 Volume of Objects Composed of Right Rectangular Prisms MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts Volume and Surface Area
for Composed Boxes
 Volume and Surface Area of a right rectangular prism (a box):  Vbox = l • w • h SAbox = 2lh + 2hw + 2lw V = volume in cubic units; SA = surface area in square units;    l = length, w = width, h = height. NGMS-7 CCSS 7  1. a) Find the volume of the figure at the right which is composed of right rectangular prisms in cubic feet. Solution:   Find the volume of each right rectangular prism (box) and add the volumes together. • Large box: (6.5)(4.8)(5.2) = 164.24 cubic feet • Small box: (4.2)(2)(2.5) = 21 cubic feet • Total volume: 164.24 + 21 = 183.24 cubic feet b) Find the surface area of the figure in square feet. Solution:   Large Box: SA = 2lh + 2hw + 2lw (SA, h, w and l are surface area, height, width and length) SA = 2lh + 2hw + 2lw = 2(6.5)(5.2) + 2(5.2)(4.8) + 2(6.5)(4.8) = 179.92 sq.ft. Small Box: SA = 2lh + 2hw + 2lw = 2(4.2)(2.5) + 2(2.5)(2) + 2(4.2)(2) = 47.8 sq.ft. The bottom of the small box is not a "surface" so subtract the area of that side from the surface area of BOTH boxes. Bottom area: 8.4 sq.ft. Total surface area: (179.92 - 8.4) + (47.8 - 8.4) = 171.82 + 39.4 = 210.92 sq. ft.  2. Ten colored boxes of the same size are stacked as shown. Each box is a cube measuring 2 feet on each side. a) Find the total volume of this stack in cubic feet. Solution:   • Volume of each box is 2 • 2 • 2 = 8 cubic feet. • Volume of 10 boxes is 10 • 8 = 80 cubic feet. b) Find the surface area of this stack in square feet. Solution:   • Area of each face of a box is 2 • 2 = 4 sq. feet. • Top of each column: 4 • 4 = 16 sq. feet • Front (and back) facing: 2(10) • 4 = 80 sq. feet • Back end view: 4 • 4 = 16 sq. feet • Front end view: 4 • 4 = 16 sq. feet • Bottom of stack: 4 • 4 = 16 sq. feet • Total surface area = 144 square feet Each box is a right rectangular prism. A cube is a right rectangular prism whose length, width and height are of equal measure.  3. a) Find the volume of this solid, composed of a right rectangular prisms, in cubic inches. Solution:   • Visualize a horizontal line on the right side of the figure forming three boxes. Find the volume of each of the boxes. • Large Bottom Box: V = 9 • 17 • 3 = 459 cubic inches • Medium Top Box: V = 9 • 7 • 3 = 189 cubic inches • Small Middle Box: V = 4 • 6 • 1 = 24 cubic inches Total Volume = 459 + 189 + 24 = 672 cubic inches b) Find the surface area of this solid, in square inches. Solution:   • Find the surface area of each of the boxes, then adjust for overlapping. • Large Bottom Box: SA = 2lh + 2hw + 2lw = 2(9)(3) + 2(3)(17) + 2(9)(17) = 462 sq. in. • Medium Top Box: SA = 2(9)(3) + 2(3)(7) + 2(9)(7) = 222 sq. in. • Small Middle Box: SA = 2(4)(1) + 2(1)(6) + 2(4)(6) = 68 sq. in. You must deal with overlapping sections. • The bottom of the medium top box and the portion it covers on the large bottom box must be subtracted. 2(7)(9) = 126 sq.in. • The bottom and back portion of the small box and portion where it covers the other boxes must be subtracted. Bottom: 2(4)(6) = 48 sq.in. Back side: 2(1)(4) = 8 sq.in. Total Surface Area = 462 + 222 + 68 - (126 + 48 + 8) = 570 square inches  4. An opening has been cut completely through the interior of this block. Both the block and the opening are right rectangular prisms. a) Find the volume of block after the hole has been cut, in cubic feet. Solution:   Be careful! This problem shows measurements in both feet and inches, but the question asks for the answer in cubic feet. We will change the inches to feet. 8 inches = 2/3 ft. 12 in = 1 ft. • Volume of the block = 3 cubic feet. • Volume of the opening = 1 cubic foot • Volume of block after cut out = 2 cubic feet b) Find the surface area of the block after the cut out, in square feet. Solution:   • Area block top (& bottom) - hole area top (bottom) = 2[(2)(1) - ( )(1)] = 2 sq. ft. • Area block front (& back) = 2[(2)(1½)] = 6 sq.ft. • Area block sides (left & right) = 2[(1)(1½)] = 3 sq.ft. We need to add in the surface areas needed to cover the interior sides of the hole. • Area hole front (& back) = 2[(1)(1½)] = 3 sq. ft. • Area hole sides (left & right) = 2[ ( )(1½)] = 2 sq. ft. • Total surface area = 2 + 6 + 3 + 3 + 2 = 16 sq. ft 