INTRODUCTION TO AREA AND PERIMETER
Simple Closed Plane Curves

Take a piece of paper and a pencil.
Put your pencil on the paper and draw anything you want, with the following conditions:

The resulting curve is called a simple closed plane curve.

a simple closed
plane curve
a simple plane curve,
that is not closed
a closed plane curve,
that is not simple
a plane curve that is
not simple and not closed
not a plane curve
a simple, closed, plane curve a simple, plane curve, that is not closed a closed, plane curve, that is not simple a plane curve that is not simple and not closed not a plane curve
        (it is drawn on both
the front face and the top face)


Every simple closed plane curve has both an inside (the region enclosed by the curve)
and an outside (the region not enclosed by the curve).
Although this seems like an obvious statement, it is actually quite difficult to prove.
Explore the ‘Jordan Curve Theorem’ if you're interested.

When you are asked for the ‘area’ of a simple closed plane curve, this means the area inside the curve.
For example, the area of a rectangle is the area inside the rectangle, and the area of a triangle is the area inside the triangle.

DEFINITION area of a simple closed plane curve
The area of a simple closed plane curve is the size of the region enclosed by the curve.

Whereas area measures the space ‘inside’ a figure, we can also ask about the distance ‘around’ a figure:

DEFINITION perimeter of a simple closed plane curve
The perimeter of a simple closed plane curve
is the length of the simple closed curve that defines it.

Quite a mouthful! Probably more than you want to hear right now!
The definition looks simpler if we specialize it to polygons:

DEFINITION perimeter of a polygon
The perimeter of a polygon is the sum of the lengths of all its sides.

Note that your teacher might ask you to ‘trace a perimeter’—that is, run your finger (or pencil) along a curve.
Or, your teacher might ask you to ‘shade an area’—that is, shade the region inside a simple closed plane curve.

UNITS FOR AREA AND PERIMETER

Perimeter is measured using units of length.
Consequently, these are all valid units for perimeter, with their common abbreviations:

Every unit of length has a corresponding unit of area, created as follows:

$\,\text{length} = 1\text{ blah}\,$ $\,\text{area} = 1 \text{ square blah} = 1{\text{ blah}}^2\,$
one blah, a unit of length one square blah, a unit of area
a unit of length a unit of area

Here are some valid units for area, with abbreviations:

Here are some basic formulas:

AREA OF A RECTANGLE:
A rectangle with length $\,\ell \,$ and width $\,w\,$ has area $\,A\,$ given by $\,A=\ell w\,$.

AREA OF A SQUARE:
A square with sides of length $\,\ell \,$ has area $\,A\,$ given by $\,A=\ell^2\,$.

PERIMETER OF A RECTANGLE:
A rectangle with length $\,\ell \,$ and width $\,w\,$ has perimeter $\,P\,$ given by $\,P=2\ell + 2w\,$.

PERIMETER OF A SQUARE:
A square with sides of length $\,\ell \,$ has perimeter $\,P\,$ given by $\,P=4\ell\,$.

Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Area Formulas: Triangle, Parallelogram, Trapezoid


On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17  
AVAILABLE MASTERED IN PROGRESS

(MAX is 17; there are 17 different problem types.)