randRange( 1, 7 )

What is the volume of a cube with side length s?

init({ range: [ [-1, 4], [-0.6, 3.1] ] }); path([ [0, 0], [0, 2], [2, 2], [2, 0], true ]); path([ [2, 0], [3, 1], [3, 3], [1, 3], [0, 2] ]); path([ [2, 2], [3, 3] ]); label( [1, 0], s, "below" );

s * s * s

The volume of a cube with side length s is V = s^3.

\qquad V = s^3

\qquad \hphantom{V} = s \cdot s \cdot s

\qquad \hphantom{V} = s * s * s

randRange( 1, 7 )

What is the surface area of a cube with side length s?

init({ range: [ [-1, 4], [-0.6, 3.1] ] }); path([ [0, 0], [0, 2], [2, 2], [2, 0], true ]); path([ [2, 0], [3, 1], [3, 3], [1, 3], [0, 2] ]); path([ [2, 2], [3, 3] ]); label( [1, 0], s, "below" );

6 * s * s

The area of each face is simply the area of a square: s^2 = s * s.

A cube has six faces so the total surface area is 6 \cdot s * s = 6 * s * s.

randRange( 2, 4 ) randRange( 5, 9 )

What is the volume of a cylinder with base radius r and height h?

init({ range: [ [-1, 4], [-2, 4] ] }); arc( [1.5, 3], [1.5, 0.4], 0, 48 ); arc( [1.5, 3], [1.5, 0.4], 70, 360 ); path([ [0, -1], [0, 3] ]); path([ [3, -1], [3, 3] ]); arc( [1.5, -1], [1.5, 0.4], 0, 180, { strokeDasharray: "- " } ); arc( [1.5, -1], [1.5, 0.4], 180, 360 ); path([ [1.5, 3], [3, 3] ]); label( [2.25, 3], r, "above" ); label( [3, 1], h, "right" );
Math.PI * r * r * h

The area of the base is simply the area of a circle: \pi r^2 = \pi \cdot r^2 = r * r \pi.

The volume of the cylinder is the area of the base times the height: B \cdot h = r * r\pi \cdot h = r * r * h\pi.

randRange( 2, 4 ) randRange( 5, 9 )

What is the surface area of a cylinder with base radius r and height h?

init({ range: [ [-1, 4], [-2, 4] ] }); arc( [1.5, 3], [1.5, 0.4], 0, 48 ); arc( [1.5, 3], [1.5, 0.4], 70, 360 ); path([ [0, -1], [0, 3] ]); path([ [3, -1], [3, 3] ]); arc( [1.5, -1], [1.5, 0.4], 0, 180, { strokeDasharray: "- " } ); arc( [1.5, -1], [1.5, 0.4], 180, 360 ); path([ [1.5, 3], [3, 3] ]); label( [2.25, 3], r, "above" ); label( [3, 1], h, "right" );
Math.PI * 2 * r * ( r + h )

The areas of the top and the base are simply the area of a circle: \pi r^2 = \pi \cdot r^2 = r * r \pi.

The lateral surface area is the same as the area of a rectangle with height h and width equal to the circumference of the base.

That circumference is 2 \pi r = 2\pi \cdot r = 2 * r\pi.

Thus, the lateral surface area is wh = 2 * r \pi \cdot h = 2 * r * h \pi.

The total surface area is r * r \pi + r * r \pi + 2 * r * h \pi = 2 * r * ( r + h )\pi.

randRange( 5, 9 ) randRange( 2, 4 ) randRange( 2, 4 )

What is the volume of a rectangular box with width w, height h, and depth d?

init({ range: [ [-1, 7], [-1, 3.5] ] }); var wv = [ 4, 0 ]; var hv = [ 0, 2 ]; var dv = [ 2, 1 ]; path([ [0, 0], hv, [wv[0] + hv[0], wv[1] + hv[1]], wv, true ]); path([ hv, [hv[0] + dv[0], hv[1] + dv[1]], [wv[0] + hv[0] + dv[0], wv[1] + hv[1] + dv[1]], [wv[0] + dv[0], wv[1] + dv[1]], wv ]); path([ [wv[0] + hv[0], wv[1] + hv[1]], [wv[0] + hv[0] + dv[0], wv[1] + hv[1] + dv[1]] ]); label( [0.5 * wv[0], 0.5 * wv[1]], w, "below" ); label( [0.5 * hv[0], 0.5 * hv[1]], h, "left" ); label( [wv[0] + 0.5 * dv[0], wv[1] + 0.5 * dv[1]], d, "above" );

w * h * d

The volume of a right rectangular prism with width w, height h, and depth d is V = whd.

In this case, the volume is w \cdot h \cdot d = w * h * d.

randRange( 5, 9 ) randRange( 2, 4 ) randRange( 2, 4 )

What is the surface area of a rectangular box with width w, height h, and depth d?

init({ range: [ [-1, 7], [-1, 3.5] ] }); var wv = [ 4, 0 ]; var hv = [ 0, 2 ]; var dv = [ 2, 1 ]; path([ [0, 0], hv, [wv[0] + hv[0], wv[1] + hv[1]], wv, true ]); path([ hv, [hv[0] + dv[0], hv[1] + dv[1]], [wv[0] + hv[0] + dv[0], wv[1] + hv[1] + dv[1]], [wv[0] + dv[0], wv[1] + dv[1]], wv ]); path([ [wv[0] + hv[0], wv[1] + hv[1]], [wv[0] + hv[0] + dv[0], wv[1] + hv[1] + dv[1]] ]); label( [0.5 * wv[0], 0.5 * wv[1]], w, "below" ); label( [0.5 * hv[0], 0.5 * hv[1]], h, "left" ); label( [wv[0] + 0.5 * dv[0], wv[1] + 0.5 * dv[1]], d, "above" );

2 * ( w * h + w * d + h * d )

The surface area of a right rectangular prism with width w, height h, and depth d is T = 2wh + 2wd + 2hd.

In this case, the surface area is 2 \cdot (w \cdot h) + 2 \cdot (w \cdot d) + 2 \cdot (h \cdot d) = 2 * ( w * h + w * d + h * d ).

randRange( 4, 7 ) randRange( 3, 4 ) randRange( 5, 9 )

What is the volume of a triangular prism of depth d whose base has width w and height h?

init({ range: [ [-1, 7], [-1, 3.5] ] }); var wv = [ 3, 0 ]; var hv = [ 2, 2 ]; var dv = [ 3, 1 ]; path([ [0, 0], dv, [hv[0] + dv[0], hv[1] + dv[1]], [wv[0] + dv[0], wv[1] + dv[1]], dv ], { strokeDasharray: "- ", stroke: "#ddd" }); path([ [0, 0], hv, wv, true ]); path([ hv, [hv[0] + dv[0], hv[1] + dv[1]], [wv[0] + dv[0], wv[1] + dv[1]], wv ]); // altitude path([ [hv[0], 0], hv ]); label( [0.5 * wv[0], 0.5 * wv[1]], w, "below" ); label( [hv[0], 0.5 * hv[1]], h, "left" ); label( [wv[0] + 0.5 * dv[0], wv[1] + 0.5 * dv[1]], d, "below right" );

w * h * d / 2

The area of the triangular base is \frac12 w h = \frac12 (w \cdot h) = w * h / 2.

The volume of the prism is the area of the base times the depth: B \cdot d = w * h / 2 \cdot d = w * h * d / 2.