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American Mathematics contest 8 (AMC 8) - Geometry problems

Try these AMC 8 Geometry Questions and check your knowledge!

**AMC 8, 2020, Problem** **18**

Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline {FE}$ as shown in the figure. Let $DA = 16$, and let $FD = AE = 9$.What is the area of $ABCD$?

(A) $240$ (B) $248$ (C) $256$ (D) $264$ (E) $272$.

**AMC 8, 2019, Problem 2**

Three identical rectangles are put together to form rectangle $ABCD$, as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is $5$ feet, what is the area in square feet of rectangle $ABCD$?

(A) $45$ (B) $75$ (C) $100$ (D) $125$ (E) $150$.

**AMC 8, 2019, Problem 4**

Quadrilateral $ABCD$ is a rhombus with perimeter $52$ meters. The length of diagonal $\overline{AC}$ is $24$ meters. What is the area in square meters of rhombus $ABCD$?

(A) $60$ (B) $90$ (C) $105$ (D) $120$ (E) $144$.

**AMC 8, 2019, Problem 24**

In triangle $ABC$, point $D$ divides side $\overline{AC}$ so that $AD:DC=1:2$. Let $E$ be the midpoint of $\overline{BD}$ and let $F$ be the point of intersection of line $BC$ and line $AE$. Given that the area of $\triangle ABC$ is $360$, what is the area of $\triangle EBF$?

(A) $24$ (B) $30$ (C) $32$ (D) $36$ (E) $40$.

**AMC 8, 2018, Problem 4**

The twelve-sided figure shown has been drawn on $1 \text{ cm}\times 1 \text{ cm}$ graph paper. What is the area of the figure in $\text{cm}^2$?

(A) $12$ (B) $12.5$ (C) $13$ (D) $13.5$ (E) $14$.

**AMC 8, 2018, Problem 9**

Bob is tiling the floor of his $12$ foot by $16$ foot living room. He plans to place one-foot by one-foot square tiles to form a border along the edges of the room and to fill in the rest of the floor with two-foot by two-foot square tiles. How many tiles will he use?

(A) $48$ (B) $87$ (C) $91$ (D) $96$ (E) $120$.

**AMC 8, 2018, Problem 15**

In the diagram below, a diameter of each of the two smaller circles is a radius of the larger circle. If the two smaller circles have a combined area of $1$ square unit, then what is the area of the shaded region, in square units?

(A) $ \frac{1}{4}$ (B) $\frac{1}{3}$ (C) $\frac{1}{2}$ (D) $1$ (E) $\frac{\pi}{2}$

**AMC 8, 2018, Problem 20**

In $\triangle ABC,$ a point $E$ is on $\overline{AB}$ with $AE=1$ and $EB=2.$ Point $D$ is on $\overline{AC}$ so that $\overline{DE} \parallel \overline{BC}$ and point $F$ is on $\overline{BC}$ so that $\overline{EF} \parallel \overline{AC}.$ What is the ratio of the area of $CDEF$ to the area of $\triangle ABC?$

(A)$ \frac{4}{9}$ (B) $\frac{1}{2}$ (C)$ \frac{5}{9}$ (D) $ \frac{3}{5}$ (E)$ \frac{2}{3}$.

**AMC 8, 2018, Problem 22**

Point $E$ is the midpoint of side $\overline{CD}$ in square $ABCD,$ and $\overline{BE}$ meets diagonal $\overline{AC}$ at $F.$ The area of quadrilateral $AFED$ is $45.$ What is the area of $ABCD?$

(A) $ 100 $ (B) $108$ (C) $120$ (D) $135$ (E) $144$.

**AMC 8, 2018, Problem 23**

From a regular octagon, a triangle is formed by connecting three randomly chosen vertices of the octagon. What is the probability that at least one of the sides of the triangle is also a side of the octagon?

(A) $ \frac{2}{7}$ (B) $ \frac{5}{42}$ (C)$ \frac{11}{14} $ (D) $\frac{5}{7}$ (E) $\frac{6}{7}$.

