Notes

DEFINITIONS:
Lateral Area- sum of the areas of the lateral faces

Surface Area- sum at the areas of all of the faces of a solid


Prisms:
Surface Area = sum of the areas of each face

2 Bases = 2(3 · 2)= 12 cm2
2 Lateral Faces = 2(6 · 2) = 24 cm2
2 Lateral Faces = 2(6 · 3) = 36 cm2
Surface Area = 72 cm2
Formula: Surface Area = 2B + ph where B is the area of the base, p is the perimeter of the base, and h is the height of the prism


Example 1: Find the surface area of the isosceles triangular prism below.

Step 1: Find the area of the two congruent bases.
Find the height of the triangular base by Pythagorean Theorem:

Find the area of the triangular base.

Step 2: Find the total area of all lateral faces.
A = p · h
A = (6.5+6.5+5) · 8
A = 144
Step 3: Find total Surface Area by adding the area of the 2 bases and the area of the lateral faces together.
SA = 2(15) + 144 = 174 cm2

Example 2: Find the surface area of the rectangular prism below.

Step 1: Find the area of the two congruent bases.
The two ends are chosen to be the bases.
2.75 · 4.2 = 11.55
Step 2: Find the total area of all lateral faces.
A = p · h
A = (4.2 + 4.2 + 2.75 + 2.75) · 15
A = 208.5
Step 3: Find total Surface Area by adding the area of the 2 bases and the area of the lateral faces together.
SA = 2(11.55) + 208.5
SA = 231.6 m2
Cylinders:
Surface Area = 2rh+2r2
Surface Area= sum of the lateral area and the area of each base

Example 1:Find the surface area of the cylinder below.

Step 1. Identify the radius and height of the cylinder.
The height is 12.2 according to the diagram.
The diameter is 6.2 and therefore the radius is 3.1
Step 2. Substitute into the formula and solve.

Pyramids:
Regular Right Pyramid:
Surface Area = add the area of the base and the area of the lateral faces


The net of this pyramid is drawn below:


Slant Height- height of each triangular face, slanted towards the APEX, represented by a cursive L
Surface AREA = 12pl+B
p = perimeter of base
l = slant height
B = area of the base

Example 1: Find the surface area of the square pyramid below.

Step 1: Find the area of the base.
A=252
A= 625 feet2
Step 2: Find the total area of all lateral faces.
A= 0.5pl
A= 0.5100 32
A = 1600 feet2
Step 3: Find total surface area by adding the area of the base and total area of all lateral faces.
Surface AREA = 625 + 1600
SA = 2225 feet2


Example 2: Find the surface area of the triangular pyramid below.

Step 1: Find the area of the base.
This is an equilateral triangle therefore the area is;

Step 2: Find the total area of all lateral faces.

Step 3: Find total surface area by adding the area of the base and total area of all lateral faces.

Cones: the base of a cone is a circle, therefore we will use the area formula of a circle and the circumference formula for a circle to help find the surface area of a cone.
Net of a cone:

Surface Area = radiusslant height+r2

r = radius; l = slant height

Example 1: Find the surface area of the cone below.

Step 1: Find slant height by using Pythagorean Theorem.

Step 2: Find total surface area by substituting the slant height and radius into the equation.

Spheres:
Surface Area = 4 r2

Example 1: Find the surface area of the sphere.

Step 1: Identify the radius of the sphere.
The raidus is 13 ft
Step 2: Substitute the radius into the equation for surface area.

Step 3: Use calculator to approximate the answer.


Hemisphere:
half of a sphere whose base is a ‘great circle’
Surface Area = 3 r2

Example 2: Find the surface area of the hemisphere below.

Radius is 4.2

Similar Figures
In the Polygons and Circles unit you learned about the scale factor of dilations and how that affects the perimeter and area of similar figures. In that unit we learned: if the similarity ratio of two similar figures is r:t, then the ratio of their perimeters is r:t, and the ratio of their areas is r2:t2. This fact is true with surface area as well. If the similarity ratio of two similar figures is r:t, then the ratio of their surface areas is r2:t2.

HELP VIDEOS:

Finding the Surface Area of a Prism:
http://my.hrw.com/math06_07/nsmedia/lesson_videos/geo/player.html?contentSrc=6580/6580.xml

Finding the Surface Area of a Cylinder:
http://my.hrw.com/math06_07/nsmedia/lesson_videos/geo/player.html?contentSrc=6805/6805.xml

Finding the Surface Area of a Pyramid:
http://my.hrw.com/math06_07/nsmedia/lesson_videos/geo/player.html?contentSrc=6581/6581.xml

Finding the Surface Area of a Cone:
http://my.hrw.com/math06_07/nsmedia/lesson_videos/geo/player.html?contentSrc=6809/6809.xml

Exploring Effects of Changing Dimensions on Surface Area:
http://my.hrw.com/math06_07/nsmedia/lesson_videos/geo/player.html?contentSrc=6807/6807.xml