American Mathematics contest 8 (AMC 8) - Geometry problems
Try these AMC 8 Geometry Questions and check your knowledge!
AMC 8, 2020, Problem 18
Rectangle is inscribed in a semicircle with diameter
as shown in the figure. Let
, and let
.What is the area of
?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2019, Problem 2
Three identical rectangles are put together to form rectangle , as shown in the figure below. Given that the length of the shorter side of each of the smaller rectangles is
feet, what is the area in square feet of rectangle
?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2019, Problem 4
Quadrilateral is a rhombus with perimeter
meters. The length of diagonal
is
meters. What is the area in square meters of rhombus
?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2019, Problem 24
In triangle , point
divides side
so that
. Let
be the midpoint of
and let
be the point of intersection of line
and line
. Given that the area of
is
, what is the area of
?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2018, Problem 4
The twelve-sided figure shown has been drawn on graph paper. What is the area of the figure in
?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2018, Problem 9
Bob is tiling the floor of his foot by
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) (B)
(C)
(D)
(E)
.
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 square unit, then what is the area of the shaded region, in square units?
(A) (B)
(C)
(D)
(E)
AMC 8, 2018, Problem 20
In a point
is on
with
and
Point
is on
so that
and point
is on
so that
What is the ratio of the area of
to the area of
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2018, Problem 22
Point is the midpoint of side
in square
and
meets diagonal
at
The area of quadrilateral
is
What is the area of
(A) (B)
(C)
(D)
(E)
.
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) (B)
(C)
(D)
(E)
.
AMC 8, 2018, Problem 24
In the cube with opposite vertices
and
and
are the midpoints of edges
and
respectively. Let
be the ratio of the area of the cross-section
to the area of one of the faces of the cube. What is
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2017, Problem 16
In the figure below, choose point on
so that
and
have equal perimeters. What is the area of
?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2017, Problem 18
In the non-convex quadrilateral shown below,
is a right angle,
,
,
, and
. What is the area of the quadrilateral
?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2017, Problem 22
In the right triangle ,
,
, and angle
is a right angle. A semicircle is inscribed in the triangle as shown. What is the radius of the semicircle?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2017, Problem 25
In the figure shown, and
are line segments each of length
, and
. Arcs
and
are each one-sixth of a circle with radius
. What is the area of the region shown?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2016, Problem 2
In rectangle ,
and
. Point
is the midpoint of
. What is the area of
?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2016, Problem 22
Rectangle below is a
rectangle with
. What is the area of the "bat wings" (shaded area)?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2016, Problem 23
Two congruent circles centered at points and
each pass through the other circle's center. The line containing both
and
is extended to intersect the circles at points
and
. The circles intersect at two points, one of which is
. What is the degree measure of
?
(A) (B)
(C)
(D)
(E)
AMC 8, 2016, Problem 25
A semicircle is inscribed in an isosceles triangle with base and height
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) (B)
(C)
(D)
(E) .
AMC 8, 2015, Problem 1
How many square yards of carpet are required to cover a rectangular floor that is feet long and
feet wide? (There are
feet in a yard.)
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2015, Problem 2
Point is the center of the regular octagon
, and
is the midpoint of the side
What fraction of the area of the octagon is shaded?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2015, Problem 6
In ,
, and
. What is the area of
?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2015, Problem 8
What is the smallest whole number larger than the perimeter of any triangle with a side of length and a side of length
?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2015, Problem 12
How many pairs of parallel edges, such as and
or
and
, does a cube have?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2015, Problem 19
A triangle with vertices as ,
, and
is plotted on a
grid. What fraction of the grid is covered by the triangle?
(A) (B)
(C)
(D)
(E)
AMC 8, 2015, Problem 21
In the given figure hexagon is equiangular,
and
are squares with areas
and
respectively,
is equilateral and
. What is the area of
?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2015, Problem 25
One-inch squares are cut from the corners of this inch square. What is the area in square inches of the largest square that can fit into the remaining space?
(A) (B)
(C)
(D)
(E)
AMC 8, 2014, Problem 9
In ,
is a point on side
such that
and
measures
. What is the degree measure of
?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2014, Problem 14
Rectangle and right triangle
have the same area. They are joined to form a trapezoid, as shown. What is
?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2014, Problem 15
The circumference of the circle with center is divided into
equal arcs, marked the letters
through
as seen below. What is the number of degrees in the sum of the angles
and
?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2014, Problem 19
A cube with -inch edges is to be constructed from
smaller cubes with
-inch edges. Twenty-one of the cubes are colored red and
are colored white. If the
-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white?
