Volume of Pyramids and Cones

Objectives/Assignment Find the volume of pyramids and cones. Find the volume of pyramids and cones in real life, such as the nautical prism in Ex. 4. Assignment: WS 12.5A

Finding Volumes of Pyramids and Cones In Lesson 12.4, you learned that the volume of a prism is equal to Bh, where B is the area of the base, and h is the height. From the figure at the left, it is clear that the volume of the pyramid with the same base area B and the same height h must be less than the volume of the prism. The volume of the pyramid is one third the volume of the prism.

Identical isosceles triangles Pyramids A Pyramid is a three dimensional figure with a regular polygon as its base and lateral faces are identical isosceles triangles meeting at a point. Identical isosceles triangles base = quadrilateral base = heptagon base = pentagon

= + + Volume of Pyramids Volume of a Pyramid: V = (1/3) Area of the base x height V = (1/3) Ah Volume of a Pyramid = 1/3 x Volume of a Prism = + +

Exercise #2 Find the volume of the pyramid. height h = 8 m apothem a = 4 m side s = 6 m Volume = 1/3 (area of base) (height) = 1/3 ( 60m2)(8m) = 160 m3 h Area of base = ½ nas a = ½ (5)(6)(4) = 60 m2 s

The Cone + + = Volume of a Cone = A Cone is a three dimensional solid with a circular base and a curved surface that gradually narrows to a vertex. + + = Volume of a Cone =

Exercise #1 = (1/3)(3.14)(1)2(2) = 3.14(1)2(2) = 2.09 m3 = 6.28 m3 Find the volume of a cylinder with a radius r=1 m and height h=2 m. Find the volume of a cone with a radius r=1 m and height h=1 m Volume of a Cylinder = base x height = pr2h = 3.14(1)2(2) = 6.28 m3 Volume of a Cone = (1/3) pr2h = (1/3)(3.14)(1)2(2) = 2.09 m3

Ex. 2: Finding the volume of a cone Find the volume of each cone.

Ex. 2: Finding the volume of a cone Find the volume of each cone.

Ex. 3: Using the Volume of a Cone

Ex. 5: Using the volume of a cone Automobiles. If oil is being poured into the funnel at a rate of 147 milliliters per second and flows out of the funnel at a rate of 42 milliliters per second, estimate the time it will take for the funnel to overflow. (1 mL = 1 cm3).

Ex. 5: Using the volume of a cone First, find the approximate volume of the funnel.

Ex. 5: Using the volume of a cone The rate of accumulation of oil in the funnel is 147 – 42 = 105 mL/s. To find the time it will take for the oil to fill the funnel, divide the volume of the funnel by the rate of accumulation of oil in the funnel as follows: