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Quadrilaterals & Polygon Areas

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

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Quadrilaterals & Polygon AreasAMC 10 Step by Step

Name: Date: October 28, 2026

20 problems, easiest first

What you need to know

Area formulas. Trapezoid: $\frac{(b_1+b_2)h}{2}$ , where $b_1$ , $b_2$ are the parallel sides and $h$ is the distance between them. Parallelogram: $bh$ . Rhombus or kite: $\frac{d_1 d_2}{2}$ , half the product of the diagonals. Example: parallel sides $5$ and $9$ with height $4$ give $28$ ; diagonals $6$ and $8$ give a rhombus of area $24$ .
Any quadrilateral. If the diagonals $d_1$ and $d_2$ cross at angle $\theta$ , the area is $\frac12 d_1 d_2\sin\theta$ . Example: diagonals $6$ and $8$ crossing at $30^\circ$ give $\frac12\cdot6\cdot8\cdot\frac12 = 12$ .
Shoelace Theorem. For corners $(x_1,y_1), \dots, (x_n,y_n)$ listed in order around the shape, area $= \frac12\left|(x_1y_2 - x_2y_1) + (x_2y_3 - x_3y_2) + \dots + (x_ny_1 - x_1y_n)\right|$ . Example: $(0,0)$ , $(4,0)$ , $(5,3)$ , $(1,4)$ give $\frac12|0 + 12 + 17 + 0| = \frac{29}{2}$ .
Regular hexagon. With side $s$ , it is six equilateral triangles, so its area is $\frac{3\sqrt3}{2}s^2$ . Its long diagonal (through the center) is $2s$ , and its short diagonal (skipping one corner) is $s\sqrt3$ . Example: $s = 2$ gives area $6\sqrt3$ and diagonals $4$ and $2\sqrt3$ .
Trapezoid with its diagonals. If the diagonals cross at $P$ , the two triangles on the parallel sides are similar, and the two triangles on the legs have equal area.
Pick's Theorem. For a polygon whose corners are lattice points (both coordinates integers), area $= I + \frac{B}{2} - 1$ , where $I$ counts the lattice points inside and $B$ those on the boundary. Example: the rectangle from $(0,0)$ to $(3,2)$ has $I = 2$ and $B = 10$ , so its area is $2 + 5 - 1 = 6$ .

How AMC 10 tests it

Early: the area of a figure on a grid. Count squares, use Shoelace or Pick, or subtract right triangles from a rectangle around it.
A trapezoid given by its bases and legs. Drop perpendiculars; often a $3$ - $4$ - $5$ triangle gives the height.
A square with points on its sides, joined to make a quadrilateral inside; find the ratio of the areas.
Regular hexagons: the area of a triangle on three of its corners, or one hexagon inside another. The answer contains $\sqrt3$ .

Standard approaches

Cut along a diagonal or drop perpendiculars to make rectangles and right triangles.
Find the area of a big simple region and subtract the unwanted pieces.
For figures on a grid or given by coordinates, use Shoelace directly.
For a trapezoid with its diagonals, get all four small areas from one, using the ratio of the bases.

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. Write $[XYZ]$ for the area of triangle $XYZ$ .

Step 1. Since $AB\parallel CD$ , triangles $ABP$ and $CDP$ have equal angles, so they are similar, with ratio $4:10 = 2:5$ . Areas go by the square of the ratio: $[CDP] = 8\cdot\left(\frac52\right)^2 = 50$ .

Step 2. The similarity also gives $PD:PB = 5:2$ . Triangles $APD$ and $ABP$ share the corner $A$ , and their bases $PD$ and $PB$ lie on line $BD$ , so they have the same height. Their areas are in the ratio of their bases: $[APD] = \frac52\cdot 8 = 20$ .

Step 3. The two side triangles have equal area, so $[BPC] = 20$ as well.

Step 4. Add the four pieces:
$8 + 50 + 20 + 20 = 98.$
The answer is $\boxed{\textbf{(C)}\ 98}$ .

Pitfalls

Using the slanted leg of a trapezoid as its height. Example: a leg of $5$ that runs $3$ sideways gives height $4$ .
Listing corners out of order in Shoelace. That measures a different, self-crossing shape.
Using $\frac{d_1 d_2}{2}$ when the 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)

1.2022 AMC 10B #2
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.)
1. A) $3\sqrt 5$
2. B) $10$
3. C) $6\sqrt 5$
4. D) $20$
5. E) $25$
2.2021 AMC Fall 10B #2
What is the area of the shaded figure shown below?
1. A) $4$
2. B) $6$
3. C) $8$
4. D) $10$
5. E) $12$
3.2017 AMC 10A #3
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?
1. A) $72$
2. B) $78$
3. C) $90$
4. D) $120$
5. E) $150$
4.2012 AMC 10B #2
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?
1. A) $50$
2. B) $100$
3. C) $125$
4. D) $150$
5. E) $200$
5.2012 AMC 10A #2
A square with side length $8$ is cut in half, creating two congruent rectangles. What are the dimensions of one of these rectangles?
1. A) $2 \text{by} 4$
2. B) $2 \text{by} 6$
3. C) $2 \text{by} 8$
4. D) $4 \text{by} 4$
5. E) $4 \text{by} 8$
6.2010 AMC 10A #2
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?
1. A) $\dfrac{5}{4}$
2. B) $\dfrac{4}{3}$
3. C) $\dfrac{3}{2}$
4. D) $2$
5. E) $3$
7.2009 AMC 10B #4
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?
1. A) $\frac {1}{8}$
2. B) $\frac {1}{6}$
3. C) $\frac {1}{5}$
4. D) $\frac {1}{4}$
5. E) $\frac {1}{3}$
8.2019 AMC 10B #8
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 the area of the shaded region?
1. A) $4$
2. B) $12 - 4\sqrt{3}$
3. C) $3\sqrt{3}$
4. D) $4\sqrt{3}$
5. E) $16 - 4\sqrt{3}$
9.2016 AMC 10A #10
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?
1. A) $1$
2. B) $2$
3. C) $4$
4. D) $6$
5. E) $8$
10.2009 AMC 10A #14
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?
1. A) $3$
2. B) $\sqrt {10}$
3. C) $2 + \sqrt2$
4. D) $2\sqrt3$
5. E) $4$
11.2008 AMC 10A #10
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 $S_3$ ?
1. A) $\frac{1}{2}$
2. B) $1$
3. C) $2$
4. D) $3$
5. E) $4$
12.2006 AMC 10A #7
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 $y$ ?
1. A) $6$
2. B) $7$
3. C) $8$
4. D) $9$
5. E) $10$
13.2003 AMC 10B #4
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 $\$2$ each, dahlias $\$2.50$ each, and Easter lilies $\$3$ each. What is the least possible cost, in dollars, for her garden?
1. A) $108$
2. B) $115$
3. C) $132$
4. D) $144$
5. E) $156$
14.2002 AMC 10A #8
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?
1. A) $B = W$
2. B) $W = P$
3. C) $B = P$
4. D) $3B = 2P$
5. E) $2P = W$
15.2022 AMC 10B #16
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?
1. A) $15\dfrac{1}{8}$
2. B) $15\dfrac{3}{8}$
3. C) $15\dfrac{1}{2}$
4. D) $15\dfrac{5}{8}$
5. E) $15\dfrac{7}{8}$
16.2022 AMC 10A #10
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 of the original index card?
1. A) $14$
2. B) $10\sqrt{2}$
3. C) $16$
4. D) $12\sqrt{2}$
5. E) $18$
17.2016 AMC 10A #11
Find the area of the shaded region.
1. A) $4\dfrac{3}{5}$
2. B) $5$
3. C) $5\dfrac{1}{4}$
4. D) $6\dfrac{1}{2}$
5. E) $8$
18.2014 AMC 10A #16
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?
1. A) $\dfrac1{12}$
2. B) $\dfrac{\sqrt3}{18}$
3. C) $\dfrac{\sqrt2}{12}$
4. D) $\dfrac{\sqrt3}{12}$
5. E) $\dfrac16$
19.2014 AMC 10A #13
Equilateral $\triangle ABC$ has side length $1$ , and squares $ABDE$ , $BCHI$ , $CAFG$ lie outside the triangle. What is the area of hexagon $DEFGHI$ ?
1. A) $\dfrac{12+3\sqrt3}4$
2. B) $\dfrac92$
3. C) $3+\sqrt3$
4. D) $\dfrac{6+3\sqrt3}2$
5. E) $6$
20.2013 AMC 10B #15
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 $\frac{a}{b}$ ?
1. A) $1$
2. B) $\frac{\sqrt{6}}{2}$
3. C) $\sqrt{3}$
4. D) $2$
5. E) $\frac{3\sqrt{2}}{2}$

Problems © Mathematical Association of America (MAA), American Mathematics Competitions. Reproduced for non-commercial educational use. The topic note, tags and key insights are original to this site.