Perimeter and Area: Chapter Review

Vocabulary and notation

Perimeter $P$ and area $A$

Perimeter measures boundary length in length units. Area measures the enclosed region in square units; equal areas do not imply equal perimeters.

Base and corresponding height

A triangle base is any chosen side. Its height is the perpendicular distance from the opposite vertex to that side’s supporting line, including an exterior foot. A parallelogram height separates its opposite parallel side lines.

Area postulates

A unit square has area one. Congruent regions have equal areas. Finitely many pieces filling a region without gaps or interior overlaps have areas adding to the whole; shared edges add no area.

Inclusive trapezoid and kite

A trapezoid has at least one pair of parallel sides, so parallelograms qualify. A kite is convex with two disjoint pairs of adjacent congruent sides, each side in one pair; rhombi and squares qualify.

Named-region area $A_{ABC}$

The subscript names the region: this notation denotes triangle ABC’s area. Subscripts on compared areas must follow the stated order of the corresponding bases or sides.

Convex regular polygon

A simple, nondegenerate polygon with congruent sides and interior angles. Its center is $O$, integer side count is $n$, and side length is $s$.

Apothem $a$ and radius $r$

The apothem runs perpendicularly from the regular polygon’s center to a side midpoint. A radius runs to a vertex. The central angle lies between radii to adjacent vertices.

Composite figure, auxiliary segment, and hole

A composite figure combines simpler regions. An auxiliary segment exposes useful pieces. A hole is an excluded region completely surrounded by retained material; its outline is an inner boundary.

Concave polygon

A simple polygon with an interior angle greater than 180 degrees. A proposed cut must remain in the region to give an interior decomposition.

Length scale factor $k$

The positive multiplier taking every original length to its corresponding image length in similar figures. Values above one enlarge, values between zero and one reduce, and one preserves size.

Formulas and theorems

• Rectangle and parallelogram area

  $$A=bh$$

  Use when Positive base and its perpendicular height, in one length unit. Rational rectangle tiling establishes the rational case; the irrational-length extension is accepted. For parallelograms, a boundary side is a height only when perpendicular to the chosen base.

• Triangle area

  $$A=\frac12bh$$

  Text description

  Triangle ABC has chosen base AB. C is beyond B, and the perpendicular CH meets the extended base line at H. A right-angle square marks H. The label b belongs only to AB; h belongs to CH, not to either slanted side.

  Use when Any nondegenerate triangle; any chosen side with its own positive perpendicular height to the supporting line. The foot may lie inside, at an endpoint, or outside the side.

• Trapezoid area

  $$A=\frac{b_1+b_2}{2}h$$

  Text description

  The convex trapezoid has a shorter upper base b one and longer lower base b two, marked parallel with matching chevrons. A dashed segment labeled h joins their lines perpendicularly, with a square at its lower foot. Neither slanted leg is labeled as the height.

  Use when A convex trapezoid with chosen parallel bases of positive lengths. The height is their perpendicular separation. Equal chosen bases include the parallelogram case; the average base also equals the trapezoid midsegment.

• Perpendicular-diagonal area

  $$A=\frac12d_1d_2$$

  Text description

  A convex kite has vertices A at the top, B at the right, C at the bottom and D at the left. The diagonals meet at a marked right angle. An outside vertical bracket labels the full diagonal AC as d one; an outside horizontal bracket labels the full diagonal BD as d two. No diagonal is marked bisected.

  Use when A convex quadrilateral with perpendicular diagonals; use both full diagonal lengths. Every kite and rhombus qualifies. Neither equal diagonals nor bisection alone supplies perpendicularity.

• Quadrilateral from a diagonal

  $$A=\frac12d(h_1+h_2)$$

  Text description

  Convex quadrilateral ABCD is split by diagonal AC, labeled d. B and D lie on opposite sides of AC. Dashed perpendiculars from B and D meet AC at separate feet, with right-angle squares, and are labeled h one and h two. The other diagonal BD is not drawn or asserted perpendicular to AC.

  Use when A convex quadrilateral split by a diagonal of length $d$. The other vertices lie on opposite sides of its line; their perpendicular distances are $h_1$ and $h_2$. The other diagonal need not be perpendicular.

