Volume & Surface Areaof Three-Dimensional Figures What’s it all about?
Volume• Volume is the three-dimensional space taken up by a polyhedron (solid figure). • It is measured in cubic units
Volume FormulasThere are two general formulas we use for the two basic shapes of three-dimensional figures: PRISMS & PYRAMIDSPrisms go straight up Pyramids come to afrom their Base to a point
Volume Formulas for Prisms • In general, the Volume formulas for any Prism is V = Bh where B is the area of the BaseRectangular Base Round Base A = lw A = r 2
Let’s try oneV = Bh or lwh V = (8)(3)(4)V = 96 ft cubed
Now try this one Find the volume of a rectangular solid (prism) shedLength: 5.5 feet Width: 7 feet Height: 9 feet If the grain to be stored in this shed sells for $ 15 per cubic foot, how much can the whole shed sell for?
Answer• A = lwh• A = (5.5)(7)(9)• A = 346.5 cubic feet• 346.5 x $15 = $5197.50
Cylinders work the same way V = Bh or r h 2 Area of the Circular Base is = (3.14) (5)^2 And then times the height V = (78.5)(9) = 706.5 cubic ins
Again, you try one1. Find the volume of a cylinder (circular prism) with the following dimensions: Radius of the circular base is 4 feet, height is 10 feet. How much grain can it hold?2. What if a supporting pole with a radius of one foot ran down the center of this flat silo? Then how much grain could it hold?
Answer1) Volume of the original cylinder V = (3.14)(4)^2 (10) = 502.4 cubic feet2) Volume of the pole V = 3.14 (1)^2 (10) = 31.4 cubic feet3) New Volume 502. 4 – 31.4 = 471 cubic feet
Now let’s look at the Volume of Pyramids (& cones) Their general formula is: V = 1/3 BhWe can take the previous examples and just divide their volume by 3. Given the sameheight and Base, the volume of pyramids and cones are 1/3 the correspondingly sized prisms.V = (1/3)(3.14)(3)^2(4)V = 37.68 cubic meters
Surface Area Surface Area is two-dimensional.It is the amount of coverage on the outside of the figure. It includes the top, bottom, and lateral sides.
Rectangular Solids are easy Add the surface areas of all six sides of this prism to get its total surface area.Top & Bottom: A = (8)(3) = 24*2 = 48 square feetFront & Back: A = (8)(4) = 32*2 = 64 square feetLeft & Right: A = (4)(3) = 12*2 = 24 square feet48+64+24 = 136 square feet total surface area
Surface Area of Square PyramidsFor any three-dimensional figure, study the shapes that make up the top, bottom & sides. Calculate the areas for each of these shapes separately and add them together at the end. What shapes do you see? How many?Side of Square base: 5 Slant height of each triangle: 8Area of Base = 25Area of triangle = (1/2)(5)(8) = 20Area of 4 lateral sides = 80Total Surface Area = 80+25 = 105 sq ins
And now the Surface Area of the CylinderCalculate the top & bottom separately from the lateral side, and then add them together. Area of circular bases: 2(3.14)(r)^2 Area of the lateral side is the circumference of the circular base * height
Let’s put some numbers in For this cylinder: Radius = 6 and Height = 8Area of Top & Bottom: 2(3.14)(6)^2 = 226.08 Area of Lateral Side: (12)(3.14)(8) = 301.44 226.08 + 301.44 = 527.52 square units
Now you tryFind the surface area of a cylinder with a radius of 3.5 inches and a height of 5 inches.
AnswerArea of top & Bottom: 2(3.14)(3.5)^2 = 76.93Area of lateral side: (3.14)(7)(5) = 109.9Add: 76.93 + 109.9 = 186.83 square units