Chapter 19
Using Formulas in Geometric Story Problems
Story problems (practical applications) appear in all types and levels of mathematics. A really nice feature of geometric story problems is that you can almost always find a formula to apply.
The Problems You'll Work On
From the beginning and through the middle and end, the geometric story problems in this chapter require the following skills:
What to Watch Out For
Don't get ahead of yourself when working these problems. Watch out for the following:
Working around the Perimeter of Rectangles
808–813 Solve the problems using the formula for the perimeter of a rectangle: P = 2(l + w).
808. What is the perimeter of a rectangular yard measuring 6 yards wide and 8 yards long?
809. If the perimeter of a rectangular plot is 400 feet, and the width is 50 feet, then what is the length of the plot?
810. A rectangular room has a length that's three times the width. The perimeter is 480 feet. What is the length?
811. A rectangular pool has a length that's 4 feet greater than the width. It has a perimeter of 200 feet. What is the length of the pool?
812. A rectangle has a width that's 30 feet more than one-third the length. If the perimeter is 420 feet, then what is the length of the rectangle?
813. You have 600 feet of fencing and need to create a rectangular pen for your llama. You want the length and width to be in a ratio of 3:2. What length and width will work?
Using the Area Formula for a Trapezoid
814–815 Solve the problems using the formula for the area of a trapezoid: A = 
h(b1 + b2).
814. What is the area of a trapezoid if the parallel bases measure 5 feet and 8 feet, and if the perpendicular distance between those bases is 4 feet?
815. The area of a trapezoid is 170 square feet. What is the length of the second base if the first base measures 16 feet and the height is 5 feet?
Tackling Area and Perimeter of a Square
816–817 Solve the problems using the formulas for the perimeter and area of a square: P = 4s and A = s2.
816. If the area of a square is 64 square feet, then what is its perimeter?
817. If the perimeter of a square is 64 inches, then what is the area of the square?
Solving Problems Using the Perimeter of a Triangle
818–819 Solve the problems using the fact that the perimeter of a triangle is equal to the sum of the measures of its sides.
818. A triangle with a perimeter of 42 inches has one side twice the length of the shortest side and the third side 6 inches greater than the shortest side. What are the lengths of the three sides?
819. If you have a fenced-in triangular garden where one side is 8 feet, the second side is three times that length, and the third side is 8 feet shorter than the second side, then what would the lengths of the sides of an equilateral triangle be, if you made it from the current fencing?
Working with the Area and Perimeter of a Rectangle
820–821 Solve the problems using the formulas for the perimeter and area of a rectangle: P = 2(l + w), A = lw.
820. A rectangle's length is 2 feet less than twice its width, and its area is 180 square feet. What is the rectangle's width?
821. A rectangle's width is 4 inches greater than half its length, and its perimeter is 248 inches. What is the rectangle's area?
Using the Formula for the Area of a Hexagon
822–823 Solve the problems using the formula for the area of a regular hexagon: A = 
x2, where x is the length of a side.
822. The perimeter of a regular hexagon is 360 cm. What is its area?
823. The area of a regular hexagon is 216
square feet. What is the hexagon's perimeter?
Getting Pythagoras Involved in the Area of a Triangle
824–827 Solve the problems using the formula for the area of a triangle, A = 
bh and the Pythagorean theorem, a2 + b2 = c2.
824. A triangle has an area of 60 square centimeters and a height of 10 centimeters. What is the length of the base of the triangle?
825. A triangle has an area of 2 square feet and a base measuring 36 inches. What is the height of the triangle? (Remember: 1 sq ft = 12 in × 12 in = 144 sq in)
826. A right triangle has a hypotenuse measuring 5 feet and a leg measuring 3 feet. What is its area?
827. The area of a right triangle is 30 square feet. If the measure of one of the legs is two more than twice the other leg, then what is the measure of the hypotenuse?
Making Use of the Volume of a Box
828–829 Solve the problems using the formula for the volume of a right rectangular prism (box), V = lwh.
828. The volume of a box is 48 cubic inches. If the length is four times the width, and the height is 3 inches, then what is the length of the box?
829. A cube has a volume of 64 cubic centimeters. What is its height?
Working with the Volume of a Cylinder
830–831 Solve the problems using the formula for the volume of a right circular cylinder (can): V = πr2h
830. What is the height of a cylinder whose volume is 54 cubic inches and whose radius is 3 inches?
831. What is the radius of a cylinder whose volume is 192π cubic feet and whose height is 12 feet?
Changing Radians to Degrees and Degrees to Radians
832–837 Solve the problems using the angle measure proportion: 
.
832. What degree measure corresponds to 
radians?
833. What radian measure corresponds to 150°?
834. What radian measure corresponds to 300°?
835. degree measure corresponds to 
radians?
836. What radian measure corresponds to −30°?
837. What degree measure corresponds to 2 radians?
Determining the Height of an Object
838–841 Solve the problems using the formula for the height, in feet, of a tennis ball shot from a launcher after t seconds: h = −16t2 + 48t.
838. How high is the tennis ball after 2 seconds?
839. How high is the tennis ball after 2.5 seconds?
840. After how many seconds is the height 20 feet?
841. After how many seconds does the ball hit the ground?
Working with Heron's Formula for the Area of a Triangle
842–845 Solve the problems using Heron's formula for the area of a triangle: A = 
, where a, b, and c are the lengths of the sides and s is the semi-perimeter.
842. What is the area of a triangle whose sides measure 5, 12, and 13 inches?
843. What is the area of a triangle whose sides measure 6, 8, and 10 feet?
844. The area of a triangle is 6 square yards. If two of the sides measure 3 and 5 yards, respectively, then what is the measure of the third side of the triangle?
845. The area of an equilateral (all sides the same length) triangle is 25
square inches. What is the perimeter of the triangle?