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 SA = (LA) + 2B 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 SA = LA + 2B or SA = 2πrh + 2πr2 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 SA = LA + B 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 SA = LA + B 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?