Topics / Geometry

Quadrilaterals & Polygon Areas

Areas of quadrilaterals/polygons, trapezoids, shoelace, decomposition, hexagons

Quadrilaterals & Polygon AreasAMC 10 Step by Step

Name: Date: September 13, 2026

Geometry · 46 problems have this as their main idea

primary-topic problems (3.5% of all)

What you need to know

Trapezoid area $= \frac{(b_1+b_2)h}{2}$ ; parallelogram $= bh$ ; rhombus or kite $= \frac{d_1 d_2}{2}$ . Any quadrilateral has area $\frac12 d_1 d_2\sin\theta$ , where $\theta$ is the angle between the diagonals.
Shoelace: for vertices $(x_1,y_1),\dots,(x_n,y_n)$ in order, area $= \frac12\left|\sum (x_i y_{i+1} - x_{i+1} y_i)\right|$ .
A regular hexagon of side $s$ is six equilateral triangles, area $\frac{3\sqrt3}{2}s^2$ , long diagonal $2s$ , short diagonal $s\sqrt3$ .
In a trapezoid with diagonals meeting at $P$ , the triangles on the bases are similar and the two side triangles have equal area.
Pick's theorem for lattice polygons: $A = I + \frac{B}{2} - 1$ .

How AMC 10 tests it

Early: area of a figure on a grid; count squares, use shoelace or Pick, or subtract right triangles from a bounding rectangle.
Trapezoids given by bases and legs: drop perpendiculars, often producing a $3$ - $4$ - $5$ triangle for the height.
A square with points on its sides joined to form an inner quadrilateral; find the ratio of areas.
Regular hexagons: area of a triangle on three vertices, or nested hexagons; the answer contains $\sqrt3$ .

Standard approaches

Cut along a diagonal or drop perpendiculars to create rectangles and right triangles.
Compute a big simple region and subtract the unwanted pieces.
For lattice or coordinate figures, use shoelace directly.
For trapezoids with diagonals, get all four sub-areas from one using the base ratio.

Worked example

Trapezoid $ABCD$ has $\overline{AB}\parallel\overline{CD}$ , $AB = 4$ , and $CD = 10$ . The diagonals meet at $P$ , and the area of triangle $ABP$ is $8$ . What is the area of the trapezoid?

(A) $84$ (B) $90$ (C) $98$ (D) $100$ (E) $112$

Solution. Since $AB\parallel CD$ , triangles $ABP$ and $CDP$ are similar with ratio $4:10 = 2:5$ , so $[CDP] = 8\cdot\left(\frac52\right)^2 = 50$ . Also $PD:PB = 5:2$ , and triangles $APD$ and $ABP$ share vertex $A$ with bases along line $BD$ , so $[APD] = \frac52\cdot 8 = 20$ ; likewise $[BPC] = 20$ . The trapezoid's area is
$8 + 50 + 20 + 20 = 98.$
The answer is $\boxed{\textbf{(C)}\ 98}$ .

Pitfalls

Using the slanted leg of a trapezoid as its height.
Listing vertices out of order in shoelace, which computes a self-intersecting polygon.
Applying $\frac{d_1 d_2}{2}$ to a quadrilateral whose diagonals are not perpendicular.

Traps that recur

Building the frame side from 3/7 alone (e.g. 1 + 6/7 or similar), which produces the decoy answers with 49 in the denominator. (2023 AMC 10A #24)
Assuming R must be parallel to the 1x3 rectangle or to the unit square, which finds only one of the two configurations. (2021 AMC Fall 10B #25)
Trying to find the trapezoid's dimensions or P's location; only the ratio is determined, and the height h cancels immediately in the differences of squares. (2022 AMC 10A #23)
Assuming the pentagon is regular, or adding four equilateral triangles plus a 'middle region' without noticing that segments FC and GD cross. (2021 AMC 10B #20)

Problems, easiest first

2022 AMC 10B · #2Quadrilaterals & Polygon Areas

In rhombus $ABCD$ , point $P$ lies on segment $\overline{AD}$ so that $\overline{BP}$ $\perp$ $\overline{AD}$ , $AP = 3$ , and $PD = 2$ . What is the area of $ABCD$ ? (Note: The figure is not drawn to scale.)

