randFromArray(metricUnits) $._("cubic ") + plural_form(UNIT)
randRange(2, 5) randRange(1, DENOMINATOR - 1)
NUMERATOR / DENOMINATOR randRange(1,4,3) DIMENSIONS[0] DIMENSIONS[1] DIMENSIONS[2] NUMERATOR * NUMERATOR * NUMERATOR DENOMINATOR * DENOMINATOR * DENOMINATOR LENGTH * WIDTH * HEIGHT max(max(LENGTH, WIDTH), HEIGHT) LENGTH/MAX WIDTH/MAX HEIGHT/MAX "#D7ED3A" "#38C77F" "#78D0EF" "#F0B63A" [ [LENGTH/MAX, WIDTH/MAX, HEIGHT/MAX], [LENGTH/MAX, WIDTH/MAX, -HEIGHT/MAX], [LENGTH/MAX, -WIDTH/MAX, -HEIGHT/MAX], [LENGTH/MAX, -WIDTH/MAX, HEIGHT/MAX], [-LENGTH/MAX, WIDTH/MAX, HEIGHT/MAX], [-LENGTH/MAX, WIDTH/MAX, -HEIGHT/MAX], [-LENGTH/MAX, -WIDTH/MAX, -HEIGHT/MAX], [-LENGTH/MAX, -WIDTH/MAX, HEIGHT/MAX] ] 4.0 [ { verts: [3, 2, 1, 0], color: COLOR1 }, { verts: [0, 1, 5, 4], color: COLOR3 }, { verts: [3, 0, 4, 7], color: COLOR2 } ]

What is the volume of this rectangular prism?

Each cube has `fraction(NUMERATOR, DENOMINATOR)` UNIT sides.

init({range: [[-2, 2], [-2, 2]], scale: [100, 100]}); addMouseLayer(); var start1 = 7; for (var i = 1; i < LENGTH;i++){ VERTICES.push([(-LENGTH+2*i)/MAX,-WIDTH/MAX,HEIGHT/MAX]); }; for (var i = 1; i < LENGTH;i++){ VERTICES.push([(-LENGTH+2*i)/MAX,WIDTH/MAX,HEIGHT/MAX]); }; for (var i = 1; i < LENGTH;i++){ VERTICES.push([(-LENGTH+2*i)/MAX,WIDTH/MAX,-HEIGHT/MAX]); }; var start2=VERTICES.length-1; for (var i = 1; i < WIDTH;i++){ VERTICES.push([-LENGTH/MAX,(-WIDTH+2*i)/MAX,HEIGHT/MAX]); }; for (var i = 1; i < WIDTH;i++){ VERTICES.push([LENGTH/MAX,(-WIDTH+2*i)/MAX,HEIGHT/MAX]); }; for (var i = 1; i < WIDTH;i++){ VERTICES.push([LENGTH/MAX,(-WIDTH+2*i)/MAX,-HEIGHT/MAX]); }; var start3 = VERTICES.length-1; for (var i = 1; i < HEIGHT;i++){ VERTICES.push([LENGTH/MAX,-WIDTH/MAX,(-HEIGHT+2*i)/MAX]); }; for (var i = 1; i < HEIGHT;i++){ VERTICES.push([LENGTH/MAX,WIDTH/MAX,(-HEIGHT+2*i)/MAX]); }; for (var i = 1; i < HEIGHT;i++){ VERTICES.push([-LENGTH/MAX,WIDTH/MAX,(-HEIGHT+2*i)/MAX]); }; //graphing the initial object graph.obj = make3dObject(VERTICES, { scale: SCALE }); _.each(FACES, function(face) { graph.obj.addFace(face); }); for (var i = 1;i<LENGTH;i++){ graph.obj.addSketch({ verts: [start1+i,start1+i+LENGTH-1,start1+i+2*LENGTH-2], opacityValue:0.5}); }; for (var i = 1;i<WIDTH;i++){ graph.obj.addSketch({ verts: [start2+i,start2+i+WIDTH-1,start2+i+2*WIDTH-2], opacityValue:0.5}); }; for (var i = 1;i<HEIGHT;i++){ graph.obj.addSketch({ verts: [start3+i,start3+i+HEIGHT-1,start3+i+2*HEIGHT-2], opacityValue:0.5}); }; graph.obj.setPos([0, 0, 5]); graph.obj.rotate(1,0 , 0, PI/3); graph.obj.rotate(0, 1, 0, 0); graph.obj.rotate(0, 0, 1, PI/6); graph.obj.doDraw(); GRAPH = graph;

N * NUMERATOR3 / DENOMINATOR3 UNIT_TEXT

The volume of a cube is equal to its side length cubed.

