Two-Dimensional Figures
Two-dimensional shapes have two dimensions, such as length and width. These include polygons, which are shapes with straight lines, as well as curved shapes, such as circles and ovals.
More specifically, polygons are coplanar closed shapes that are formed by straight line segments (sides). Some polygons you’ve probably familiar with are rectangles, squares, and triangles. Circles, for example, are not polygons, since they are not formed by straight lines.
A polygon can be convex, where no points of any of the lines are in the interior of the shape, or concave, where some of the lines are in the interior. I like to identify concave polygons by looking for a “cave” or indentation in the shape.
Here is an example of a concave polygon, which has $ 5$ sides (pentagon). See how there’s a “cave” or indentation on the left-hand side? 
. Here is an example of a convex pentagon; notice any line segment between any two points on the figure would be either inside the figure or on the boundary: 
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Note that this polygon has $ 5$ congruent sides and $ 5$ congruent angles; it is both equilateral, meaning that all sides are the same lengths, and equiangular, meaning that all angles are congruent. Thus, it is a regular polygon.
Here are the names of polygons. Notice also that a larger number of sides is just called a ” n”-gon. For example, a polygon with $ 15$ sides is a $ 15$-gon.
| Number of Sides | Polygon | Number of Sides | Polygon | |
| $ 3$ | Triangle | $ 8$ | Octagon | |
| $ 4$ | Quadrilateral | $ 9$ | Nonagon | |
| $ 5$ | Pentagon | $ 10$ | Decagon | |
| $ 6$ | Hexagon | $ 11$ | Hendecagon | |
| $ 7$ | Heptagon | $ 12$ | Dodecagon |
Perimeter and Area
Perimeter is the distance all the way around a shape, such as a polygon, and area is the space inside the shape. Note that for a circle, we say the perimeter is the circumference.
Here are some basic formulas that you may have seen before:
| Square | Rectangle | Triangle | Circle |
$ s $ is a side
$ P=4s$ $ \displaystyle A= {{s}^{2}}$ |
$ l=$ length, $ w=$ width
$ P=2l+2w$ $ A=l\times w$ |
 
$ h=$ height, $ b=$ base (perpendicular to height)
$ \displaystyle \begin{align}P&=\text{add up all sides}\\A&=\frac{{b\times h}}{2}=\frac{1}{2}bh\end{align}$ |
$r=$ radius, $ d=$ diameter
$ \displaystyle C= 2\pi r$ or $ \displaystyle \pi d$ $ \displaystyle A= \pi {{r}^{2}}\,$
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Three-Dimensional Figures (Solids)
Three-dimensional shapes, or solids, have three dimensions, such as length, width, and height.
A polyhedron is a solid with all flat surfaces (not circular or curved) with just one region of space inside. Each flat side is called a face, and each one of these is a two-dimensional polygon. The corners are called vertices (singular is vertex), and the line segments connecting the corners are called edges.
Polyhedrons are either prisms or pyramids. Prisms have two congruent faces (called bases) that are connected by parallelogram faces. (A parallelogram is a quadrilateral with two sets of parallel lines). A pyramid has one polygon base from which three or more triangular faces connect at a point (vertex).
The circular or curved figures include a cylinder, which has has two connected circular parallel bases, a cone, which has one circular base and tapers to one point (vertex), and a sphere, which is a round figure whose points all all equidistant from a center.
Here are some examples of polyhedrons:
Hexagonal Prism |
 Cone |
Square Pyramid |
Sphere |
Cylinder |
Rectangular Prism |
Problems and Solutions
Problem: Name each two-dimensional figure by it’s number of sides and then classify it as concave or convex and regular or irregular:
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| Irregular Quadrilateral, Convex | Irregular Dodecagon ($ 12$-sided), Concave | Regular Octagon, Convex | Irregular Triangle, Convex |
Problem: Find the perimeter and area of each two-dimensional figure. Notice how we indicate a two-dimensional area in the units.
| Square | Circle | Triangle | Rectangle |
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| $ \displaystyle \begin{align}P&=4s=4\times 5\\&=20\text{ yards}\\\\A&={{s}^{2}}={{5}^{2}}\\&=25\text{ yard}{{\text{s}}^{2}}\end{align}$ | $\displaystyle \begin{align}C&=2\pi r=2\pi \times 6\\&=12\pi \approx 37.7\text{ feet}\\\\A&=\pi {{r}^{2}}=\pi \times {{6}^{2}}\\&=36\pi \approx 113.1\text{ fee}{{\text{t}}^{2}}\end{align}$ | $\displaystyle \begin{align}P&=10+10.5+4.5\\&=25\text{ feet}\\\\A&=\frac{{b\times h}}{2}=\frac{{4.5\times 7}}{2}\\&=15.75\text{ fee}{{\text{t}}^{2}}\end{align}$ | $ \displaystyle \begin{align}P&=2l+2w\\&=\left( {2\times 12} \right)+\left( {2\times 3} \right)\\&=30\text{ yards}\\\\A&=l\times w=12\times 3\\&=36\text{ yard}{{\text{s}}^{2}}\end{align}$ |
We will address two- and three-dimensional figures in greater detail in the Area and Volume sections