2021 AMC Fall 10B · #2Quadrilaterals & Polygon Areas

What is the area of the shaded figure shown below?

2017 AMC 10A · #3Quadrilaterals & Polygon Areas

Tamara has three rows of two $6$ -feet by $2$ -feet flower beds in her garden. The beds are separated and also surrounded by $1$ -foot-wide walkways, as shown on the diagram. What is the total area of the walkways, in square feet?

2012 AMC 10A · #2Quadrilaterals & Polygon Areas

A square with side length $8$ is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?

2012 AMC 10B · #2Quadrilaterals & Polygon Areas

A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle?

2010 AMC 10A · #2Quadrilaterals & Polygon Areas

Four identical squares and one rectangle are placed together to form one large square as shown. The length of the rectangle is how many times as large as its width?

2009 AMC 10B · #4Quadrilaterals & Polygon Areas

A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths $15$ and $25$ meters. What fraction of the yard is occupied by the flower beds?

2019 AMC 10B · #8Quadrilaterals & Polygon Areas

The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length $2$ and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is …

2016 AMC 10A · #10Quadrilaterals & Polygon Areas

A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the length in feet of the inner rectangle?

2009 AMC 10A · #14Quadrilaterals & Polygon Areas

Four congruent rectangles are placed as shown. The area of the outer square is $4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side?

2008 AMC 10A · #10Quadrilaterals & Polygon Areas

Each of the sides of a square $S_1$ with area $16$ is bisected, and a smaller square $S_2$ is constructed using the bisection points as vertices. The same process is carried out on $S_2$ to construct an even smaller square $S_3$ . What is the area of $S_3$ ?

2006 AMC 10A · #7Quadrilaterals & Polygon Areas

The $8\times18$ rectangle $ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $y$ ?

2003 AMC 10B · #4Quadrilaterals & Polygon Areas

Rose fills each of the rectangular regions of her rectangular flower bed with a different type of flower. The lengths, in feet, of the rectangular regions in her flower bed are as shown in the figure. She plants one flower per square foot in each region. Asters cost $1 $each, begonias$ 1.50 $each, cannas$ 2$ …

2002 AMC 10A · #8Quadrilaterals & Polygon Areas

Betsy designed a flag using blue triangles, small white squares, and a red center square, as shown. Let $B$ be the total area of the blue triangles, $W$ the total area of the white squares, and $P$ the area of the red square. Which of the following is correct?

2022 AMC 10A · #10Quadrilaterals & Polygon Areas

Daniel finds a rectangular index card and measures its diagonal to be $8$ centimeters. Daniel then cuts out equal squares of side $1$ cm at two opposite corners of the index card and measures the distance between the two closest vertices of these squares to be $4\sqrt{2}$ centimeters, as shown below. What is the area …

2022 AMC 10B · #16Quadrilaterals & Polygon Areas

The diagram below shows a rectangle with side lengths $4$ and $8$ and a square with side length $5$ . Three vertices of the square lie on three different sides of the rectangle, as shown. What is the area of the region inside both the square and the rectangle?

2016 AMC 10A · #11Quadrilaterals & Polygon Areas

Find the area of the shaded region.

2014 AMC 10A · #13Quadrilaterals & Polygon Areas

Equilateral $\triangle ABC$ has side length $1$ , and squares $ABDE$ , $BCHI$ , $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$ ?

2014 AMC 10A · #16Quadrilaterals & Polygon Areas

In rectangle $ABCD$ , $AB=1$ , $BC=2$ , and points $E$ , $F$ , and $G$ are midpoints of $\overline{BC}$ , $\overline{CD}$ , and $\overline{AD}$ , respectively. Point $H$ is the midpoint of $\overline{GE}$ . What is the area of the shaded region?

