Examples on Surface Area and Volume

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Volume Problems

We live in a three-dimensional world surrounded by objects of various shapes. Each of these products can be estimated regarding length, broadness, and tallness. For instance, the house that we live in has measurements. The surface area is the external aspect of the figure, while volume is the internal part. This article contains fathomed models on surface region and volume.


Definition of Surface Area Volume Problems

The total region of the outside of a three dimensional solid can be named as surface territory. It is denoted by "S" and estimated in square units. The amount of a three-dimensional shape or the space taken by it is called volume. Volume is indicated by "V" and estimated in cubic units.

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The picture above represents Surface Area Volume Problems


Formulae for Surface Area Volume Problems

Shapes

Curved Surface Area

Total Surface Area

Volume

Cube

4 (l2)

6 (l2)

l3 

Cuboid

2h ( l+ b)

2(lb + lh + bh)

lh

Cylinder

2πrh

2πr (r + h)

π (r2)h

Cone

πrl = π(h2+r2)1/2

πr (r + l)

⅓ π(r2)h

Hollow Cylinder

2 πh (R + r)

2 πh (R + r) + 2πh (R2 – r2)

π (R2)h - π (r2)h

Frustum of Cone

πl (R + r)

πR2 + πr2 + πl (R + r)

⅓ πh [R2 + r2 + Rr]

Sphere

4 πr2

4 πr2

4/3 πr3

Pyramid

½ (perimeter of base) * Slant Height

Curved Surface Area + Area of the Base

⅓ Area of Base * Height

Prism

Perimeter is of Base * Height

Curved Surface Area + 2 (Area of the End Surface)

Area of the Base * Height


Examples on Surface Area Volume Problems

Model 1: A customary hexagonal crystal has its border of the base as 600 cm and tallness 200 cm. Discover the weight of petroleum that it can hold if the thickness is 0.8 g/cc. (Take √3 = 1.73) 

Side of hexagon = (Perimeter)/(Number of Sides) = 600/6 = 100 cm. 

Territory of ordinary hexagon = (3√3)/2 x 100 x 100 = 25,950 sq.cm. 

Volume = Base Area × stature = 25950 x 200 = 5,190,000 cu.cm 

Weight of petroleum = Volume x Density = 5,190,000 cc × 0.8 g/cc = 4,152,000 gm. = 4,152 kg. 


Model 2: Find the volume of the most prominent right round cone that can be removed of a 3D shape of edge 42 cm. 

The cone base will be hover recorded in the face of the 3D shape, and its tallness will be equivalent to the edge of the 3D square. Range of cone = 21 cm. Stature = 42 cm. 

Volume of cone = 1/3πr2h = 1/3x 22/7 x 21 x 21 x 42 = 19,404 cu. cm. 


Model 3: An iron bundle of the distance across 6 inches is dropped into a round, hollow vessel of measurement 1ft loaded up with water. Discover the ascent at the water level. 

Radius of the vessel = 12 inch/2 = 6 inch. 

Volume of water that has risen = Volume of circle 

πR2h = 4/3 πr3 ⇒ 6 x 6 x h = 4/3 x 33 ⇒ h = 1 inch. 


Model 4: The measurement of a circle is 9 cm. It is softened and brought into a wire of width 3 cm. Discover the length of the wire. 

Vol. of circle = 4/3πr3 = 4/3π(4.5)3 = 121.5πcm3 

A wire is of round and hollow shape 

∴πR2H = 121.5π ⇒ (1.5) 2 H = 121.5 

H = 54 cm 


Model 5: The area of a square pyramid is 36 m2. The volume of the pyramid is 288mk3. What is the tallness of the square pyramid? 

Since the zone of the base = 36 m2

Volume of the pyramid = (1/3) * (base zone) * (tallness) 

288 = (1/3) * (36) * (H) 

H=24m 


Model 6: A crystal with symmetrical three-sided base (side= 12 cm) and Altitude (H)= 17cm. Locate the Lateral surface Area, Total surface territory, and Volume? 

Suppose the triangle XYZ is symmetrical. 

Edge (P) =12+12+12 = 36cm 

The zone of XYZ = (√3/4) a2 = √3/4 x 12 x 12 = 36√3cm2 

CSA = Perimeter * Altitude = 36 x 17 = 612 cm2 

TSA = CSA + 2 x Area of XYZ = 612+72√3 cm2 

Volume = Area x elevation = 36√3 x 17 = 612√3 cm3


FAQ (Frequently Asked Questions)

1. How is the Surface Area and Volume of a Cube Related?

A surface area is a space taken by the object. For a 3D square, it's the surface area of every one of the six sides of the block. Mathematical shapes have explicit equations you can use to ascertain their surface zones. A 3D cube has a surface zone of 6s2. These equivalent three-dimensional mathematical shapes additionally have equations for volume. The volume is the space inside the shape. Just three-dimensional shapes have volume. Two-dimensional shapes, for example, a line or a square, don't have volume. The equation for the volume of a 3D cube is s3. To estimate the relationship between surface area and volume, we divide the surface area by the volume. This ratio comes to 6/s. The ratio is called the surface area to volume ratio.

2. How Can We Implement Surface Area Volume Problems in Real Life

The surface area can be utilized for discovering things that are relative to the surface region. For example, The amount of paint needed to cover an object. The wallpaper required for a room. The amount of material needed for making clothes. The number of tiles needed to cover a roof. For the implication of volume, we can consider how much water fills a pool. The volume of air required to fill a balloon. The amount of sand that can fit inside a bucket. The heaviness of an object. The number of cups that can be filled from a tea-pot. The amount of wood is required to build a solid structure.