11.5 – Surface Area

Key Terms

• Base Area (BA) – The sum of the areas of all base surfaces in a three-dimensional 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 three-dimensional 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 three-dimensional solid consisting of two parallel congruent discs not directly above each other and all the points between them.
• Oblique Prism – A three-dimensional 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 three-dimensional 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 three-dimensional solid consisting of two parallel congruent discs directly above each other and all the points between them.
• Right Prism – A three-dimensional solid consisting of two parallel congruent polygons directly above each other and all the points between them.
• Slant Height (s) – The height of a two-dimensional figure that is a lateral face of a three-dimensional figure.
• Square Pyramid – A three-dimensional 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 one-dimensional segments together.
• Building 3D Objects – A two-dimensional figure is bounded by one-dimensional line segments and a three-dimensional solid is bounded by two-dimensional faces.
• Area of a Parallelogram = Length of the base times the height
• Written A=bh
• Area of a Triangle = 1/2 times base times height
• Written $A=\frac{1}{2}bh$
• 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
1. Find the area of each of the solid’s faces
• Formula for Surface Area
• Surface Area = Base Area (BA) + Lateral Area (LA)
Base Area & Lateral Area (Examples)
• Prisms
• The bases are congruent polygons
• There are 2 bases
• 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
• There is only 1 base
• 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.
• $SA=BA+LA$
• $SA=BA+ph$
• $SA=2t^{2}+4t\bullet h$

The older (longer) way of calculating SA

The newer (faster) way of calculating SA

Surface Area of Regular Pyramids
• In regular pyramids, the lateral faces are all the same: congruent isosceles triangles.
• LA = the area of one isosceles triangle times the number of lateral faces
• The altitude of each triangular lateral face is slanted.
• The length of this slanted altitude is called the slant height of each face.

• Example 1
• The unfolded figure of a pyramid (below) covers is the same area as its total surface area when it is folded up into a square pyramid.
• $SA=BA+LA$
• $SA=n^{2}+4(\frac{1}{2})ns$
• $SA=n^{2}+2ns$

• Example 2

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: $A=\frac{1}{2}bh$ 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 three-dimensional 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: $SA=BA+\frac{1}{2}ps$ and $SA=BA+LA$ 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: $SA=BA+LA$ and $SA=BA+ph$

Surface Area of Circular Solids
• Uses the same formula for Surface Area: $SA=BA+LA$
Surface Area of a Right Cylinder Surface Area of a Right Cone
• Finding the surface area of a cylinder is similar to finding the SA of a prism.
• Each has 2 congruent bases and a lateral area.
• The center of a circular base is aligned with the center of the other base.
• The base area is equal to the total area of the two bases (circles).
• The lateral area is equal to the circumference of one circular base times the cylinder’s height.

• How to Find Surface Area of a Right Cylinder
• Step 1: Write the surface area formula.
• Step 2: Find the area of its two circular bases.
• Step 3: Find the area of its rectangular lateral face.
• Step 4: Add the two areas.

• Surface Area = Base Area + Lateral Area
• $SA=2\pi r^{2}+2\pi rh$
• $\pi \approx 3.14$
• h: altitude (height of cylinder)

• The LATERAL surface area of a cylinder is TWICE the LATERAL surface area of a cone with the same radius.
• 2 (Cone LA) = 1 (Cylinder LA)
• You’d have to double (multiply by 2) the measure of the cone to equal 1 cylinder.
• The total SURFACE area of a cylinder is 3 times the SURFACE area of a cone with the same radius.
• 3 (Cone SA) = 1 (Cylinder SA)
• You’d have to triple (multiply by 3) the measure of the cone to equal 1 cylinder.

• Finding the surface area of a cone is similar to finding the SA of a pyramid.
• Each has one base, and the centers of their bases are each aligned with their respective vertices.

• How to Find Surface Area of a Right Cone
• Step 1: Write the surface area formula.
• Step 2: Find the area of its single circular base.
• Step 3: Find the area of its lateral face.
• Step 4: Add the two areas.

• The lateral area of a right cone is one half the product of the circumference (C) of its base, and it’s slant height is s:
• $LA=\frac{1}{2}Cs$
• $LA=\frac{1}{2}(2\pi r)s$
• $\pi rs$ (the 1/2 and the 2 cancel each other)

• Surface Area = Base Area + Lateral Area
• $SA=\pi r^{2}+\pi rs$

• The LATERAL surface area of a cone is 1/2 the LATERAL surface area of a cylinder with the same radius.
• 1 (Cone LA) = 1/2 (Cylinder LA)
• The total SURFACE area of a cone is 1/3 the total SURFACE area of a cylinder with the same radius.
• 1 (Cone SA) = 1/3 (Cylinder SA)

Formulas for Surface Area
• SA of a Right Cylinder

• Example

• SA of a Right Cone

• Example

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? $352 \pi \;units^{2}$ Ex 2. Which of the following is the surface area of the right cylinder below? $198 \pi \;units^{2}$ Ex 3. What is the surface area of the right cone below? $176 \pi \;units^{2}$ Ex 4. It is true that the lateral surface area of cone A is exactly $\frac{1}{2}$ 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: $LA=\pi rs$ Ex 7. Given a right cylinder where h is the height and r is the radius, what does the expression $2\pi rh$ represent? Answer: lateral area 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? $2\pi r^{2}+2\pi rh$ $BA+2\pi rh$