Geometric
Problems
To solve word problems involving geometric shapes, write down the formula or formulas referred to in the problem. For example, after reading ‘‘The perimeter of a rectangular room . . . ’’ write P = 2L + 2W. Then fill in the information given about the formula. For example, after reading ‘‘The peri- meter of the room is 50 feet . . . ’’ write P = 50 and 50 = 2L + 2W. ‘‘Its width is two-thirds its length.’’ Write W = ⅔ L and 50 = 2L + 2W becomes 50 = 2L + 2(⅔ L).
The formulas you will need in this section
are listed below.
Rectangle
Formulas
·
Area A =
LW
·
Perimeter
(the length around its sides) P = 2L + 2W
·
Diagonal
D2 = L2 + W2
Triangle
Formulas
·
Area A =
½ BH
·
Perimeter
P = a + b + c (for any triangle)
·
Pythagorean
Theorem a2 + b2 = c2 (for right triangles only)
Miscellaneous
Shapes
· Volume
of a right circular cylinder V = πr2h, where r is the cylinder’s radius,
h is the cylinder’s height
· Surface
area of a sphere (ball) is SA = 4πr2, where r is the sphere’s radius
· Area of
a circle A = πr2,
where r is the circle’s radius
· Volume
of a rectangular box V = LWH, where L is the box’s length, W is the box’s
width, and H is the box’s height
In many of the examples and practice
problems, there will be two solutions to the equation but only one solution to
the geometric problem. The extra solutions come from solving quadratic
equations.
Examples
A square has a diameter of 50 cm. What is the
length of each side?
Let x represent the length of each side.
The diagonal formula for a rectangle is D2 ¼ L2
þW2. In this example, D = 50, L = x, and W = x. D2 = L2 +W2
becomes x2 + x2 = 502.
Each side is 25√2 cm long.
A rectangle is one inch longer than it is
wide. Its diameter is five inches. What are its dimensions?
L = W + 1
The diagonal formula for a rectangle is D2
= L2 +W2. In this example,
= 5
and L = W + 1. D2 = L2 +W2 becomes 52
= (W þ 1)2 += W2.
The width is 3 inches and the length is 3 + 1
= 4 inches.
Practice
1. The
diameter of a square is 60 feet. What is the length of its sides?
2. A
rectangle has one side 14 cm longer than the other. Its diameter is 34 cm. What
are its dimensions?
3. The
length of a rectangle is 7 inches more than its width. The diagonal is 17
inches. What are its dimensions?
4. The
width of a rectangle is three-fourths its length. The diagonal is 10 inches.
What are its dimensions?
5. The
diameter of a rectangular classroom is 34 feet. The room’s length is 14 feet
longer than its width. How wide and long is the classroom?
Solutions
When there is more than one solution to an
equation and one of them is not valid, only the valid solution will be given.
1. Let x represent
the length of each side (in feet). The diagonal is 60 feet, so D ¼ 60. The
formula D2 = L2 +W2 becomes 602 = x2
+ x2.
The length of the square’s sides is 30 √2 feet, or approximately 42.4 feet.
2. The
length is 14 cm more than the width, so L = W + 14. The diameter is 34 cm, so D
= 34. The formula D2 = L2 + W2 becomes 342
= (W + 14)2 + W2.
The width is 16 cm and the length is 16 + 14 =
30 cm.
3. The
length is 7 inches more than the width, so L = W + 7. The diagonal is 17
inches. The formula D2 = L2 + W2 becomes 172 =
(W + 7)2 + W2.
The rectangle’s width is 8 inches and its
length is 8 + 7 = 15 inches.
4. The
width is three-fourths its length, so W = L. The diagonal is 10 inches, so the formula D2 = L2
+ W2 becomes 102 = L2 + (¾ L)2.
The rectangle’s length is 8 inches and its
width is 8(¾) = 6 inches.
5. The
classroom’s length is 14 feet more than its width, s L = W + 14. The diameter
is 34 feet. The formula D2 = L2 + W2 becomes
342 = (W + 14)2 + W2.
The classroom is 16 feet wide and 16 + 14 = 30
feet long.
“Sumber Informasi”
Thanks for reading Geometric Problems. Please share...!