2013 AMC 10B · #15Quadrilaterals & Polygon Areas

A wire is cut into two pieces, one of length $a$ and the other of length $b$ . The piece of length $a$ is bent to form an equilateral triangle, and the piece of length $b$ is bent to form a regular hexagon. The triangle and the hexagon have equal area. What is $\frac{a}{b}$ ?

2012 AMC 10B · #19Quadrilaterals & Polygon Areas

In rectangle $ABCD$ , $AB=6$ , $AD=30$ , and $G$ is the midpoint of $\overline{AD}$ . Segment $AB$ is extended 2 units beyond $B$ to point $E$ , and $F$ is the intersection of $\overline{ED}$ and $\overline{BC}$ . What is the area of $BFDG$ ?

2012 AMC 10B · #14Quadrilaterals & Polygon Areas

Two equilateral triangles are contained in a square whose side length is $2\sqrt 3$ . The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. What is the area of the rhombus?

2011 AMC 10B · #20Quadrilaterals & Polygon Areas

Rhombus $ABCD$ has side length $2$ and $\angle B = 120^\circ$ . Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$ ?

2007 AMC 10A · #18Quadrilaterals & Polygon Areas

Consider the $12$ -sided polygon $ABCDEFGHIJKL$ , as shown. Each of its sides has length $4$ , and each two consecutive sides form a right angle. Suppose that $\overline{AG}$ and $\overline{CH}$ meet at $M$ . What is the area of quadrilateral $ABCM$ ?

2006 AMC 10A · #17Quadrilaterals & Polygon Areas

In rectangle $ADEH$ , points $B$ and $C$ trisect $\overline{AD}$ , and points $G$ and $F$ trisect $\overline{HE}$ . In addition, $AH=AC=2$ , and $AD=3$ . What is the area of quadrilateral $WXYZ$ shown in the figure?

2006 AMC 10B · #15Quadrilaterals & Polygon Areas

Rhombus $ABCD$ is similar to rhombus $BFDE$ . The area of rhombus $ABCD$ is $24$ and $\angle BAD = 60^\circ$ . What is the area of rhombus $BFDE$ ?

2005 AMC 10A · #20Quadrilaterals & Polygon Areas

An equiangular octagon has four sides of length $1$ and four sides of length $\sqrt{2}/2$ , arranged so that no two consecutive sides have the same length. What is the area of the octagon?

2002 AMC 10B · #17Quadrilaterals & Polygon Areas

A regular octagon $ABCDEFGH$ has sides of length two. Find the area of $\triangle ADG$ .

2001 AMC 10 · #18Quadrilaterals & Polygon Areas

The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to

2001 AMC 10 · #15Quadrilaterals & Polygon Areas

A street has parallel curbs $40$ feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is $15$ feet and each stripe is $50$ feet long. Find the distance, in feet, between the stripes.

2000 AMC 10 · #18Quadrilaterals & Polygon Areas

Charlyn walks completely around the boundary of a square whose sides are each $5$ km long. From any point on her path she can see exactly $1$ km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the …

2022 AMC 10A · #23Quadrilaterals & Polygon Areas

Isosceles trapezoid $ABCD$ has parallel sides $\overline{AD}$ and $\overline{BC},$ with $BC < AD$ and $AB = CD.$ There is a point $P$ in the plane such that $PA=1, PB=2, PC=3,$ and $PD=4.$ What is $\tfrac{BC}{AD}?$

2021 AMC 10B · #20Quadrilaterals & Polygon Areas

The figure is constructed from $11$ line segments, each of which has length $2$ . The area of pentagon $ABCDE$ can be written as $\sqrt{m} + \sqrt{n}$ , where $m$ and $n$ are positive integers. What is $m + n ?$

2021 AMC Fall 10B · #18Quadrilaterals & Polygon Areas

Three identical square sheets of paper each with side length $6{ }$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$ -sided polygon shown in the figure below. The area of this …