• Common-height triangle ratios

  $$\frac{A_1}{A_2}=\frac{b_1}{b_2}$$

  Text description

  Triangle ABC is divided by AD, with D strictly between B and C. BD is labeled b one and DC is labeled b two. A dashed perpendicular from A to line BC is labeled h and has a square at its foot. Both small triangles use this same distance to the base line.

  Use when Nondegenerate triangles with equal corresponding perpendicular heights. Match each area to its own base. Equal base lengths then give equal areas; the converse holds when the common height is positive.

• Common-base triangle ratios

  $$\frac{A_1}{A_2}=\frac{h_1}{h_2}$$

  Use when Nondegenerate triangles with equal positive chosen base lengths. Heights are measured perpendicularly to their respective base lines. Equal areas then force equal corresponding heights.

• Equal-included-angle area ratios

  $$\frac{A_1}{A_2}=\frac{ab}{cd}$$

  Text description

  Two separate nondegenerate triangles are shown schematically. The first, in a solid blue outline, has sides a and b meeting at a single-arc angle. The second, in a dashed red outline, has sides c and d meeting at an angle with the same single arc. The unlabeled third sides are not part of the products.

  Use when The first triangle’s positive sides $a,b$ enclose an angle equal to the angle enclosed by the second triangle’s positive sides $c,d$. Both triangles are nondegenerate. Valid for acute, right, or obtuse included angles.

• Medians and centroid areas

  $$\begin{gathered} A_{ABD}=\frac12A_{ABC}\\ A_{ADC}=\frac12A_{ABC}\\ A_{GBD}=\frac16A_{ABC} \end{gathered}$$

  Text description

  In triangle ABC, D is the midpoint of BC, E the midpoint of CA, and F the midpoint of AB. Matching one, two and three tick patterns identify the separate midpoint pairs. All three medians meet at G. The six triangular regions remain visible; none is marked congruent to another.

  Use when In nondegenerate ABC, D is the midpoint of BC. For the one-sixth relation, G is the centroid and all three medians form six equal-area triangles. Conversely, a cut AD with D strictly inside BC halves area only when D is the midpoint.

• Regular-polygon area

  $$\begin{gathered} P=ns\\ A=\frac12Pa \end{gathered}$$

  Text description

  A convex regular hexagon illustrates the center-triangle configuration. Segments from the interior center O to every vertex partition the polygon into six congruent triangles. The bottom side is labeled s. A dashed apothem a reaches the midpoint of that side with a right-angle mark. The formula applies to any allowed side count, not just six.

  Use when A convex regular polygon with integer side count at least three and positive side length and apothem. Its center triangles have disjoint interiors and fill the polygon. A perimeter and one arbitrary center-to-side distance do not justify the rule for an arbitrary polygon.

• Regular-polygon half-triangle

  $$\begin{gathered} \text{central angle}=\frac{360^\circ}{n}\\ \theta=\frac{180^\circ}{n}\\ a=\frac{s}{2\tan\theta}\\ a=r\cos\theta\\ \frac{s}{2}=r\sin\theta\\ r^2=a^2+(s/2)^2 \end{gathered}$$

  Text description

  Right triangle OMB is half a center triangle of a convex regular polygon. O is the center and M is a side midpoint. OM is the apothem a, MB is the half-side s divided by two, and OB is the radius r. A square at M marks perpendicularity. The angle at O, between a and r, is theta.

  Use when A convex regular polygon with integer side count at least three and positive lengths. Use degree mode; theta is the positive half-central angle, at most 60 degrees. The radius is the hypotenuse, so it is longer than the apothem. Exact dimensions must also yield an integer side count.

• Exact regular-polygon special cases

  $$\begin{gathered} a_{\text{equilateral}}=\frac{s\sqrt3}{6}\\ A_{\text{equilateral}}=\frac{s^2\sqrt3}{4}\\ a_{\text{square}}=\frac{s}{2}\\ A_{\text{square}}=s^2\\ a_{\text{hexagon}}=\frac{s\sqrt3}{2}\\ A_{\text{hexagon}}=\frac{3s^2\sqrt3}{2} \end{gathered}$$

  Use when Positive side length $s$. The hexagon is regular; its radius equals its side. The equilateral triangle’s apothem starts at its center and is not its full altitude.

