Print this Page

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

GeoB 11.5 SA Right Cylinder Parts

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
  2. 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
      • 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)

Alg2B 11.5 - PrismSA

  • 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)

Alg2B 11.5 - PyramidSA

 

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)
GeoB 11.5 SA Right Prism1
  • Example: SA of a Triangular Prism

GeoB 11.5 SA Reg Prism Ex1


  • 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 SAGeoB 11.5 SA RectPrism

The newer (faster) way of calculating SAGeoB 11.5 SA RectPrismNet

 

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.

GeoB 11.5 SA Regular Pyramid1

  • 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

GeoB 11.5 SA Reg Pyramids


  • Example 2

GeoB 11.5 SA Reg Pyramid Ex1

 

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?

GeoB 11.5 Q1-2c

Answer: 736 Units Squared

  • Ex 2. What is the surface area of the regular pyramid below?

GeoB 11.5 Q1-3c

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.

GeoB 11.5 SA Right Cylinder2


  • 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
    • r: radius
    • 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.

GeoB 11.5 SA Right Cone2


  • 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

GeoB 11.5 SA Right Cylinder


  • Example

GeoB 11.5 SA Right Cylinder Ex

  • SA of a Right Cone

GeoB 11.5 SA Right Cone


  • Example

GeoB 11.5 SA Right Cone Ex

 

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!

GeoB 11.5 SA ObliqueCylinder

  • 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!

GeoB 11.5 SA ObliqueCone

Examples

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

GeoB 11.5 SA Q2-3c

352 \pi \;units^{2}

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

GeoB 11.5 SA Q2-4c

198 \pi \;units^{2}

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

GeoB 11.5 SA Q2-5c

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.

GeoB 11.5 SA Q2-6c

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

GeoB 11.5 SA Q2-7c

  • 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

Permanent link to this article: http://newvillagegirlsacademy.org/math/?page_id=4692