GRE: Three Dimensional Figures (Uniform Solids)

Volume

The volume of a solid is the amount of space enclosed by the solid.

Surface area

In general, the surface area of a solid is equal to the sum of the areas of the solid’s faces.

Rectangular Solid

A solid with six rectangular faces ( all edges meet at right angles)

Volume= Area of Base x Height
= l x w x h

Surface Area = Sum of areas of faces
=2(lw + wh + hl )

Cube

A special rectangular solid with all edges equal (l=w=h) is called a cube. For Example:- Die

Volume = L3

Surface area = Sum of areas of faces
= 6 x L2

Cylinder

A uniform solid whose base is a circle is called a cylinder.

Lateral Surface area (or area of rest of shell)
= 2 π r h
Volume = Area of base X Height = π r2h
Total Surface area = (2 X Area of base) + Lateral Surface Area

= 2 π r+ 2 π r h = 2 π r ( r + h)

Sphere

A sphere is made up of all the points in space at a certain distance from a center point, it is also called a three-dimensional circle.
The difference from the center to a point on the sphere is the radius of the sphere.

Volume of sphere = 4/3 π r3

Total Surface area = 4 π r2

 

Problem 1) The solid shown is half a rectangular solid. What is the volume of the solid shown?

 

 

 

 

 

 

 

Sol. In Triangle ABC,

AB2 + BC2 = AC2

AB+ 9= 25

AB=4
Area of base = (1/2) x 4 x 3 = 6 Sq. units
Volume = 6 x 4 = 24 Cubic units

 

Problem 2)  The height of a cylinder is twice its radius. If the volume of the cylinder is 128 π. What is the radius?

Sol. Let radius be r
Height of cylinder (h) = 2 r
Volume = π r2 h
128 π = π r2 ( 2 r )
64= r3
r = 4 units.

 

Problem 3) A cube of ice has edges of length 10. What is the volume of the largest cylinder that can be carved from the cube?

Sol. The largest cylinder would have diameter 10 and height 10 ( each equal to the edge of the cube)

Volume = π r2 h
= π (100/4) (10)

=1000 π / 4
= 250 π Cubic units

 

Problem 4) A solid metal cylinder with a radius of 6 and a height of 3 is melted down and all of the metal is used to recast a new solid cylinder with a radius of 3. What is the height of the new cylinder?

Sol. Volume of solid metal cylinder
=Volume of new solid cylinder

Let radius and height of solid metal cylinder be and new solid cylinder be

π R12H1 = π R22H2

62. (3) = 32. h

H = (36 x 3) / (3 x 3) = 12 units

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