Quadrilaterals and Polygons

The formulas are short and the trap is consistent. Test writers know which number you will grab, and they put the slanted side in the question.

The family tree

A quadrilateral is any four-sided figure. The named ones nest inside each other.

The inheritance runs downward only. Every square is a rectangle; not every rectangle is a square. Every rectangle is a parallelogram; not every parallelogram is a rectangle.

Questions phrased "which statement is always true" are almost always about this direction, and the wrong choice is the true statement read backward.

Two properties worth carrying:

• In any parallelogram, opposite sides are equal and opposite angles are equal, and the diagonals bisect each other. Consecutive angles - two next to each other - are supplementary: they add to 180.
• In a rectangle the diagonals are also equal to each other. In a rhombus they meet at right angles.

Perimeter

Add the sides. For a rectangle that is 2 times length plus 2 times width, and for anything else it is just addition.

The only thing worth saying about perimeter is the unit. Perimeter is a length, so it is in feet, not square feet. Checking the unit of the answer choices tells you which quantity a question wants before you read the words twice.

Area

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Every one of these is base times height with an adjustment, which is worth seeing because it means there are not really five formulas here.

A parallelogram is a rectangle with its top slid sideways, and sliding does not change the area, so it is base times height. A triangle is half a parallelogram cut along a diagonal. A trapezoid is a parallelogram whose top is a different length, so you use the average of the two parallel sides as the base.

That last one is the useful reading: the trapezoid formula is the average of the parallel sides, times the height. One half times (b1 + b2) is exactly an average, and thinking of it that way makes it hard to misremember.

The height, again

The figure exists for this. In a parallelogram with a base of 12, a slanted side of 5 and a perpendicular height of 4, the area is 12 times 4, which is 48. Using the 5 gives 60, and 60 is always among the choices.

The rule for spotting it: if a question gives you three numbers for a parallelogram or a trapezoid, one of them is there to be ignored, and it is usually the slant.

Composite shapes

An odd shape on this subtest is two rectangles, added or subtracted.

An L-shaped room is a 20 by 14 rectangle with a 6 by 5 corner removed. What is its floor area?

• Full rectangle: 280 square feet.
• Removed corner: 30.
• Floor: 250 square feet.

Subtracting is usually fewer operations than dissecting, and there is one less place to go wrong.

Regular polygons

Interior angle sum: (n - 2) x 180. Divide by n for one angle of a regular polygon.

The faster route for one angle, especially with many sides: exterior angles always sum to 360, so each exterior angle is 360 / n and the interior angle is 180 minus that.

Perimeter of a regular polygon is simply n times the side length.

Area of a regular polygon uses the apothem, the distance from the center straight to the middle of a side: area = one half x perimeter x apothem. A regular pentagon with side 4 and apothem 2.75 has a perimeter of 20, so its area is half of 20 x 2.75, which is 27.5. A regular hexagon is six equilateral triangles, so its apothem is the altitude of one of them: half the side times root 3.

What you can skip

Across the 39 questions on this topic:

• Diagonal counts. One question asks how many diagonals a polygon has. If it comes up: n times (n - 3), divided by 2.
• Rarely named shapes. Kites come up twice; heptagons never.
• Interior angle sums for large polygons need no table beyond the formula above; (n - 2) x 180 covers every case.

Where people lose points

Using the slanted side as the height.

Forgetting the one half on a triangle or a trapezoid.

Adding rather than averaging the two parallel sides of a trapezoid, which doubles the answer.

Reading the family tree backward - assuming every parallelogram is a rectangle.

Confusing area and perimeter, which the unit of the answer choices settles.

Scaling area by the length factor instead of its square.

Work one in under a minute

A trapezoid has parallel sides of 9 and 15 and a height of 8. What is its area?

Average the parallel sides: (9 + 15) / 2 is 12.

Multiply by the height: 12 times 8 is 96.

Check by the formula as written: one half times 24 times 8 is 96. Same answer, and averaging first kept the numbers small.

Where this leads

The same perpendicular height rule governs triangle area, and the base-times- height idea extends directly into volume as base area times height.