Geometry

The sides of a square region, measured to the nearest centimeter, are 6 centimeters long. The least possible value of the actual area of the square region is

In the figure, the 6 small squares are identical, each with sides of length 1. What is the outer perimeter (shown in bold) of the entire figure?

The figure shown above consists of a shaded 9-sided polygon and 9 unshaded isosceles triangles.For each isosceles triangle,the longest side is a side of the shaded polygon and the two sides of equal length are extensions of the two adjacent sides of the shaded polygon.What is the value of a ?

The smaller rectangle in the figure above represents the original size of a parking lot before its length and width were each extended by w feet to make the larger rectangular lot shown. If the area of the enlarged lot is twice the area of the original lot, what is the value of w?

The outer dimensions of a closed rectangular cardboard box are 8 centimeters by 10 centimeters by 12 centimeters, and the six sides of the box are uniformly 1/2 centimeter thick. A closed canister in the shape of a right circular cylinder is to be placed inside the box so that it stands upright when the box rests on one of its sides. Of all such canisters that would fit, what is the outer radius, in centimeters, of the canister that occupies the maximum volume?

Three-fourths of the area of a rectangular lawn 30 feet wide by 40 feet long is to be enclosed by a rectangular fence. If the enclosure has full width and reduced length rather than full length and reduced width, how much less fence will be needed?

A border of uniform width is placed around a rectangular photograph that measures 8 inches by 10 inches. If the area of the border is 144 square inches, what is the width of the border, in inches?

In the floor plan of an executive's beach house above, the north and south walls of the living room are parallel. What is the floor area, in square feet, of the bedroom?

In the figure shown, the triangle is inscribed in the semicircle. If the length of line segment AB is 8 and the length of line segment BC is 6, what is the length of arc ABC?

In the figure shown, what is the value of x ?

Abdul, Barb, and Carlos all live on the same straight road, on which their school is also located. The school is halfway between Abdul’s house and Barb’s house. Barb’s house is halfway between the school and Carlos’s house. If the school is 4 miles from Carlos’s house, how many miles is Abdul’s house from Carlos’s house?

For a party, three solid cheese balls with diameters of 2 inches, 4 inches, and 6 inches, respectively, were combined to form a single cheese ball. What was the approximate diameter, in inches, of the new cheese ball?

In the figure shown, if the area of the shaded region is 3 times the area of the smaller circular region, then the circumference of the larger circle is how many times the circumference of the smaller circle?

The annual budget of a certain college is to be shown on a circle graph. If the size of each sector of the graph is to be proportional to the amount of the budget it represents , how many degrees of the circle should be used to represent an item that is 15 percent of the budget?

In the figure above, ABCD is a parallelogram and E is the midpoint of side AD. The area of triangular region ABE is what fraction of the area of the quadrilateral region BCDE?

In triangle ABC above, what is x in terms of z ?

The interior of a rectangular carton is designed by a certain manufacturer to have a volume of x cubic feet and a ratio of length to width to height of 3:2:2. In terms of x, which of the following equals the height of the carton, in feet?

In quadrilateral ABCD above, what is the length of AB ?

Tanks A and B are each in the shape of a right circular cylinder. The interior of Tank A has a height of 10 meters and a circumference of 8 meters, and the interior of tank B has a height of 8 meters and a circumference of 10 meters. The capacity of tank A is what percent of the capacity of tank B?

If a sphere with radius r is inscribed in a cube with edges of length e, which of the following expresses the relationship between r and e ?