• Composite and overlapping areas

  $$\begin{gathered} A_{\text{retained}}=A_{\text{whole}}-A_{\text{removed}}\\ A_{\text{union}}=A_1+A_2-A_{\text{common}} \end{gathered}$$

  Use when Use a common square unit. Removed regions must belong to the whole. Add pieces with disjoint interiors; if separately subtracting several holes, their interiors must not overlap. In the union formula, subtract the common area once.

• Boundary accounting

  $$\begin{gathered} P=\text{sum of boundary lengths}\\ P_{\text{joined}}=P_1+P_2-2s\\ P_{\text{total}}=P_{\text{outer}}+P_{\text{holes}} \end{gathered}$$

  Use when The joined formula assumes two simple polygons with disjoint interiors meeting along exactly one length-$s$ segment and no other contact. The last formula includes all requested inner boundaries. Auxiliary cuts are internal, and outside-only perimeter omits hole edges.

• Linear and square unit conversions

  If one larger length unit equals $c$ smaller length units, one larger square unit equals $c^2$ smaller square units.

  Use when Positive conversion factor. Multiply numerical measures by $c$ for lengths or by its square for areas when converting larger units to smaller units. Reverse direction uses reciprocal factors; changing units does not resize the region.

• Similarity: lengths, perimeter, and area

  $$\begin{gathered} k=\frac{\text{image length}}{\text{original length}}\\ P_{\text{image}}=kP_{\text{original}}\\ A_{\text{image}}=k^2A_{\text{original}}\\ k=\sqrt{\frac{A_{\text{image}}}{A_{\text{original}}}} \end{gathered}$$

  Use when Established similarity, positive $k$, and simple polygons of positive area. Compare matching measurements in consistent units and image-over-original order. Corresponding parts and polygonal holes carried by the same similarity receive the same squared area factor; part-to-whole area fractions stay unchanged.

Problem types, step by step

Exam traps

• Trap Using the slanted side as a height

  Fix Find the perpendicular distance to the chosen base line. An exterior foot is valid; a base extension is not added to the original side.

• Trap Keeping the old height after changing the base

  Fix The area stays fixed, but each chosen side needs its own perpendicular height.

• Trap Halving an already halved diagonal

  Fix The diagonal area rule uses two full diagonals. In a general kite, only the diagonal joining equal-side vertices must bisect the other.

• Trap Inferring area equality from just one matched factor

  Fix Equal bases alone or equal heights alone are insufficient. Match both, or use the applicable ordered ratio.

• Trap Treating side products as absolute areas

  Fix Equal included angles give a ratio of products. Without the angle’s height information, those products do not determine each area.

• Trap Assuming a median is an altitude or that equal-area pieces are congruent

  Fix A median reaches a midpoint. It halves area without requiring perpendicularity, angle bisection, or congruent pieces.

• Trap Substituting a radius for an apothem

  Fix Use the perpendicular center-to-side distance as the height. The right half-triangle uses half the central angle and half a side.

• Trap Multiplying by the side count twice

  Fix The full perimeter already includes every center-triangle base. Delay rounding until the final requested results.

• Trap Counting overlaps, joins, or holes incorrectly

  Fix Count each included area once. Trace actual boundary components separately; a shared internal edge contributes no perimeter.

• Trap Using a linear factor for square units or similar areas

  Fix Square the appropriate length factor. Unit conversion renames one region; a similarity resizes it. A dimensionless area multiplier does not change square units into fourth-power units.

• Trap Reversing only one ratio or confusing “more” with “of”

  Fix Keep the same comparison direction. An increase of p percent gives a factor of one plus p divided by 100, before any square root.

• Trap Inferring similarity or congruence from equal areas

  Fix Equal area alone determines neither shape nor corresponding lengths. If positive-area polygons are already similar and also have equal areas, their positive length factor is one and they are congruent.