**AMC 8, 2018, Problem 24**

In the cube $ABCDEFGH$ with opposite vertices $C$ and $E,$ $J$ and $I$ are the midpoints of edges $\overline{FB}$ and $\overline{HD},$ respectively. Let $R$ be the ratio of the area of the cross-section $EJCI$ to the area of one of the faces of the cube. What is $R^2?$

(A) $\frac{5}{4}$ (B) $ \frac{4}{3} $ (C) $ \frac{3}{2} $ (D) $ \frac{25}{16}$ (E) $\frac{9}{4}$.

**AMC 8, 2017, Problem 16**

In the figure below, choose point $D$ on $\overline{BC}$ so that $\triangle ACD$ and $\triangle ABD$ have equal perimeters. What is the area of $\triangle ABD$?

(A) $\frac{3}{4}$ (B) $\frac{3}{2}$ (C) $2$ (D) $\frac{12}{5}$ (E)$\frac{5}{2}$.

**AMC 8, 2017, Problem 18**

In the non-convex quadrilateral $ABCD$ shown below, $\angle BCD$ is a right angle, $AB=12$, $BC=4$, $CD=3$, and $AD=13$. What is the area of the quadrilateral $ABCD$?

(A) $12$ (B) $24$ (C) $26$ (D) $30$ (E) $36$.

**AMC 8, 2017, Problem 22**

In the right triangle $ABC$, $AC=12$, $BC=5$, and angle $C$ is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?

(A) $\frac{7}{6}$ (B) $\frac{13}{5}$ (C) $\frac{59}{18}$ (D) $\frac{10}{3}$ (E) $\frac{60}{13}$.

**AMC 8, 2017, Problem 25**

In the figure shown, $\overline{US}$ and $\overline{UT}$ are line segments each of length $2$, and $m\angle TUS = 60^{\circ}$. Arcs ${TR}$ and ${SR}$ are each one-sixth of a circle with radius $2$. What is the area of the region shown?

(A) $3\sqrt{3}-\pi$ (B) $4\sqrt{3}-\frac{4\pi}{3}$ (C) $2\sqrt{3}$ (D) $4\sqrt{3}-\frac{2\pi}{3}$ (E) $4+\frac{4\pi}{3}$.

**AMC 8, 2016, Problem 2**

In rectangle $ABCD$, $AB=6$ and $AD=8$. Point $M$ is the midpoint of $\overline{AD}$. What is the area of $\triangle AMC$?

(A) $12$ (B) $15$ (C) $18$ (D) $20$ (E) $24$.

**AMC 8, 2016, Problem 22**

Rectangle $DEFA$ below is a $3 \times 4$ rectangle with $DC=CB=BA$. What is the area of the "bat wings" (shaded area)?

(A) $2$ (B) $2 \frac{1}{2}$ (C) $3$ (D) $3 \frac{1}{2}$ (E) $5$.

**AMC 8, 2016, Problem 23**

Two congruent circles centered at points $A$ and $B$ each pass through the other circle's center. The line containing both $A$ and $B$ is extended to intersect the circles at points $C$ and $D$. The circles intersect at two points, one of which is $E$. What is the degree measure of $\angle CED$?

(A) $90$ (B) $105$ (C) $120$ (D) $135$ (E) $150$

**AMC 8, 2016, Problem 25**

A semicircle is inscribed in an isosceles triangle with base $16$ and height $15$ so that the diameter of the semicircle is contained in the base of the triangle as shown. What is the radius of the semicircle?

(A) $4 \sqrt{3}$ (B) $\frac{120}{17}$ (C) $10$ (D) $\frac{17\sqrt{2}}{2}$

(E) $\frac{17\sqrt{3}}{2}$.

**AMC 8, 2015, Problem 1**

How many square yards of carpet are required to cover a rectangular floor that is $12$ feet long and $9$ feet wide? (There are $3$ feet in a yard.)

(A) $12$ (B)$36$ (C) $108$ (D) $324$ (E) $972$.

