Volume and Surface Area Unit 28 Remember Volume of a solid figure is calculated in cubic units and measures three dimensions.

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1 Volume and Surface Area Unit 28 Remember Volume of a solid figure is calculated in cubic units and measures three dimensions. Surface Area is calculated in square units and measures two dimensions. Prisms and Cylinders have parallel, identical bases. Volume is calculated using the formula V = (Base area) x h (a cross product) When using the volume formulas remember that the height measurement is always made perpendicular to the base ) Find the volume of a right prism with a 4-5 right triangle base and a height of 0.0cm 0.0cm.0cm 4.0cm 2) Find the volume of the cylinder r = 5cm ) A PVC water pipe has an inside diameter of.029. If water flows through the pipe with a velocity of ft/s, How many gallons of water flow through the pipe in 0 minutes?

2 Surface Area The surface of a shape includes the outside of the top and bottom and the sides Surface area is calculated adding up the surface on the sides, called the lateral area, and the area of both bases The area of the sides of a prism or cylinder is called the Lateral area 4) Find the surface area of the prism 4-5 right triangle base and a height of 0.0cm 0.0cm.0cm 4.0cm 5) Find the surface area of the cylinder r = 5cm 6) Consider two different.000 liter cylindrical containers. One has a diameter of 2.80 cm and the other has a diameter of 0.40 cm. Which container has less surface area? If the cost of material if $2.00 per square meter and your company uses 6,000,000 of these containers each year, how much can you save by using the more efficient container?

3 7) Refer to #2 on page 58 in the text. The figure shown is the cross section of a road. It is shaped like a rectangle with a segment of a circle on top. The road is 2 miles long. a) Find the area in square yards of the impervious surface (you would need this to calculate the amount of water runoff you will need to control) b) Find the volume in cubic yards of concrete need to build the road way.

4 Pyramids and Cones Units 29 Pyramids and cones have a geometric base that rises to a point or vertex Cones have a circular base and Pyramids have regular polygon base Volumes To find the volume of a pyramid use the formula V = B h The volume of cone is the same formula as for a pyramid. V = The h, or height of the pyramid or cone is perpendicular to the base.. Find the volume of the pyramid B h 2cm 2. Find the volume of the cone 2cm radius = 5cm. A water tank is in the shape of an inverted cone. If the tank has capacity of 000gallons and a top diameter of 8 ft, How tall is the tank? 4. How many gallons are left in the tank, when the water level is half way down?

5 Surface Areas The surface area of pyramid or cone is found by adding up the area of the base and each of the triangular sides. The surface area of a cone is found the same way. Notice that the side unwraps as a triangle and A πrh s 2 the side or lateral area of a cone is = (2 ) where h s is the slant height along the face of the cone The slant height h s can be calculated using Pythagorean Theorem and the radius and perpendicular height 5. Find the Surface area of the pyramid 2cm 6. Find the surface area of the cone 2cm radius = 5cm 7. If we want to fabricate the water tank from the last page out of sheet metal, how many square feet do we need

6 A part of a pyramid or a cone is called a frustum The volume of a frustum can be found by creating a picture of the pyramid or cone before the top is cut off and then subtracting the cut off tip from the volume of the whole cone or pyramid To do this you need to use similar triangles to find the length of the cut off tip 8. Find the volume of a tapered steel shaft which is 9.00 in long and has large end diameter of.75 inches and a small end Diameter of.25 inches. Side view of shaft Cut off tip There are formulas to use instead. Volume V = h( A + A + A A ) B b B b Where A B is the area of the large base and A b is the area of the small base 2 The lateral area LA = h ( C + C ) s B b Where C B is the circumference of the large base and C b is the circumference of the small base 9. Find the lateral area of the shaft above.

7 Spheres and Composite figures Unit 0 Spheres are balls. The volume and surface area of spheres are developed using calculus so just look up the formulas when you need them. Volume = Surface Area = 4 π r 4π r 2 0. Find the surface area and volume of the sphere r =.5cm. If the sphere above is a steel ball bearing and steels weighs 7.88g / cubic cm, How many ball bearings to make kg? Composite figures Many shapes are made up from other shapes. Just like in areas you can add up the parts to get to the whole or subtract out void from a whole.. Find the capacity of the Grain Hopper in cubic feet. Assume that the dimensions given are inside dimensions. Top Inside Diameter = 8.0 ft 6.0 ft 4.0 ft 2. Find the square ft of sheet metal required to fabricate the container