**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 =** L ^{3}**

Surface area = Sum of areas of faces

= **6 x L ^{2}**

**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 r ^{2}h**

Total Surface area = (2 X Area of base) + Lateral Surface Area

= **2 π r ^{2 }+ 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 r ^{3}**

Total Surface area = **4 π r ^{2}**

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

**Sol.** In Triangle ABC,

AB^{2} + BC^{2} = AC^{2}

AB^{2 }+ 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 r^{2} h

128 π = π r^{2} ( 2 r )

64= r^{3}

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 r^{2} 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 R_{1}^{2}H_{1} = π R_{2}^{2}H_{2}

6^{2}. (3) = 3^{2}. h

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