**AMC 8, 2015, Problem 2**

Point $O$ is the center of the regular octagon $ABCDEFGH$, and $X$ is the midpoint of the side $\overline{AB}.$ What fraction of the area of the octagon is shaded?

(A) $\frac{11}{32}$ (B) $\frac{3}{8}$ (C) $\frac{13}{32}$ (D) $\frac{7}{16}$ (E) $\frac{15}{32}$.

**AMC 8, 2015, Problem 6**

In $\bigtriangleup ABC$, $AB=BC=29$, and $AC=42$. What is the area of $\bigtriangleup ABC$?

(A)$100$ (B) $420$ (C) $500$ (D) $609$ (E) $701$.

**AMC 8, 2015, Problem 8**

What is the smallest whole number larger than the perimeter of any triangle with a side of length $5$ and a side of length $19$?

(A) $24$ (B) $29$ (C) $43$ (D) $48$ (E) $57$.

**AMC 8, 2015, Problem 12**

How many pairs of parallel edges, such as $\overline{AB}$ and $\overline{GH}$ or $\overline{EH}$ and $\overline{FG}$, does a cube have?

(A) $6$ (B) $12$ (C) $18$ (D) $ 24$ (E) $ 36$.

**AMC 8, 2015, Problem 19**

A triangle with vertices as $A=(1,3)$, $B=(5,1)$, and $C=(4,4)$ is plotted on a $6\times5$ grid. What fraction of the grid is covered by the triangle?

(A) $\frac{1}{6}$ (B)$\frac{1}{5}$ (C) $\frac{1}{4}$ (D) $\frac{1}{3}$ (E) $\frac{1}{2}$

**AMC 8, 2015, Problem 21**

In the given figure hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\triangle KBC$?

(A) $6\sqrt{2}$ (B)$9$ (C) $12$ (D) $9\sqrt{2}$ (E) $32$.

**AMC 8, 2015, Problem 25**

One-inch squares are cut from the corners of this $5$ inch square. What is the area in square inches of the largest square that can fit into the remaining space?

(A) $ 9$ (B) $12\frac{1}{2}$ (C) $15$ (D)$15\frac{1}{2}$ (E)$17$

**AMC 8, 2014, Problem 9**

In $\triangle ABC$, $D$ is a point on side $\overline{AC}$ such that $BD=DC$ and $\angle BCD$ measures $70^{\circ}$. What is the degree measure of $\angle ADB$?

(A) $100$ (B)$120$ (C) $135$ (D) $140$ (E) $150$.

**AMC 8, 2014, Problem 14**

Rectangle $ABCD$ and right triangle $DCE$ have the same area. They are joined to form a trapezoid, as shown. What is $DE$?

(A) $12$ (B) $13$ (C) $14$ (D) $15$ (E) $16$.

**AMC 8, 2014, Problem 15**

The circumference of the circle with center $O$ is divided into $12$ equal arcs, marked the letters $A$ through $L$ as seen below. What is the number of degrees in the sum of the angles $x$ and $y$?

(A) $75$ (B) $80$ (C) $90$ (D) $120$ (E) $150$.

**AMC 8, 2014, Problem 19**

A cube with $3$-inch edges is to be constructed from $27$ smaller cubes with $1$-inch edges. Twenty-one of the cubes are colored red and $6$ are colored white. If the $3$-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white?

(A) $\frac{5}{54}$ (B) $\frac{1}{9}$ (C) $\frac{5}{27}$ (D)$\frac{2}{9}$

(E) $\frac{1}{3}$

**AMC 8, 2014, Problem 20**

Rectangle $ABCD$ has sides $CD=3$ and $DA=5$. A circle with a radius of $1$ is centered at $A$, a circle with a radius of $2$ is centered at $B$, and a circle with a radius of $3$ is centered at $C$. Which of the following is closest to the area of the region inside the rectangle but outside all three circles?

(A) $3.5$ (B) $4.0$ (C) $4.5$ (D) $5.0$ (E) $5.5$.

**AMC 8, 2013, Problem 18**

Isabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?

(A)$ 204$ (B) $ 280$ (C) $320$ (D) $340$ (E) $600$.

