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 thevertex 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?