Octants

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Definition

\(1 \over 8\) of the 3D region divided up by 3 planes (X,Y,Z). Think about a rubik's cube with 8 individual cubes. 4 at the top and 4 at the bottom, forming a giant cube. It can also be defined as \(1 \over 8\) of a full circle. It would have an angle of 45° (360°/ 8 = 45)

Worked examples

\(\)Octant I: \( (+x, +y, +z)\quad\)Octant II: \( (-x, +y, +z)\)

The 3D coordinate system divides space into 8 octants by the xy-, xz-, and yz-planes.

\(\frac{360^\circ}{8} = 45^\circ\)

One octant of a circle is one-eighth of the full rotation, so it spans 45 degrees.

Common mistakes

\(\)There are 4 octants in 3D space\(\) → \(\)There are 8 octants in 3D space\(\) In 2D there are 4 quadrants; adding the z-axis doubles that to 8 octants in 3D.
\(\)An octant is \( \frac{1}{4} \) of a circle\(\) → \(\)An octant is \( \frac{1}{8} \) of a circle (or sphere)\(\) One-fourth is a quadrant; one-eighth is an octant.

Where you'll use it next

Octants are essential when graphing in three dimensions, analyzing surfaces and solids in multivariable calculus, and solving problems in vector calculus and 3D geometry.

Found in 1 StudyPug lesson

Mastering 3D Coordinate Systems: Navigate the Third Dimension

Multivariable Calculus

Unlock the power of 3D coordinate systems! Learn to visualize points, planes, and shapes in three dimensions. Discover real-world applications in physics, engineering, and computer graphics.

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