**AMC 8, 2013, Problem 20**

A $1\times 2$ rectangle is inscribed in a semicircle with longer side on the diameter. What is the area of the semicircle?

(A)$ \frac{\pi}{2}$ (B) $ \frac{2\pi}{3} $ (C)$ \pi $ (D)$ \frac{4\pi}{3}$ (E)$ \frac{5\pi}{3}$.

**AMC 8, 2013, Problem 21**

Samantha lives $2$ blocks west and $1$ block south of the southwest corner of City Park. Her school is $2$ blocks east and $2$ blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?

(A)$3$ (B) $6$ (C) $ 9$ (D) $ 12$ (E) $18$.

**AMC 8, 2013, Problem 23**

Angle $ABC$ of $\triangle ABC$ is a right angle. The sides of $\triangle ABC$ are the diameters of semicircles as shown. The area of the semicircle on $\overline{AB}$ equals $8\pi$, and the arc of the semicircle on $\overline{AC}$ has length $8.5\pi$. What is the radius of the semicircle on $\overline{BC}$?

(A)$ 7$ (B)$ 7.5 $ (C)$8 $ (D)$ 8.5 $ (E)$9$

**AMC 8, 2013, Problem 24**

Squares $ABCD$, $EFGH$, and $GHIJ$ are equal in area. Points $C$ and $D$ are the midpoints of sides $IH$ and $HE$, respectively. What is the ratio of the area of the shaded pentagon $AJICB$ to the sum of the areas of the three squares?

(A)$\frac{1}{4}$ (B)$\frac{7}{24}$ (C)$\frac{1}{3}$ (D)$\frac{3}{8}$ (E)$\frac{5}{12}$.

**AMC 8, 2013, Problem 25**

A ball with diameter $4$ inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are $R_1 = 100$ inches, $R_2 = 60$ inches, and $R_3 = 80$ inches, respectively. The ball always remains in contact with the track and does not slip. What is the distance the center of the ball travels over the course from $A$ to $B$?

(A)$ 238\pi$ (B)$240\pi$ (C)$260\pi$ (D)$280\pi$ (E)$500\pi$.

**AMC 8, 2012, Problem 5**

In the diagram, all angles are right angles and the lengths of the sides are given in centimeters. Note the diagram is not drawn to scale. What is the length in $X$, in centimeters?

(A)$1$ (B) $2$ (C) $3$ (D) $4$ (E) $5$.

**AMC 8, 2012, Problem 6**

A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures $8$ inches high and $10$ inches wide. What is the area of the border, in square inches?

(A) $36$ (B) $40$ (C) $64$ (D) $72$ (E) $88$

**AMC 8, 2012, Problem 17**

A square with integer side length is cut into $10$ squares, all of which have integer side length and at least $8$ of which have area $1$. What is the smallest possible value of the length of the side of the original square?

(A) $3$ (B) $4$ (C) $5$ (D) $6$ (E) $7$

**AMC 8, 2012, Problem 21**

Marla has a large white cube that has an edge of $10$ feet. She also has enough green paint to cover $300$ square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square feet?

(A) $5\sqrt2$ (B) $10$ (C) $10\sqrt2$ (D) $50$ (E) $50\sqrt2$.

**AMC 8, 2012, Problem 23**

An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is $4$, what is the area of the hexagon?

(A) $4$ (B) $5$ (C) $6$ (D) $4\sqrt3$ (E) $6\sqrt3$.

**AMC 8, 2012, Problem 24**

A circle of radius $2$ is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?

(A) $\frac{4-\pi}{\pi}$ (B) $\frac{1}{\pi}$ (C) $\frac{\sqrt2}{\pi}$ (D) $\frac{\pi-1}{\pi}$ (E) $\frac{3}{\pi}$

**AMC 8, 2009, Problem 7**

The triangular plot of $ACD$ lies between Aspen Road, Brown Road and a railroad. Main Street runs east and west, and the railroad runs north and south. The numbers in the diagram indicate distances in miles. The width of the railroad track can be ignored. How many square miles are in the plot of land $ACD$?

