What is the volume of this object? Drag on the object to rotate it.
DEPTH * BASE * HEIGHT / 2 cubic units
The volume of a prism is the area of the base \times depth.
In this object, one of the triangles is the base.
The area of a triangle is
\frac{1}{2}\ \text{BASE_TEXT} \times \text{HEIGHT_TEXT}.
Orient the triangle like this, and find the base and height from the object:
From the object, we find that the base of the triangle
is BASE and the height is
HEIGHT.
The area of the triangle is then
\frac{1}{2}BASE \times HEIGHT
= BASE * HEIGHT / 2.
From the object, we can then find that the depth is
DEPTH.
So, the final volume is
BASE * HEIGHT / 2 \times DEPTH
= BASE * HEIGHT * DEPTH / 2
cubic units.
What is the volume of this box? Drag on the box to rotate it.
LENGTH * WIDTH * HEIGHT cubic units
The volume of a box is the length \times width \times height.
From the figure, we can find that the lengths of the sides are
LENGTH, WIDTH, and HEIGHT.
LENGTH \times WIDTH \times HEIGHT =
LENGTH * WIDTH * HEIGHT
So, the volume of the box is LENGTH * WIDTH * HEIGHT cubic units.
What is the surface area of this box? Drag on the box to rotate it.
2 * (LENGTH * WIDTH + LENGTH * HEIGHT + WIDTH * HEIGHT) square units
To find the surface area, find the areas of each of the faces, and add all the areas up.
To see this better, let's try unwrapping the box:
We can group the faces together based on size and color:
We can calculate the area of each of the pieces now:
2 \left(WIDTH \times
HEIGHT\right) +
2 \left(LENGTH \times
WIDTH\right) +
2 \left(HEIGHT \times
LENGTH\right)
\qquad= 2 * WIDTH * HEIGHT +
2 * LENGTH * WIDTH +
2 * HEIGHT * LENGTH
\qquad=
2 * WIDTH * HEIGHT +
2 * LENGTH * WIDTH +
2 * HEIGHT * LENGTH
So, the total surface area of the box is
2 * WIDTH * HEIGHT + 2 * LENGTH * WIDTH + 2 * HEIGHT * LENGTH
square units.
What is the volume of this cube? Drag on the cube to rotate it.
The volume of a box is the
length \times
width \times
height.
Since this is a cube, all the sides are the same length.
From the figure, we can find that the side length is
WIDTH.
To find the volume, we just cube this side length.
WIDTH^3 =
WIDTH \times
WIDTH \times
WIDTH =
WIDTH * WIDTH * WIDTH
So, the volume of the cube is WIDTH * WIDTH * WIDTH cubic units.
What is the surface area of this cube? Drag on the cube to rotate it.
Here, we can see that all of the six sides have the same size.
We can now calculate the total area:
6 \left(WIDTH \times
WIDTH\right)
\qquad= 6 \times WIDTH * WIDTH
\qquad= 6 * WIDTH * WIDTH
So, the total surface area of the cube is
6 * WIDTH * WIDTH
square units.
What is the volume of a cylinder with base radius
r and height
h?
Math.PI * r * r * h cubic units
The volume of a cylinder is the area of the base \times height.
The area of a circle is \pi r^2.
So the area the base is \pi \cdot r^2 = r * r \pi.
The height of the cylinder is h, so the volume is
r * r\pi \cdot h = r * r * h\pi cubic units.
What is the surface area of a cylinder with
base radius r and
height h?
Math.PI * 2 * r * ( r + h ) square units
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
square units.