Problems

A unit square is rotated $45^\circ$ about its center. What is the area of the region swept out by the interior of the square?

Six spheres of radius $1$ are positioned so that their centers are at the vertices of a regular hexagon of side length $2$ . The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to …

In $\triangle ABC$ , $AB = 86$ , and $AC=97$ . A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$ . Moreover $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is $BC$ ?

Six points are equally spaced around a circle of radius 1. Three of these points are the vertices of a triangle that is neither equilateral nor isosceles. What is the area of this triangle?

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 $\frac{a}{b}$ ?

In triangle $ABC$ , medians $AD$ and $CE$ intersect at $P$ , $PE=1.5$ , $PD=2$ , and $DE=2.5$ . What is the area of $AEDC$ ?

In triangle $ABC$ , $AB = 13$ , $BC = 14$ , and $CA = 15$ . Distinct points $D$ , $E$ , and $F$ lie on segments $\overline{BC}$ , $\overline{CA}$ , and $\overline{DE}$ , respectively, such that $\overline{AD} \perp \overline{BC}$ , $\overline{DE} \perp \overline{AC}$ , and $\overline{AF} \perp \overline{BF}$ . The …

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

Let $\angle ABC = 24^\circ$ and $\angle ABD = 20^\circ$ . What is the smallest possible degree measure for angle $CBD$ ?

Externally tangent circles with centers at points $A$ and $B$ have radii of lengths $5$ and $3$ , respectively. A line externally tangent to both circles intersects ray $AB$ at point $C$ . What is $BC$ ?

Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of $\triangle ABC$ ?

The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\frac{2\pi}{3}$ , where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2$ . What is the area enclosed by the curve?

Let points $A = (0 ,0 ,0)$ , $B = (1, 0, 0)$ , $C = (0, 2, 0)$ , and $D = (0, 0, 3)$ . Points $E$ , $F$ , $G$ , and $H$ are midpoints of line segments $\overline{BD},\text{ } \overline{AB}, \text{ } \overline {AC},$ and $\overline{DC}$ respectively. What is the area of $EFGH$ ?

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?

The point in the xy-plane with coordinates $(1000, 2012)$ is reflected across the line $y = 2000$ . What are the coordinates of the reflected point?

Point $B$ is due east of point $A$ . Point $C$ is due north of point $B$ . The distance between points $A$ and $C$ is $10\sqrt 2$ , and $\angle BAC = 45^\circ$ . Point $D$ is $20$ meters due north of point $C$ . The distance $AD$ is between which two integers?

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?

Three circles with radius 2 are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure?

Jesse cuts a circular paper disk of radius $12$ along two radii to form two sectors, the smaller having a central angle of $120$ degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?

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 $BFDG$ ?

Four distinct points are arranged on a plane so that the segments connecting them have lengths $a$ , $a$ , $a$ , $a$ , $2a$ , and $b$ . What is the ratio of $b$ to $a$ ?

A solid tetrahedron is sliced off a wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face …

A rectangular region is bounded by the graphs of the equations $y=a, y=-b, x=-c,$ and $x=d$ , where $a,b,c,$ and $d$ are all positive numbers. Which of the following represents the area of this region?

Square $EFGH$ has one vertex on each side of square $ABCD$ . Point $E$ is on $\overline{AB}$ with $AE=7\cdot EB$ . What is the ratio of the area of $EFGH$ to the area of $ABCD$ ? {{}}

Circles $A, B,$ and $C$ each have radius 1. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}$ . What is the area inside Circle $C$ but outside circle $A$ and circle $B$  ?