(A)$ 2$ (B) $3$ (C) $ 4.5$ (D) $6$ (E) $ 9$.

**AMC 8, 2009, Problem 9**

Construct a square on one side of an equilateral triangle. On one non-adjacent side of the square, construct a regular pentagon, as shown. On a non-adjacent side of the pentagon, construct a hexagon. Continue to construct regular polygons in the same way, until you construct an octagon. How many sides does the resulting polygon have?

(A)$ 21$ (B)$23$ (C)$25$ (D)$27$ (E)$29$.

**AMC 8, 2009, Problem 18**

The diagram represents a $7$-foot-by-$7$-foot floor that is tiled with $1$-square-foot black tiles and white tiles. Notice that the corners have white tiles. If a $15$-foot-by-$15$-foot floor is to be tiled in the same manner, how many white tiles will be needed?

(A) $49$ (B) $57$ (C) $64$ (D) $96$ (E) $126$.

**AMC 8, 2009, Problem 20**

How many non-congruent triangles have vertices at three of the eight points in the array shown below?

(A)$ 5$ (B) $6$ (C) $ 7$ (D) $8$ (E) $ 9$.

**AMC 8, 2009, Problem 25**

A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is $1/2$ foot from the top face. The second cut is $1/3$ foot below the first cut, and the third cut is $1/17$ foot below the second cut. From the top to the bottom the pieces are labeled $A, B, C$, and $D$. The pieces are then glued together end to end as shown in the second diagram. What is the total surface area of this solid in square feet?

(A)$6$ (B)$7$ (C)$\frac{419}{51}$ (D)$\frac{158}{17}$ (E)$11$.

**AMC 8, 2008, Problem 16**

A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units?

(A) $1 : 6$ (B) $7 : 36$ (C) $1 : 5$ (D) $7 : 30$ (E) $6 : 25$.

**AMC 8, 2008, Problem 18**

Two circles that share the same center have radii $10$ meters and $20$ meters. An aardvark runs along the path shown, starting at $A$ and ending at $K$. How many meters does the aardvark run?

(A) $10\pi + 201$ (B) $ 10\pi + 30$ (C) $10 \pi +40$ (D) $20\pi + 20$ (E) $20\ pi +40$.

**AMC 8, 2008, Problem 21**

Jerry cuts a wedge from a $6$-cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters?

(A) $48$ (B) $75$ (C) $151$ (D) $192$ (E) $603$.

**AMC 8, 2008, Problem 23**

In square $ABCE, A F=2 F E$ and $C D=2 D E$. What is the ratio of the area of $\triangle B F D$ to the area of square $A B C E ?$

(A) $\frac{1}{6}$ (B) $\frac{2}{9}$ (C) $\frac{5}{18}$ (D) $\frac{1}{3}$

(E) $\frac{7}{20}$.

**AMC 8, 2008, Problem 25**

Margie's winning art design is shown. The smallest circle has radius $2$ inches, with each successive circle's radius increasing by $2$ inches. Which of the following is closest to the percent of the design that is black?

(A) $41.7$ (B) $44$ (C) $45$ (D) $46$ (E) $48$.

**AMC 8, 2007, Problem 8**

In trapezoid $ABCD$, $AD$ is perpendicular to $DC$, $AD$ = $AB$ = $3$, and $DC$ = $6$. In addition, $E$ is on $DC$, and $BE$ is parallel to $AD$. Find the area of $\triangle BEC$.

(A) $3$ (B) $4.5$ (C) $6$ (D) $9$ (E) $18$

**AMC 8, 2007, Problem 12**

A unit hexagram is composed of a regular hexagon of side length $1$ and its $6$ equilateral triangular extensions, as shown in the diagram. What is the ratio of the area of the extensions to the area of the original hexagon?

(A) $1: 1$ (B) $6: 5$ (C) $3: 2$ (D) $2: 1$ (E) $3: 1$

**AMC 8, 2007, Problem 14**

The base of isosceles $\triangle ABC$ is $24$ and its area is $60$. What is the length of one of the congruent sides?

