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Surface Area and Volume
Prisms & Cylinders
Surface Area
Prior Knowledge
• A polyhedron is a three – dimensional figure,
whose surfaces are polygons. Each polygon is
a face of the polyhedron
• An edge Is a segment that is formed by the
intersection of two faces.
• A vertex is a point where three of more edges
intersect
Prism
• A Prism is a polyhedron with two congruent parallel faces, called bases.
The other faces are lateral faces.
– You name a prism using the shape of the base
• The altitude of a prism is the perpendicular segment that joins the planes
of the bases, the height is the length of the altitude.
Oblique vs. Right
In a right prism the lateral faces are rectangles,
and the altitude is a lateral edge.
In an oblique prism some of the lateral faces are
non-rectangular,
* in this class you can assume that all prisms are
right unless otherwise stated
Lateral Area Vs Surface Area
• Lateral Area (LA) is the sum of the areas of the
lateral faces
• Surface Area (SA) is the sum of the lateral area
and the area of the two bases
Formulas
• LA = ph
– Where p is the perimeter of the bases and h is the
height of the prism
• SA = (LA) + 2B
– Where LA is the lateral area and B is the area of
the Base
Example 1
• Find the Lateral Area and Surface Area
Example 2
• Find the Lateral Area and Surface Area
Cylinder
• A cylinder is a solid that has two congruent //
bases that are circles
– An altitude of a cylinder is a perpendicular
segment that joins the planes of the bases.
– The height (h) of a cylinder is the length of the
altitude
Oblique vs. Right
• In a right cylinder the segment joining the
centers of the bases is an altitude
• In an oblique cylinder the segment joining the
centers in not perpendicular to the planes
containing the base.
* in this class you can assume that all prisms are
right unless otherwise stated
Formulas
• LA = 2πrh or LA = πdh
– Where r is the radius and
h is the height
• SA = LA + 2B or SA = 2πrh + 2πr2
– Where LA is the lateral area, B is the area of the
base, r is the radius and h is the height
Example 1
• Find the Lateral Area and Surface Area
Example 2
Prisms & Cylinders
Volume
Volume
• Volume (V) is the space that a figure occupies,
it is measured in cubic units
Volume of a Prism
• V = Bh
• Where B is the Area of the base
and h is the height
Example 2
• What is the volume of the rectangular prism?
Example 3
• What is the volume of the triangular Prism
Volume of a Cylinder
• V = Bh or V = πr2h
• Where B is the area of the base, h is the height and r is
the radius
Example 1
• Find the volume of the cylinder
Example 2
• Find the volume of the cylinder
Composite Figures
• Find the Volume of this figure
Pyramids and Cones
Surface Area
Pyramid
• A pyramid is a polyhedron in which one face,
the base, can be any polygon and the other
faces, lateral faces, are triangles that meet at a
common vertex called the vertex of the
pyramid
• The altitude of a pyramid is a perpendicular segment from the
vertex of the pyramid to the plane of the base – the length of
the altitude = height
Regular Pyramid
• A pyramid whose base is a regular polygon
and whose lateral faces are congruent
isosceles triangles.
• The slant height, l , is the length of the
altitude of a lateral face of the pyramid.
(In this class all pyramids are regular unless otherwise stated)
Formulas For Pyramids
• LA = ½ p l
– Where p is the perimeter of the base and l is the
slant height of the pyramid
• SA = LA + B
– Where B is the area of the base of the pyramid
Example 1
• A square pyramid has base edges of 5 m and a
slant height of 3 m. What is the surface area
of the pyramid?
Example 2
• Find the Surface Area of the Pyramid
Example 3
Cone
• A cone is a solid that has one base and a
vertex that is not in the same plane as the
base
– The base of a cone in a circle
– In a right cone the altitude is a
perpendicular segment from the
vertex to the center of the base, the height = length
of the altitude
– The slant height l is the distance from the vertex to a
point on the edge of the base
Formulas For Cones
• LA = ½ 2πrl
or LA = πrl
– Where r is the radius, and l is the slant height
• SA = LA + B
– Where is B is the area of the base
Example 1
• The radius of the base of a cone is 16 m. Its
slant height is 28 m. What is the surface area
in terms of π?
Example 2
Example 3
Pyramids and Cones
Volume
Volume of a Pyramid
• V = ⅓Bh
– Where B is the Area of the base and h is the
height
Example 1
• A sports arena shaped like a pyramid has a
base area of about300,000 ft2 and a height of
321 ft. What is the approximate volume of the
arena?
Example 2
Example 3
Volume of a Cone
• V = ⅓Bh or V=⅓πr2h
– Where B is the Area of
the Base, h is the height,
and r is the radius
Example 1
Example 2
Example 3
• A small child’s teepee is 6 ft high with a base
diameter of 7 ft. What is the volume of the
child’s teepee to the nearest cubic foot?