Get down with the lingo
Area
The amount of space within the boundaries of a two-dimensional shape, reported in square units (like miles2 or feet2). Area is essentially space, but don't go around saying things like "area-ships," "area-cadets," or the "area-bar" on your keyboard.
Area Addition Postulate
The assumption that we can add areas together. This is what allows us to find the areas of weird-looking polygons by breaking them up into simpler ones. The total area is equal to the sum of its individual parts. Just make sure the parts don't overlap or leave any gaps.
Base
One of the sides of a polygon that we use to calculate area. We choose bases based on the type of shape we're dealing with.
Circle
A curved shape defined by all the points that are a set distance (that's the radius) from a center. A circle's area is A = πr2 and its circumference is C = 2πr, where r is the radius. By far the most kid-friendly of shapes, since there aren't any corners or sharp edges.
Height
The straight perpendicular line from the topmost point of a shape (either a side or a vertex) down to the base. Remember, the base may need to be extended to find the height.
Kite
A quadrilateral with two adjacent pairs of congruent sides. It's also got perpendicular diagonals and an area formula of A = ½d1d2, where d1 and d2 are the diagonals. Colorful tail and windy days not included.
Parallelogram
A quadrilateral with two sets of parallel sides. To track down a parallelogram's area, we bust out the formula A = bh, where b is the base length and h is the height. Careful though; the height might not necessarily be one of the sides. Tricky, tricky.
Rectangle
A quadrilateral with four right angles. These guys have a pretty simple area formula: A = lw, where l is the length and w is the width. It doesn't get much more chill than that.
Rhombus
A fancy-shmancy type of parallelogram with four equal sides and perpendicular diagonals. We can either use our trusty parallelogram area formula, A = bh, or the slightly cooler diagonal area formula, A = ½d1d2. It just depends on what the problem gives us.
Sector
A "slice" of a circle whose area can be calculated if the radius r and central angle θ is known.
Square
A quadrilateral with four right angles and four equal side lengths. It's technically a special type of rectangle, so we use a modified version of the rectangle area formula. The length and width are identical twins (fun identical, not creepy-horror-movie identical), so a square's area is just A = s2, where s is the side length.
Trapezoid
A quadrilateral with only one set of parallel sides. Trapezoids refuse to follow the crowd, and they've even got their own area formula: A = ½(b1 + b2)h, where b1 and b2 are the parallel bases and h is the height.
Triangle
A three-sided, 2D shape whose area is given by the formula A = ½bh. Just multiply the base length by the height, then slice that product in half. Preferably with a hand-forged samurai sword.