(A) $5$ (B) $8$ (C) $13$ (D) $14$ (E) $18$

**AMC 8, 2007, Problem 23**

What is the area of the shaded pinwheel shown in the $5 \times 5$ grid?

(A) $ 4$ (B) $6$ (C) $8$ (D) $10$ (E) $12$

**AMC 8, 2006, Problem 5**

Points $A, B, C$ and $D$ are midpoints of the sides of the larger square. If the larger square has area $60$, what is the area of the smaller square?

(A) $15$ (B)$20$ (C) $24$ (D) $30$ (E) $40$

**AMC 8, 2006, Problem 6**

The letter $T$ is formed by placing two $2 \times 4$ inch rectangles next to each other, as shown. What is the perimeter of the $T$, in inches?

(A) $12$ (B) $16$ (C) $20$ (D) $22$ (E) $ 24$

**AMC 8, 2006, Problem 7**

Circle $X$ has a radius of $\pi$. Circle $Y$ has a circumference of $8 \pi$. Circle $Z$ has an area of $9 \pi$. List the circles in order from smallest to largest radius.

(A) $ X, Y, Z$ (B) $Z, X, Y$ (C) $Y, X, Z$ (D) $Z, Y, X$ (E) $ X, Z, Y$

**AMC 8, 2006, Problem 18**

A cube with $3$-inch edges is made using $27$ cubes with $1$-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white?

(A) $\frac{1}{9}$ (B) $\frac{1}{4}$ (C)$\frac{4}{9}$ (D)$\frac{5}{9}$ (E)$\frac{19}{27}$

**AMC 8, 2006, Problem 7**

Triangle $ABC$ is an isosceles triangle with $\overline{AB}=\overline{BC}$. Point $D$ is the midpoint of both $\overline{BC}$ and $\overline{AE}$, and $\overline{CE}$ is $11$ units long. Triangle $ABD$ is congruent to triangle $ECD$. What is the length of $\overline{BD}$?

(A) $4$ (B) $4.5$ (C) $5$ (D) $5.5$ (E) $6$

**AMC 8, 2005, Problem 3**

What is the minimum number of small squares that must be colored black so that a line of symmetry lies on the diagonal $\overline{B D}$ of square $A B C D ?$

(A)$1$ (B) $2$ (C) $3$ (D) $4$ (E) $5$

**AMC 8, 2005, Problem 9**

In quadrilateral $A B C D,$ sides $\overline{A B}$ and $\overline{B C}$ both have length $10,$ sides $\overline{C D}$ and $\overline{D A}$ both have length $17,$ and the measure of angle $A D C$ is $60^{\circ} .$ What is the length of diagonal $\overline{A C} ?$

(A) $13.5$ (B) $14$ (C) $15.5$ (D) $17$ (E) $18.5$

**AMC 8, 2005, Problem 13**

The area of polygon $A B C D E F$ is $52$ with $A B=8, B C=9$ and $F A=5$. What is $D E+E F ?$

(A) $47$ (B) $8$ (C) $9$ (D) $10$ (E) $11$

**AMC 8, 2005, Problem 19**

What is the perimeter of trapezoid $A B C D ?$

(A) $180$ (B) $188$ (C) $196$ (D) $200$ (E) $204$

**AMC 8, 2005, Problem 23**

Isosceles right triangle $A B C$ encloses a semicircle of area $2 \pi$. The circle has its center $O$ on hypotenuse $\overline{A B}$ and is tangent to sides $\overline{A C}$ and $\overline{B C}$. What is the area of triangle $A B C$ ?

(A) $6$ (B) $8$ (C) $3 \pi$ (D) $10$ (E) $4 \pi$

**AMC 8, 2005, Problem 25**

A square with side length $2$ and a circle share the same center. The total area of the regions that are inside the circle and outside the square is equal to the total area of the regions that are outside the circle and inside the square. What is the radius of the circle?