The side length of one cube is `fraction(NUMERATOR, DENOMINATOR)` UNIT, so the volume of one cube is `\left(fraction(NUMERATOR, DENOMINATOR)\right)^3 = fraction(NUMERATOR3, DENOMINATOR3)` plural_form(UNIT).

The volume of a shape is measured by counting the number of cubic plural_form(UNIT).

Carefully count the cubes. Some of the cubes might be hidden behind other cubes. Try to visualize all of the cubes.

There are a total of `N` cubes.

The volume is `N \times fraction(NUMERATOR3, DENOMINATOR3) = fractionReduce(N * NUMERATOR3, DENOMINATOR3)` UNIT_TEXT.

randRange(2, 5) randRange(1, DENOMINATOR - 1) NUMERATOR * NUMERATOR * NUMERATOR DENOMINATOR * DENOMINATOR * DENOMINATOR NUMERATOR / DENOMINATOR randRange(3,8) "#D7ED3A" "#38C77F" "#78D0EF" "#F0B63A" [] 2.0 [] 0

What is the volume of this shape?

Each cube has `fraction(NUMERATOR, DENOMINATOR)` UNIT sides.

You can click and drag to rotate the shape.

init({range: [[-2, 2], [-2, 2]], scale: [100, 100]}); addMouseLayer(); //cubes stores the positions of the cubes var cubes = [[0,0,0]]; // inArray checks whether a point is in the array already. // For example inArray([0,0,0],[[0,0,0,],[0,0,1]]) = true, // but inArray([0,0,2],[[0,0,0],[0,0,1]]) = false. This is // only used for an ordered triple and an array of ordered // triples. var inArray = function(point,array) { for (var i=0; i< array.length;i++) { if (point[0] == array[i][0] && point[1] == array[i][1] && point[2] == array[i][2]){ return true; } } return false; } //This will store the boundary of the current set of cubes //Add cubes to cubes from the current boundary until we have N cubes for (var i = 0; i < N-1; i++){ var boundary = []; // Figuring out the current boundary for (var j = 0; j<cubes.length;j++){ var cube0 = [cubes[j][0]+1,cubes[j][1],cubes[j][2]]; var cube1 = [cubes[j][0],cubes[j][1]+1,cubes[j][2]]; var cube2 = [cubes[j][0],cubes[j][1],cubes[j][2]+1]; var cube3 = [cubes[j][0]-1,cubes[j][1],cubes[j][2]]; var cube4 = [cubes[j][0],cubes[j][1]-1,cubes[j][2]]; var cube5 = [cubes[j][0],cubes[j][1],cubes[j][2]-1]; if (!inArray(cube0,cubes) && ! inArray(cube0,boundary)){ boundary.push(cube0); } if (!inArray(cube1,cubes) && ! inArray(cube1,boundary)){ boundary.push(cube1); } if (!inArray(cube2,cubes) && ! inArray(cube2,boundary)){ boundary.push(cube2); } if (!inArray(cube2,cubes) && ! inArray(cube2,boundary)){ boundary.push(cube2); } if (!inArray(cube3,cubes) && ! inArray(cube3,boundary)){ boundary.push(cube3); } if (!inArray(cube4,cubes) && ! inArray(cube4,boundary)){ boundary.push(cube4); } } cubes.push(boundary[randRange(0,boundary.length-1)]); } // Addind vertices and faces for (var i = 0; i < N; i++) { var index = VERTICES.length; VERTICES.push([cubes[i][0]+0.5,cubes[i][1]+0.5,cubes[i][2]-0.5]); VERTICES.push([cubes[i][0]+0.5,cubes[i][1]-0.5,cubes[i][2]-0.5]); VERTICES.push([cubes[i][0]-0.5,cubes[i][1]-0.5,cubes[i][2]-0.5]); VERTICES.push([cubes[i][0]-0.5,cubes[i][1]+0.5,cubes[i][2]-0.5]); VERTICES.push([cubes[i][0]+0.5,cubes[i][1]+0.5,cubes[i][2]+0.5]); VERTICES.push([cubes[i][0]+0.5,cubes[i][1]-0.5,cubes[i][2]+0.5]); VERTICES.push([cubes[i][0]-0.5,cubes[i][1]-0.5,cubes[i][2]+0.5]); VERTICES.push([cubes[i][0]-0.5,cubes[i][1]+0.5,cubes[i][2]+0.5]); //only