(A) (B)
(C)
(D)
(E)
AMC 8, 2014, Problem 20
Rectangle has sides
and
. A circle with a radius of
is centered at
, a circle with a radius of
is centered at
, and a circle with a radius of
is centered at
. Which of the following is closest to the area of the region inside the rectangle but outside all three circles?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2013, Problem 18
Isabella uses one-foot cubical blocks to build a rectangular fort that is feet long,
feet wide, and
feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2013, Problem 20
A rectangle is inscribed in a semicircle with longer side on the diameter. What is the area of the semicircle?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2013, Problem 21
Samantha lives blocks west and
block south of the southwest corner of City Park. Her school is
blocks east and
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) (B)
(C)
(D)
(E)
.
AMC 8, 2013, Problem 23
Angle of
is a right angle. The sides of
are the diameters of semicircles as shown. The area of the semicircle on
equals
, and the arc of the semicircle on
has length
. What is the radius of the semicircle on
?
(A) (B)
(C)
(D)
(E)
AMC 8, 2013, Problem 24
Squares ,
, and
are equal in area. Points
and
are the midpoints of sides
and
, respectively. What is the ratio of the area of the shaded pentagon
to the sum of the areas of the three squares?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2013, Problem 25
A ball with diameter inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are
inches,
inches, and
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
to
?
(A) (B)
(C)
(D)
(E)
.
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 , in centimeters?
(A) (B)
(C)
(D)
(E)
.
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 inches high and
inches wide. What is the area of the border, in square inches?
(A) (B)
(C)
(D)
(E)
AMC 8, 2012, Problem 17
A square with integer side length is cut into squares, all of which have integer side length and at least
of which have area
. What is the smallest possible value of the length of the side of the original square?
(A) (B)
(C)
(D)
(E)
AMC 8, 2012, Problem 21
Marla has a large white cube that has an edge of feet. She also has enough green paint to cover
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) (B)
(C)
(D)
(E)
.
AMC 8, 2012, Problem 23
An equilateral triangle and a regular hexagon have equal perimeters. If the area of the triangle is , what is the area of the hexagon?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2012, Problem 24
A circle of radius 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) (B)
(C)
(D)
(E)
AMC 8, 2009, Problem 7
The triangular plot of 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
?
(A) (B)
(C)
(D)
(E)
.
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) (B)
(C)
(D)
(E)
.
AMC 8, 2009, Problem 18
The diagram represents a -foot-by-
-foot floor that is tiled with
-square-foot black tiles and white tiles. Notice that the corners have white tiles. If a
-foot-by-
-foot floor is to be tiled in the same manner, how many white tiles will be needed?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2009, Problem 20
How many non-congruent triangles have vertices at three of the eight points in the array shown below?
(A) (B)
(C)
(D)
(E)
.
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 foot from the top face. The second cut is
foot below the first cut, and the third cut is
foot below the second cut. From the top to the bottom the pieces are labeled
, and
. 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) (B)
(C)
(D)
(E)
.
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) (B)
(C)
(D)
(E)
.
AMC 8, 2008, Problem 18
Two circles that share the same center have radii meters and
meters. An aardvark runs along the path shown, starting at
and ending at
. How many meters does the aardvark run?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2008, Problem 21
Jerry cuts a wedge from a -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) (B)
(C)
(D)
(E)
.
AMC 8, 2008, Problem 23
In square and
. What is the ratio of the area of
to the area of square
(A) (B)
(C)
(D)
(E) .
AMC 8, 2008, Problem 25
Margie's winning art design is shown. The smallest circle has radius inches, with each successive circle's radius increasing by
inches. Which of the following is closest to the percent of the design that is black?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2007, Problem 8
In trapezoid ,
is perpendicular to
,
=
=
, and
=
. In addition,
is on
, and
is parallel to
. Find the area of
.
(A) (B)
(C)
(D)
(E)
AMC 8, 2007, Problem 12
A unit hexagram is composed of a regular hexagon of side length and its
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) (B)
(C)
(D)
(E)
AMC 8, 2007, Problem 14
The base of isosceles is
and its area is
. What is the length of one of the congruent sides?
(A) (B)
(C)
(D)
(E)
AMC 8, 2007, Problem 23
What is the area of the shaded pinwheel shown in the grid?
(A) (B)
(C)
(D)
(E)
AMC 8, 2006, Problem 5
Points and
are midpoints of the sides of the larger square. If the larger square has area
, what is the area of the smaller square?
(A) (B)
(C)
(D)
(E)
AMC 8, 2006, Problem 6
The letter is formed by placing two
inch rectangles next to each other, as shown. What is the perimeter of the
, in inches?
(A) (B)
(C)
(D)
(E)
AMC 8, 2006, Problem 7
Circle has a radius of
. Circle
has a circumference of
. Circle
has an area of
. List the circles in order from smallest to largest radius.