(A) $\frac{2}{\sqrt{\pi}}$ (B) $\frac{1+\sqrt{2}}{2}$ (C) $\frac{3}{2}$

(D) $\sqrt{3}$ (E) $\sqrt{\pi}$

**AMC 8, 2004, Problem 14**

What is the area enclosed by the geoboard quadrilateral below?

(A) $15$ (B) $18 \frac{1}{2}$ (C) $22 \frac{1}{2}$ (D) $27$ (E) $41$.

**AMC 8, 2004, Problem 24**

In the figure, $A B C D$ is a rectangle and $E F G H$ is a parallelogram. Using the measurements given in the figure, what is the length $d$ of the segment that is perpendicular to $\overline{H E}$ and $\overline{F G}$ ?

(A) $6.8$ (B) $7.1$ (C) $7.6$ (D) $7.8$ (E) $8.1$

**AMC 8, 2004, Problem 25**

Two $4 \times 4$ squares intersect at right angles, bisecting their intersecting sides, as shown. The circle's diameter is the segment between the two points of intersection. What is the area of the shaded region created by removing the circle from the squares?

(A) $16-4 \pi$ (B) $16-2 \pi$ (C) $28-4 \pi$ (D) $28-2 \pi$ (E) $32-2 \pi$

**AMC 8, 2003, Problem 6**

Given the areas of the three squares in the figure, what is the area of the interior triangle?

(A) $13$ (B) $30$ (C) $60$ (D) $300$ (E) $1800$

**AMC 8, 2003, Problem 8**

Bake Sale

Four friends, Art, Roger, Paul and Trisha, bake cookies, and all cookies have the same thickness. The shapes of the cookies differ, as shown.

Art's cookies are trapezoids.

Roger's cookies are rectangles.

Paul's cookies are parallelograms.

Trisha's cookies are triangles.

Each friend uses the same amount of dough, and Art makes exactly $12$ cookies. Who gets the fewest cookies from one batch of cookie dough?

(A) Art (B) Roger (C) Paul (D) Trisha (E) There is a tie for fewest.

**AMC 8, 2003, Problem 21**

The area of trapezoid $A B C D$ is $164 \mathrm{~cm}^{2}$. The altitude is $8 \mathrm{~cm}, A B$ is $10 \mathrm{~cm},$ and $C D$ is 17 $\mathrm{cm}$. What is $B C$, in centimeters?

(A) $9$ (B) $10$ (C) $12$ (D) $15$ (E) $20$

**AMC 8, 2003, Problem 22**

The following figures are composed of squares and circles. Which figure has a shaded region with largest area?

(A) A only (B) B (C) C only (D) both A and B (E) all are equal

**AMC 8, 2003, Problem 25**

In the figure, the area of square $W X Y Z$ is $25 \mathrm{~cm}^{2}$. The four smaller squares have sides 1 $\mathrm{cm}$ long, either parallel to or coinciding with the sides of the large square. In $\triangle A B C$, $A B=A C,$ and when $\triangle A B C$ is folded over side $\overline{B C}$, point $A$ coincides with $O,$ the center of square $W X Y Z$. What is the area of $\triangle A B C$, in square centimeters?

(A) $\frac{15}{4}$ (B) $\frac{21}{4}$ (C) $\frac{27}{4}$ (D) $\frac{21}{2}$

(E) $\frac{27}{2}$

**AMC 8, 2002, Problem 20**

The area of triangle $X Y Z$ is $8$ square inches. Points $A$ and $B$ are midpoints of congruent segments $\overline{X Y}$ and $\overline{X Z}$. Altitude $\overline{X C}$ bisects $\overline{Y Z}$. The area (in square inches) of the shaded region is

(A) $1 \frac{1}{2}$ (B) $2$ (C) $2 \frac{1}{2}$ (D) $3$ (E) $3 \frac{1}{2}$

**AMC 8, 2001, Problem 16**

A square piece of paper, 4 inches on a side, is folded in half vertically. Both layers are then cut in half parallel to the fold. Three new rectangles are formed, a large one and two small ones. What is the ratio of the perimeter of one of the small rectangles to the perimeter of the large rectangle?

