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11.5 – Surface Area
Key Terms
 Base Area (BA) – The sum of the areas of all base surfaces in a threedimensional figure.
 The base area added to the lateral area of the figure equals its surface area.
 Base area is sometimes denoted with a big B instead of BA.
 Height – The measurement taken from the bottom to the top of an object.
 The perpendicular distance between the base of a geometric figure and the opposite vertex or parallel base.
 Also called the altitude.
 Lateral Area (LA) – The sum of the areas of all nonbase surfaces in a threedimensional figure.
 The lateral area added to the base area of the figure equals its surface area.
 Oblique Cone – A cone whose vertex is not directly over the center of its base.
 Oblique Cylinder – A threedimensional solid consisting of two parallel congruent discs not directly above each other and all the points between them.
 Oblique Prism – A threedimensional solid consisting of two parallel congruent polygons not directly above each other and all the points between them.
 Oblique Pyramid – A pyramid whose vertex is not directly over the center of its regular polygon base.
 Rectangular Prism – A threedimensional solid consisting of two parallel congruent rectangles and all the points between them.
 Regular Pyramid – A right pyramid whose vertex is directly over the center of a regular polygon base.
 Right Cone – A cone whose vertex is directly over the center of its circular base.
 Right Cylinder – A threedimensional solid consisting of two parallel congruent discs directly above each other and all the points between them.
 Right Prism – A threedimensional solid consisting of two parallel congruent polygons directly above each other and all the points between them.
 Slant Height (s) – The height of a twodimensional figure that is a lateral face of a threedimensional figure.
 Square Pyramid – A threedimensional solid consisting of a square base, a point not in the same plane as the square base, and all the points between them.
 Surface Area (SA) – The total area of the exterior surface of a solid figure.
Review
Polygons to Polyhedrons 
 Perimeter – To find the perimeter of a figure, add the lengths of its onedimensional segments together.
 Building 3D Objects – A twodimensional figure is bounded by onedimensional line segments and a threedimensional solid is bounded by twodimensional faces.
 Area of a Parallelogram = Length of the base times the height
 Area of a Triangle = 1/2 times base times height
 Written
 Area Formulas

Notes
Surface Area of Polygons 
 The area of a polygon is the amount of space it covers.
 All the faces of a polyhedron are polygons.
 So the surface area of a polyhedron is the amount of space all its polygon faces cover.
 To understand surface area, think about wrapping a gift with wrapping paper.

How to Calculate Surface Area 
 Find the area of each of the solid’s faces
 Add them all together

 Formula for Surface Area
 Surface Area = Base Area (BA) + Lateral Area (LA)

Base Area & Lateral Area (Examples) 
 Prisms
 The bases are congruent polygons
 The lateral area is formed by a set of rectangles
 The number of sides of the base polygon shape is the same number of rectangles (3 in this case)

 Pyramids
 The base is a single polygon
 The lateral area is formed by a set of triangles
 The number of sides of the base polygon shape is the same number of triangles (4 in this case)

Surface Area of Right Prisms 
 Formula Options
 Surface Area = BA + LA, or…
 Surface Area = BA + ph
 LA can be replaced by “ph”
 p: perimeter of ONE base
 h: prism’s altitude (height)


 Example: SA of a Triangular Prism
 Example: SA of a Right Rectangular Prism
 The unfolded figure of a prism (below) covers is the same area as its total surface area when it is folded up into a rectangular prism.
The older (longer) way of calculating SA
The newer (faster) way of calculating SA

Oblique Prisms & Pyramids 
 Surface Area of an Oblique Prism
 These cannot be calculated using a quick formula.
 2BA: The bases of an oblique prism are congruent, but they are not aligned.
 So, you can find the BA and multiply it by 2.
 The lateral faces will have different measures, so calculate them separately and add them to the total base area (2BA).
 They are parallelograms, so use the formula: A = bh for each of the polygons that make up the lateral faces.

 Surface Area of an Oblique Pyramid
 LA: the triangular faces are not congruent; so, you have to calculate them separately.
 Use the formula: to find the area of each triangle that makes up the lateral faces.
 Then, add the LA to the BA
 SA = LA + BA

Examples
 Ex 1. What is the surface area of the rectangular prism below?
Answer: 736 Units Squared

 Ex 2. What is the surface area of the regular pyramid below?
Answer: 864 Units Squared

 Ex 3. It is true that in order to find the surface area of a threedimensional figure, you must find the area of each of its faces and then add the areas.

 Ex 4. It is true that a regular pyramid has a regular polygon base and a vertex over the center of the base.

 Ex 5. Which formulas can be used to find the surface area of a regular pyramid with a square base where the perimeter of the base is equal to p, s is the slant height, BA is the base area, and LA is the lateral area?
 Answers: and

 Ex 6. Given a right prism where p is the perimeter of the base, h is the height, BA is the area of the bases, and LA is the lateral area, what is the surface area?
 Answers: and

Oblique Cylinders & Cones 
 Surface Area of an Oblique Cylinder
 There is one way to estimate the surface area of an oblique cylinder.
 1. Sketch a rectangle around the lateral face that closely matches its size. Find its area.
 2. Sketch a circle on one base that closely matches its size. Find its area (times 2).
 3. Add the results from steps 1 and 2.
 You will NOT be asked to do this in this course!

 Surface Area of an Oblique Cone
 Because of its unusual shape, there is no quick formula for finding the surface area of an oblique cone.
 A general BA + LA formula will work.
 You will NOT be asked to do this in this course!

Examples
 Ex 1. Which of the following is the surface area of the right cylinder below?

 Ex 2. Which of the following is the surface area of the right cylinder below?

 Ex 3. What is the surface area of the right cone below?

 Ex 4. It is true that the lateral surface area of cone A is exactly the lateral surface area of cylinder B.

 Ex 5. It is true that the cone and the cylinder below have equal surface area.

 Ex 6. What is the formula used to find the lateral area of a right cone where r is the radius and s is the slant height?
 Answer:

 Ex 7. Given a right cylinder where h is the height and r is the radius, what does the expression represent?

 Ex 8. What two formulas would find the surface area of a right cylinder where h is the height, r is the radius, and BA is the base area?

Important!
Practice (Apex Study 11.5)
 Try practice problems on Pg 14, 25
 Mandatory: write and answer problems on Pg 15, 26
 2 Quizzes
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