(A) (B)
(C)
(D)
(E)
AMC 8, 2006, Problem 18
A cube with -inch edges is made using
cubes with
-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) (B)
(C)
(D)
(E)
AMC 8, 2006, Problem 19
Triangle is an isosceles triangle with
. Point
is the midpoint of both
and
, and
is
units long. Triangle
is congruent to triangle
. What is the length of
?
(A) (B)
(C)
(D)
(E)
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 of square
(A) (B)
(C)
(D)
(E)
AMC 8, 2005, Problem 9
In quadrilateral sides
and
both have length
sides
and
both have length
and the measure of angle
is
What is the length of diagonal
(A) (B)
(C)
(D)
(E)
AMC 8, 2005, Problem 13
The area of polygon is
with
and
. What is
(A) (B)
(C)
(D)
(E)
AMC 8, 2005, Problem 19
What is the perimeter of trapezoid
(A) (B)
(C)
(D)
(E)
AMC 8, 2005, Problem 23
Isosceles right triangle encloses a semicircle of area
. The circle has its center
on hypotenuse
and is tangent to sides
and
. What is the area of triangle
?
(A) (B)
(C)
(D)
(E)
AMC 8, 2005, Problem 25
A square with side length 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) (B)
(C)
(D) (E)
AMC 8, 2004, Problem 14
What is the area enclosed by the geoboard quadrilateral below?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2004, Problem 24
In the figure, is a rectangle and
is a parallelogram. Using the measurements given in the figure, what is the length
of the segment that is perpendicular to
and
?
(A) (B)
(C)
(D)
(E)
AMC 8, 2004, Problem 25
Two 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) (B)
(C)
(D)
(E)
AMC 8, 2003, Problem 6
Given the areas of the three squares in the figure, what is the area of the interior triangle?
(A) (B)
(C)
(D)
(E)
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 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 is
. The altitude is
is
and
is 17
. What is
, in centimeters?
(A) (B)
(C)
(D)
(E)
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 is
. The four smaller squares have sides 1
long, either parallel to or coinciding with the sides of the large square. In
,
and when
is folded over side
, point
coincides with
the center of square
. What is the area of
, in square centimeters?
(A) (B)
(C)
(D)
(E)
AMC 8, 2002, Problem 20
The area of triangle is
square inches. Points
and
are midpoints of congruent segments
and
. Altitude
bisects
. The area (in square inches) of the shaded region is
(A) (B)
(C)
(D)
(E)
AMC 8, 2001, Problem 11
Points and
have these coordinates:
and
. The area of quadrilateral
is
(A)
(B)
(C)
(D)
(E)
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) (B)
(C)
(D)
(E) .
AMC 8, 2000, Problem 6
Figure is a square. Inside this square three smaller squares are drawn with the side lengths as labeled. The area of the shaded
-shaped region is
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2000, Problem 13
In triangle we have
and
If
bisects
then
(A) (B)
(C)
(D)
(E) .
AMC 8, 2000, Problem 15
Triangles and
are all equilateral. Points
and
are midpoints of
and
respectively. If
what is the perimeter of figure
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2000, Problem 19
Three circular arcs of radius units bound the region shown. Arcs
and
are quarter circles, and arc
is a semicircle. What is the area, in square units, of the region?
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2000, Problem 22
A cube has edge length . Suppose that we glue a cube of edge length
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) (B)
(C)
(D)
(E)
.
AMC 8, 2000, Problem 24
If and
, then
(A) (B)
(C)
(D)
(E)
.
AMC 8, 2000, Problem 25
The area of rectangle is 72 . If point
and the midpoints of
and
are joined to form a triangle, the area of that triangle is
(A) (B)
(C)
(D)
(E)
.
AMC 8, 1999, Problem 5
A rectangular garden feet long and
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) (B)
(C)
(D)
(E)
.
AMC 8, 1999, Problem 14
In trapezoid , the sides
and
are equal. The perimeter of
is
(A) (B)
(C)
(D)
(E)
AMC 8, 1999, Problem 21
The degree measure of angle is
(A) (B)
(C)
(D)
(E)
.
AMC 8, 1999, Problem 23
Square has sides of length
Segments
and
divide the square's area into three equal parts. How long is segment
(A) (B)
(C)
(D)
(E)
.
AMC 8, 1999, Problem 25
Points and
are midpoints of the sides of right triangle
Points
are midpoints of the sides of triangle
, etc. If the dividing and shading process is done
times (the first three are shown) and
, then the total area of the shaded triangles is nearest
(A) (B)
(C)
(D)
(E)
.