add a face if there is not another cube next to it if (!inArray([cubes[i][0]+1,cubes[i][1],cubes[i][2]],cubes)) { FACES.push({ verts: [index,index+4,index+5,index+1], color: COLOR1 }); surfaceArea++; } if (!inArray([cubes[i][0]-1,cubes[i][1],cubes[i][2]],cubes)) { FACES.push({ verts: [index+7,index+3,index+2,index+6], color: COLOR1 }); surfaceArea++; } if (!inArray([cubes[i][0],cubes[i][1]+1,cubes[i][2]],cubes)) { FACES.push({ verts: [index,index+3,index+7,index+4], color: COLOR2 }); surfaceArea++; } if (!inArray([cubes[i][0],cubes[i][1]-1,cubes[i][2]],cubes)) { FACES.push({ verts: [index+5,index+6,index+2,index+1], color: COLOR2 }); surfaceArea++; } if (!inArray([cubes[i][0],cubes[i][1],cubes[i][2]+1],cubes)) { FACES.push({ verts: [index+4,index+7,index+6,index+5], color: COLOR3 }); surfaceArea++; } if (!inArray([cubes[i][0],cubes[i][1],cubes[i][2]-1],cubes)) { FACES.push({ verts: [index+3,index,index+1,index+2], color: COLOR3 }); surfaceArea++; } } // graphing the initial object graph.obj = make3dObject(VERTICES, { scale: SCALE, facesTransparent: false, faceBorder:true }); _.each(FACES, function(face) { graph.obj.addFace(face); }); graph.obj.setPos([0, 0, 5]); graph.obj.rotate(1,0 , 0, PI/3); graph.obj.rotate(0, 1, 0, 0); graph.obj.rotate(0, 0, 1, PI/8); graph.obj.doDraw(); GRAPH = graph; var mouseTarget = mouselayer.rect(0, 0, 400, 400).attr({ fill: "#000", opacity: 0.0 }); $(mouseTarget[0]).bind("vmousedown", function(event) { event.preventDefault(); GRAPH.lastX = event.pageX; GRAPH.lastY = event.pageY; $(document).bind("vmousemove vmouseup", function(event) { event.preventDefault(); var dx = (event.pageX - GRAPH.lastX) / 200 * PI; var dy = (event.pageY - GRAPH.lastY) / 200 * PI; GRAPH.lastX = event.pageX; GRAPH.lastY = event.pageY; var xAxis = normalize(GRAPH.obj.perspective[1]); var yAxis = normalize(GRAPH.obj.perspective[0]); GRAPH.obj.rotate(xAxis[0], xAxis[1], xAxis[2], -dx / 2); GRAPH.obj.rotate(yAxis[0], yAxis[1], yAxis[2], -dy / 2); GRAPH.obj.doDraw(); if (event.type === "vmouseup") { $(document).unbind("vmousemove vmouseup"); } }) });

N * NUMERATOR3 / DENOMINATOR3 UNIT_TEXT

The volume of a cube is equal to its side length cubed.

The side length of one cube is `fraction(NUMERATOR, DENOMINATOR)` UNIT, so the volume of one cube is `\left(fraction(NUMERATOR, DENOMINATOR)\right)^3 = fraction(NUMERATOR3, DENOMINATOR3)` plural_form(UNIT).

The volume of a shape is measured by counting the number of cubic plural_form(UNIT).

Carefully count the cubes. Some of the cubes might be hidden behind other cubes. Try to visualize all of the cubes.

There are a total of `N` cubes.

The volume is `N \times fraction(NUMERATOR3, DENOMINATOR3) = fractionReduce(N * NUMERATOR3, DENOMINATOR3)` UNIT_TEXT.