Finding Volume of Rectangular Prisms

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1 MA.FL.7.G.2.1 Justify and apply formulas for surface area and volume of pyramids, prisms, cylinders, and cones. MA.7.G.2.2 Use formulas to find surface areas and volume of three-dimensional composite shapes. VOLUME Volume - the number of cubic units (un³) used to fill a three-dimensional figure. To find volume, simply use the formula on the reference sheet. **Remember that B is equal to the area of the base** Finding Volume of Rectangular Prisms V= bwh Simply multiply these 3 dimensions. 3m 3x10x2=30x2=60m 3 2 m 10 m m **Remember a cube has all 3 dimensions congruent. 2cm 2x2x2=4x2=8cm 3 Volume of Triangular Prisms Volume of Cylinders

2 Use the formula: V = πr 2 h where π = 3.14 r = radius of the circular base h = height of the cylinder 3in V = πr 2 h V = πr 2 h V=3.14 x 3 2 x 8 8m V = 3.14 x 4 2 x 3 V=3.14 x 9 x 8 V = 3.14 x 16 x 3 8in V= in 3 3m V = m 3 **Worth noting: Volume of Prisms and Cylinders can be found the same way find the area of the base, whatever shape it is, then multiply by the height** Volume of Pyramids Volume of Cones **Volume of Pyramids and Cones can be found the same way as with prisms and cones with one additional step find the area of the base, whatever shape it is, multiply by the height, then divide by 3** SURFACE AREA The surface area is the area all around a 3-dimensional object. Find the area of each face and add. Surface Area of a Cube = 6a 2 (a is the length of the side of each edge of the cube) The surface area of a cube is the area of the six squares that cover it. The area of one of them is a x a, or a 2. Since these are all the same, you can multiply one of them by six, so the surface area of a cube is 6 times one of the sides squared.

3 Surface Area of a Rectangular Prism = 2bh + 2bw + 2hw (b, h, and w are the lengths of the 3 sides) b The surface area of a rectangular prism is the area of the six rectangles that cover it. But we don't have to figure out all six because we know that the top and bottom are the same, the front and back are the same, and the left and right sides are the same. The area of the top and bottom (side lengths b and w) = b x w. Since there are two of them, you get 2bw. The front and back have side lengths of b and h. The area of one of them is b x h, and there are two of them, so the surface area of those two is 2bh. The left and right side have side lengths of h and w, so the surface area of one of them is h x w. Again, there are two of them, so their combined surface area is 2hw. Then add them together and you get the formula: 2bh + 2bw + 2hw Surface Area of Triangular Prism = 2B + Ph Surface Area = Areas of the triangular top and triangular bottom + (Perimeter of the base x height of the prism) Surface Area = 2(Area of triangular top) + Ph Formula for area of a triangle = ½ bh. Multiply the base and height of the triangle, then divide by 2 - After you have found the area of the triangle, multiply it by 2 since there are 2 of them (top and bottom) - Next find P, or the perimeter of the base, by adding all three sides of the base together. After you have found P, multiply by h, or the height of the prism. - The final step is to add these two values together: The area of the top and bottom (2B) + the product of the perimeter and height (Ph) Surface Area of a Cylinder = 2 pi r h + 2 pi r 2 (h is the height of the cylinder, r is the radius of the top) Surface Area = Areas of top and bottom + Area of the side Surface Area = 2(Area of top) + (circumference of top) x height Surface Area = 2(pi r 2 ) + (2 pi r) x h In words, the easiest way is to think of a can. The surface area is the areas of all the parts needed to cover the can. That's the top, the bottom, and the paper label that wraps around the middle. You can find the area of the top (or the bottom). That's the formula for area of a circle (pi r 2 ). Since there is both a top and a bottom, that gets multiplied by two. The side is like the label of the can. If you peel it off and lay it flat it will be a rectangle. The area of a rectangle is the product of the two sides. One side is the height of the can, the other side is the perimeter of the circle, since the label wraps once around the can. So the area of the rectangle is (2 pi r)* h. Add those two parts together and you have the formula for the surface area of a cylinder. Surface Area = 2(pi r 2 ) + (2 pi r)* h

4 Surface Area of a Pyramid = Area of the base + Area of all the triangular sides If there is more than one shape combine in a figure, then you will need to find the volume of each part of the figure and add. MA.FL.7.G.2.1/G.2.2 Practice Problems 1. Find the volume of the following rectangular prisms: c. 8m 5ft 7mm 2m 3ft 4mm 2m 8ft 5mm 2. Find the volume of the following triangular prisms: 3. Find the volume of the following cylinder: 4. Find the volume of the following pyramids: 5. Find the volume of the following cones:

5 6. Find the surface area of these rectangular prisms: A box used to ship computers from the manufacturer to stores is 20 inches long, 24 inches wide, and 12 inches high. What is the surface area of the box? 7. Find the surface area of these triangular prisms: 8. Find the surface area of these cylinders: 9. Find the surface area of these pyramids:

6 10. Volume = Surface Area = 11. Volume = Rina is building a table as shown in the figure. Find the volume of wood she needs for the table. The table leags are 24 inches tall with a diameter of 2 inches.

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