = 36.
Surface area = 6 x 36^2 = 7776 sq. Cm.

Cuboid

Volume = length x breadth x height
Surface area = 2 (length x breadth + breadth x height + length x height)

1. The length, breadth and height of a parallelepiped are 5 cm, 8 cm and 12 cm respectively. Find its total surface area and volume.
Solution :-
Surface area = 2 (5 x 8 + 8 x 12 + 5 x 12)
=392 sq cm
Volume = 5 x 8 x 12 = 480 cu. cm.

Prism

Area of polygon = (n/4) x (side)^2 Cot (180/n)
n = no of sides in polygon

1. Find the volume of a prism with a regular pentagonal base. Side of its base is 3.81 cm and its height is 8 cm. cot (36°) = 1.3763
Area of pentagonal base = (5/4) (3.81)^2 Cot (180 / 5)
= (5/4) (3.81)^2 x 1.3763
= 25 sq cm (approximately)
Volume = area x height
= 25 x 8
= 200 cu. cm

2. Find the volume of a prism with a regular hexagonal base of side 12 cm and height 15 cm.
Area of hexagonal base = (6/4) x (12)^2 x Cot (180/6)
=216( 3)^1/2 .
Volume = 216(3)^1/2 x 15 = 3240 (3)^1/2.

3. Find the surface area of the vertical faces of a prism with a regular pentagonal base of side 50 cm and height 14 cm.
Perimeter = 50 x 5 = 250.
Surface area = perimeter x height = 250 x 14 = 3500.

Pyramid

1. The side of a regular hexagonal base of a pyramid is 18 cm. The slant height and height of the pyramid are 30 cm and 28 cm respectively. Find its volume and total surface area.
Solution :-
Area of hexagonal base = (6/4) x (18)^2 x Cot (180/6)
=841.75 sq cm.
Volume of pyramid = (1/3) x base area x height
= (1/3) x 841.75 x 28
=7856.33 cu cm.
Surface area = 1/2 x perimeter x slant height + base area
=2461.75 sq cm.

2. The volume and surface area of the lateral faces of a pyramid with a regular pentagonal base of side 12 cm is 1984 cu. cm and 900 sq. cm. What are its height and slant height?
cot (36°) = 1.3763
Solution :-
Base area = (5/4) x (12)^2 x Cot (180/5) = 248.4 sq cm.
Volume = 1/3 x base area x height
Height = 3 x 1984 / 248.4 = 24 cm (approximately)
Surface area = 1/2 x perimeter x slant height + base area
900 – 248.4 = 1/2 x (5 x 12) x slant height
Slant height = 30 cm.

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