Area & Volume The student will learn about: area postulates, Cavalieri’s Principle, 1 and the areas and volume of basic shapes.

Perimeter 2 The idea and formulas for perimeter of all plane figures is assumed knowledge of the students. Ask now if you have any questions.

§4.6 Area and Volume Historic idea of area of a rectangle. 3 Unit square (2)  (3) (2.5)  (1.75) However irrational measures and irregular shapes were a problem.

Greek’s Use of Area in Algebra a (b + c) = ab + ac distributive law a bc Completing the square.

Modern Idea of Area Historic idea of area of a rectangle. 5 Continue with ever finer grids and use a limit process.

Postulates - Hilbert 6 Area Postulate - To each polygonal region there corresponds a unique positive real number. Congruence Postulate - If two triangles are congruent, then the triangular regions determined by them have the same area. Area Addition Postulate - If two polygonal regions intersect only in edges and vertices (or do not intersect al all), then the area of their union is the sum of their areas.

Postulates - Hilbert 7 Unit Postulate - The area of a square region is the square of the length of its side. Cavalieri’s Principle – next slide please.

Cavalieri’s Principle for Area 8 L x L’ x If, in two shapes of equal altitude, the sections made by lines at the same distance from their respective bases are always equal in length, then the areas of the shapes are equal.

9 Playing with Sketchpad

10 Area of a Parallelogram. w l Remove triangle from one end and slide it to the other end giving an area of, K = l x w.

11 Area of a Trapezoid. Average base Remove two triangle from the bottom and swivel them to the top giving an area of, K = ½ ( b 1 + b 2 ) x h.

12 Area of a Triangle. Remove two triangle from the top and swivel them to the bottom giving an area of, K = b x ½ h = ½ bh.

13 Area of a Regular Polygon. Rearrange the triangles to form a parallelogram giving an area of, K = ½ ap. ½ p a

14 Area of a Circle. Rearrange the triangles to form a parallelogram giving an area of, K = π r 2. ½ C = π r r

Circular Sector θ

Circular Segment A = sector - triangle

Postulates - Hilbert 17 Unit Postulate - The volume of a rectangular parallelepiped is the product of the altitude and the area of the base. Cavalieri’s Principle – next slide please.

Cavalieri’s Principle for Volmue 18 If, in two solids of equal altitude, the sections made by planes parallel to and at the same distance from their respective bases are always equal, then the volumes of the two solids are equal

19 Prisms. A prism is the figure formed when the corresponding vertices of two congruent polygons, lying in parallel planes are joined. The lines joining the corresponding vertices are called lateral edges. The congruent polygons are called the bases, and the other surfaces are called the lateral faces, or as a group, the lateral surfaces.

20 Prisms. If the lateral edges are perpendicular to the plane of the bases, the prism is a right prism; otherwise, it is an oblique prism.

21 Volume of a Prism. Theorem. The volume of any prism is the product of the altitude and the area of the base. Use Cavalieri’s Principle and the unit postulate.

22 Pyramids. Given a polygonal region R in a plane E, and a point V not in E. the pyramid with base R and vertex V is the union of all segments VQ for which Q belongs to R. The altitude of the pyramid is the perpendicular distance from V to E. R E V

23 Theorem. Theorem. If two pyramids have the same altitude and the same base area, and their bases lie in the same plane, then they have the same volume. Use Cavalieri’s Principle.

24 Theorem. Theorem. The volume of a triangular pyramid is one-third the product of its altitude and its base. In the figure the prism is divided into three pyramids of equal volume.

25 Theorem. Theorem. The volume of any pyramid is one- third the product of its altitude and its base. Use Cavalieri’s Principle on a triangular pyramid with area the same base as the given pyramid.

26 Truncated Pyramids. Volume of a truncated pyramid is that of the full pyramid minus the pyramid cut off the top.

27 Cylinders. Use Cavalieri’s Principle a cylinder may be treated the same as a prism. Hence the volume of a cylinder is the product of its altitude and the area of its base.

28 Cones. Use Cavalieri’s Principle a cone may be treated the same as a pyramid. Hence the volume of a cone is one-third the product of its altitude and the area of its base.

Sphere The Problem is to find a shape with a known volume that compares to a sphere – What do you thinks works?

30 Cavalieri’s Method r r h r h h 2π r 2π(r – h) P Hemisphere only! V HS = V P = r 2 2π r = π r 3 S rh Hence the volume of the sphere is

Sphere The derivative of the volume! The volume of the sphere is The surface area of the sphere is 4 π r 2.

Platonic Solids You should be able to find the volume of the first three of these solids. You should be able to find the surface area of all five of these solids.

34 Summary. We learned about area postulates. We learned about the area of basic shapes. We learned Cavalieri’s Principle for area. We learned about volume postulates. We learned Cavalieri’s Principle for volume. We learned about the volume of basic shapes.

Assignment: 11.1 Hand out Pick’s Theorem