Lesson 12-3 Cylinders and Cones (page 490) Essential Question How can you calculate the area and volume of a cylinder and a cone?

Cylinders

Prism

Cylinder

Cylinders A cylinder is like a prism, but is bases are circles instead of polygons. Therefore, the formulas for areas and volumes are very similar. Just like prisms are either right or oblique, cylinders can be right or oblique. NOTE We will study right prisms and right cylinders.

… Cylinders The segment joining the centers of the bases is an altitude. The length of an altitude is called a height (h). The radius of a base is called a radius of the cylinder.

Right Cylinder altitude center radius

… Cylinders The lateral area of a cylinder is a rectangle. You can easily see this if you take the bases off and unroll the lateral area.

The lateral area of a cylinder equals the circumference of a base times the height of the cylinder. Theorem 12-5 L.A. = C  h L.A. = 2 πrh

TOTAL AREA of a cylinder: T.A. = L.A. + 2B B = base area which is a circle

The volume of a cylinder equals the area of a base times the height of the cylinder. Theorem 12-6 V = B  h

Cones

rectangular pyramid

cone

Cones A cone is like a pyramid, but is bases are circles instead of polygons. Therefore, the formulas for areas and volumes are very similar. Just like pyramids are either regular or oblique, cones can be right or oblique. NOTE We will study regular pyramids and right cones.

… Cones The segment joining the vertex of the cone to the center of its base is the altitude. The length of an altitude is called a height (h), of the cone. The slant height ( ℓ ) is the hypotenuse of a right triangle formed by the altitude and a radius.

right cone altitude center radius slant height vertex

The lateral area of a cone equals half the circumference of a base times the slant height. Theorem 12-7

TOTAL AREA of a Cone: T.A. = L.A. + B B = base area which is a circle

The volume of a cone equals one-third the area of the base times the height of the cone. Theorem 12-8 V = ⅓ B  h

Class Demonstration: Cylinder and Cone with equal height and equal radius.

Example #1: Find L.A., T.A., and volume of a cylinder with h = 8 cm and r = 6 cm. h = 8 cm r = 6 cm

Example #1: Find L.A., T.A., and volume of a cylinder with h = 8 cm and r = 6 cm. h = 8 cm r = 6 cm

Example #1: Find L.A., T.A., and volume of a cylinder with h = 8 cm and r = 6 cm. h = 8 cm r = 6 cm

Example #2: Find L.A., T.A., and volume of a cone with h = 12” and slant height = 13”. h = 12” ℓ = 13” r = 5”

Example #1: Find L.A., T.A., and volume of a cylinder with h = 8 cm and r = 6 cm. h = 12” 13” r = 5”

Example #1: Find L.A., T.A., and volume of a cylinder with h = 8 cm and r = 6 cm. h = 12” 13” r = 5”

CONIC SECTION or Conic : The intersection of a plane with a right circular cone or a right circular cylinder.

This is actually a double-napped cone.

State the names of the main conics. The degenerative conics are given a straight line 6.2 intersecting lines 7.2 parallel lines 8.a point

When the plane intersects one cone and is parallel to the bases the conic section is a CIRCLE. 1.Circle

When the plane intersects one cone and is NOT parallel to the bases the conic section is an ELLIPSE. ircle 2.Ellipse

When the plane intersects one cone and passes through its base the conic section is a PARABOLA. 1.Circle 2.Ellipse 3.Parabola

When the plane intersects the double-napped cone and passes through its bases the conic section is a HYPERBOLA. 1.Circle 2.Ellipse 3.Parabola 4.Hyperbola

The Main Conic Sections 1.Circle 2.Ellipse 3.Parabola 4.Hyperbola

State the names of the main conics. The degenerative conics are given. 5.a straight line 6.2 intersecting lines 7.2 parallel lines 8.a point 1.Circle 2.Ellipse 3.Parabola 4.Hyperbola

Class Activity: Tall vs. Short Cylinder Problem Given equal lateral area, which cylinder has more volume … or are the volumes equal?

BONUS: #18 on page 493 “What in the World” BONUS Unscramble the letters to form “cone” words. Elliptical Chamber BONUS Name of the room. Name of the building it is located. Name the president associated with this room. How can you calculate the area and volume of a cylinder and a cone?

Right Cylinder altitude center radius L.A. = C h T.A. = L.A. + 2B V = B  h

right cone altitude center radius slant height vertex T.A. = L.A. + B V = ⅓ B  h

Assignment Written Exercises on pages 492 & 493 REQUIRED: 3, 9, 10, 15, 17 What are the formulas that are associated with cylinders & cones?

Lesson 12-5 Areas & Volumes of Similar Solids (page 497) Problems #9, #10, and #17 on page 493 verify the main part of this lesson! Don’t miss this theorem!

If the scale factor of two similar solids is a: b, then: (1) the ratio of corresponding perimeters is a : b. (2) the ratio of the base areas, of the lateral areas, and of the total areas is a 2 : b 2. (3) the ratio of the volumes is a 3 : b 3. Theorem 12-11

(1)Two similar pyramids have heights of 12 m and 18 m. Find the ratios. (a)the ratio of the base areas = ___________________ (b)the ratio of the lateral areas = ___________________ (c)the ratio of the total areas = ___________________ (d)the ratio of the volumes = ___________________

(2)Two similar pyramids have volumes of 3 cubic units and 375 cubic units. Find the ratios. (a)the ratio of the heights = ___________________ (b)the ratio of the base areas = ___________________ (c)the ratio of the total areas = ___________________