What is the volume of a cube with side length s
?
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
What is the surface area of a cube with side length s
?
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
.
What is the volume of a cylinder with base radius r
and height 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
.
What is the surface area of a cylinder with base radius r
and height 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
.
What is the volume of a rectangular box with width w
, height h
, and depth d
?
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
.
What is the surface area of a rectangular box with width w
, height h
, and depth d
?
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 )
.
What is the volume of a triangular prism of depth d
whose base has width w
and height h
?
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
.