2020 AMC 10A · #20Quadrilaterals & Polygon Areas

Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC=20,$ and $CD=30.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD?$

2020 AMC 10B · #21Quadrilaterals & Polygon Areas

In square $ABCD$ , points $E$ and $H$ lie on $\overline{AB}$ and $\overline{DA}$ , respectively, so that $AE=AH.$ Points $F$ and $G$ lie on $\overline{BC}$ and $\overline{CD}$ , respectively, and points $I$ and $J$ lie on $\overline{EH}$ so that $\overline{FI} \perp \overline{EH}$ and …

2018 AMC 10B · #24Quadrilaterals & Polygon Areas

Let $ABCDEF$ be a regular hexagon with side length $1$ . Denote by $X$ , $Y$ , and $Z$ the midpoints of sides $\overline {AB}$ , $\overline{CD}$ , and $\overline{EF}$ , respectively. What is the area of the convex hexagon whose interior is the intersection of the interiors of $\triangle ACE$ and $\triangle XYZ$ ?

2016 AMC 10B · #23Quadrilaterals & Polygon Areas

In regular hexagon $ABCDEF$ , points $W$ , $X$ , $Y$ , and $Z$ are chosen on sides $\overline{BC}$ , $\overline{CD}$ , $\overline{EF}$ , and $\overline{FA}$ respectively, so lines $AB$ , $ZW$ , $YX$ , and $ED$ are parallel and equally spaced. What is the ratio of the area of hexagon $WCXYFZ$ to the area of hexagon …

2014 AMC 10B · #21Quadrilaterals & Polygon Areas

Trapezoid $ABCD$ has parallel sides $\overline{AB}$ of length $33$ and $\overline{CD}$ of length $21$ . The other two sides are of lengths $10$ and $14$ . The angles at $A$ and $B$ are acute. What is the length of the shorter diagonal of $ABCD$ ?

2010 AMC 10A · #19Quadrilaterals & Polygon Areas

Equiangular hexagon $ABCDEF$ has side lengths $AB=CD=EF=1$ and $BC=DE=FA=r$ . The area of $\triangle ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $r$ ?

2007 AMC 10A · #19Quadrilaterals & Polygon Areas

A paint brush is swept along both diagonals of a square to produce the symmetric painted area, as shown. Half the area of the square is painted. What is the ratio of the side length of the square to the brush width?

2005 AMC 10B · #23Quadrilaterals & Polygon Areas

In trapezoid $ABCD$ we have $\overline{AB}$ parallel to $\overline{DC}$ , $E$ as the midpoint of $\overline{BC}$ , and $F$ as the midpoint of $\overline{DA}$ . The area of $ABEF$ is twice the area of $FECD$ . What is $AB/DC$ ?

2003 AMC 10B · #23Quadrilaterals & Polygon Areas

A regular octagon $ABCDEFGH$ has an area of one square unit. What is the area of the rectangle $ABEF$ ?

2002 AMC 10A · #25Quadrilaterals & Polygon Areas

In trapezoid $ABCD$ with bases $AB$ and $CD$ , we have $AB = 52$ , $BC = 12$ , $CD = 39$ , and $DA = 5$ . The area of $ABCD$ is

2023 AMC 10A · #24Quadrilaterals & Polygon Areas

Six regular hexagonal blocks of side length $1$ unit are arranged inside a regular hexagonal frame. Each block lies along an inside edge of the frame and is aligned with two other blocks, as shown in the figure below. The distance from any corner of the frame to the nearest vertex of a block is $\frac{3}{7}$ unit. What …

2021 AMC Fall 10B · #25Quadrilaterals & Polygon Areas

A rectangle with side lengths $1{ }$ and $3,$ a square with side length $1,$ and a rectangle $R$ are inscribed inside a larger square as shown. The sum of all possible values for the area of $R$ can be written in the form $\tfrac mn$ , where $m$ and $n$ are relatively prime positive integers. What is $m+n?$