(A) $\frac{1}{3}$ (B) $\frac{1}{2}$ (C) $\frac{3}{4}$ (D) $\frac{4}{5}$

(E) $\frac{5}{6}$.

**AMC 8, 2000, Problem 6**

Figure $A B C D$ is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded $L$ -shaped region is

(A) $7$ (B) $10$ (C) $12.5$ (D) $14$ (E) $15$.

**AMC 8, 2000, Problem 13**

In triangle $C A T,$ we have $\angle A C T=\angle A T C$ and $\angle C A T=36^{\circ} .$ If $\overline{T R}$ bisects $\angle A T C$ then $\angle C R T=$

(A) $36^{\circ}$ (B) $54^{\circ}$ (C) $72^{\circ}$ (D) $90^{\circ}$

(E) $108^{\circ}$.

**AMC 8, 2000, Problem 15**

Triangles $A B C, A D E,$ and $E F G$ are all equilateral. Points $D$ and $G$ are midpoints of $\overline{A C}$ and $\overline{A E},$ respectively. If $A B=4,$ what is the perimeter of figure $A B C D E F G ?$

(A) $12$ (B) $13$ (C) $15$ (D) $18$ (E) $21$.

**AMC 8, 2000, Problem 19**

Three circular arcs of radius $5$ units bound the region shown. Arcs $A B$ and $A D$ are quarter circles, and arc $B C D$ is a semicircle. What is the area, in square units, of the region?

(A) $25$ (B) $10+5 \pi$ (C) $50$ (D) $50+5 \pi$ (E) $25 \pi$.

**AMC 8, 2000, Problem 22**

A cube has edge length $2$ . Suppose that we glue a cube of edge length $1$ on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The percent increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is closest to

(A) $10$ (B) $15$ (C) $17$ (D) $21$ (E) $25$.

**AMC 8, 2000, Problem 24**

If $\angle A=20^{\circ}$ and $\angle A F G=\angle A G F$, then $\angle B+\angle D=$

(A) $48^{\circ}$ (B) $60^{\circ}$ (C) $72^{\circ}$ (D) $80^{\circ}$ (E) $90^{\circ}$.

**AMC 8, 2000, Problem 25**

The area of rectangle $A B C D$ is 72 . If point $A$ and the midpoints of $\overline{B C}$ and $\overline{C D}$ are joined to form a triangle, the area of that triangle is

(A) $21$ (B) $27$ (C) $30$ (D) $36$ (E) $40$.

**AMC 8, 1999, Problem 5**

A rectangular garden $60$ feet long and $20$ feet wide is enclosed by a fence. To make the garden larger, while using the same fence, its shape is changed to a square. By how many square feet does this enlarge the garden?

(A) $100$ (B) $200$ (C) $300$ (D) $400$ (E) $500$.

**AMC 8, 1999, Problem 14**

In trapezoid $A B C D$, the sides $A B$ and $C D$ are equal. The perimeter of $A B C D$ is

(A) $27$ (B) $30$ (C) $32$ (D) $34$ (E) $48$

**AMC 8, 1999, Problem 21**

The degree measure of angle $A$ is

(A) $20$ (B) $30$ (C) $35$ (D) $40$ (E) $45$.

**AMC 8, 1999, Problem 23**

Square $A B C D$ has sides of length $3 .$ Segments $C M$ and $C N$ divide the square's area into three equal parts. How long is segment $C M ?$

(A) $\sqrt{10}$ (B) $\sqrt{12}$ (C) $\sqrt{13}$ (D) $\sqrt{14}$ (E) $\sqrt{15}$.

**AMC 8, 1999, Problem 25**

Points $B, D,$ and $J$ are midpoints of the sides of right triangle $A C G .$ Points $K, E, I$ are midpoints of the sides of triangle $J D G$, etc. If the dividing and shading process is done $100$ times (the first three are shown) and $A C=C G=6$, then the total area of the shaded triangles is nearest

(A) $6$ (B) $7$ (C) $8$ (D) $9$ (E) $10$.

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