# 7-7 Filling and Wrapping - Concepts and Explanations

## Surface Area of Rectangular Prisms

Surface area is the sum of the areas of the faces.

Surface Area = (area of the front x 2) + (area of the side x 2) + (area of the top x 2)

Or

Surface Area = (area of the front + area of the side + area of the top) X 2 = (*w* X *h* + *w* X *l* + *l* X *h*) X 2.

# Example

There are three sets of two congruent faces:

4 cm by 3 cm (area is 12 cm^{2});

4 cm by 5 cm (area is 20 cm^{2});

3 cm by 5 cm (area is 15 cm^{2}).

Surface area = 94 cm^{2}

## Volume of Rectangular Prism

The volume (the total number of unit cubes) of a rectangular prism is the area of its base (the number of unit cubes in the first layer) multiplied by its height (the total number of layers).

Volume = Area of the base x height = *Bh* = *lwh*

# Example

3 X 2 = 6 cubes on the base

5 layers of cubes (height);

Volume = 6 X 5 = 30 cubic units

## Volume of Prisms

The same layering strategy is used to generalize the method for finding the volume of any prism. The volume of any prism is the area of its base multiplied by its height.

Volume = Area of the base X height = *Bh*

# Example

## Area of Circles

Students begin by finding the number of “radius squares” with side lengths that are equal to the radius, that cover the circle. It is a little more than 3, or pi.

# Example

The area of a circle is pi X a "radius square" or pi X radius X radius = π X *r* X *r*

= π*r*^{2}

## Perimeter of Circles (Circumferences)

Students count the number of diameter lengths needed to surround the circle. It is a little more than 3, or pi.

# Example

The circumference of a circle is pi X diameter, or πd.

## Surface Area of Cylinders

By folding a flat pattern to form a cylinder, students discover that the surface area of the cylinder is the area of the rectangle that forms the lateral surface (2 π*rh*) plus the areas of the two circular ends (2π*r*^{2}). Surface Area = 2π*r*^{2} + 2π*rh*

# Example

Use 3.14 for π. r = 4 h = 5.

2π x 4^{2} + 2π x 4(5) = 100.48 + 125.6

≈ 226.08 square units.

## Volume of Cylinders

The volume of a cylinder is the number of unit cubes in one layer (the area of the circular base, π*r*^{2}) multiplied by the number of layers (the height *h*) needed to fill the cylinder. Volume = *Bh* = π*r*^{2}*h*.

# Example

Area of base *B* = π *r*^{2}

≈ 3.14 X 2.5^{2}

= 19.625 square units

V = Bh

= 19.625 X 7

= 137.375 cubic units

## Volumes of Cones and Spheres

If a cylinder, a cone, and a sphere all have the same radius and the same height (the height being equal to two radii), then it takes 3 cones to fill the cylinder, and 1 1/2 spheres to fill the cylinder.

Volume_{cone} = 1/3 x Volume_{cylinder} = 1/3πr^{2}h

Volume_{sphere} = 2/3 x Volume_{cylinder} = 2/3πr^{2}h.

# Example

Volume_{cylinder} = 628 cm^{3}

Volume_{cone} ≈ 209 cm^{3}

Volume_{sphere} ≈ 419 cm^{3}