Problem Solving in Geometry

Introduction to Geometry in Algebra

Geometry is the study of shapes, sizes, and properties of space. In College Algebra, geometric concepts are often applied to solve real-world problems involving linear equations and inequalities. Understanding perimeter, area, and volume is essential for interpreting and solving these problems.

Perimeter and Area

Definitions and Units

• Perimeter: The total length around a two-dimensional shape. Measured in linear units (e.g., feet, meters).

• Area: The amount of surface covered by a shape. Measured in square units (e.g., square feet, square meters).

Common Formulas for Perimeter and Area

• Square:

  • Perimeter:

  • Area:

• Rectangle:

  • Perimeter:

  • Area:

• Triangle:

  • Perimeter:

  • Area:

• Trapezoid:

  • Area:

Example: Area of a Triangle

A sailboat has a triangular sail with an area of 24 square feet and a base of 4 feet. To find the height:

• Use

• Substitute:

• Solve: ft

Circles: Area and Circumference

Key Properties and Formulas

• Radius (r): Distance from the center to any point on the circle.

• Diameter (d): Twice the radius ().

• Area:

• Circumference:

The circumference is the perimeter of a circle.

Example: Area and Circumference of a Circle

Find the area and circumference of a circle with diameter 40 ft:

• Radius: ft

• Area: sq. ft

• Circumference: ft

Example: Comparing Pizza Values

To compare two pizzas (18-inch and 14-inch diameters):

• Large pizza: in, sq. in

• Medium pizza: in, sq. in

• Price per square inch (large):

• Price per square inch (medium):

The large pizza is the better buy.

Volume of Three-Dimensional Figures

Definition and Units

Volume is the amount of space occupied by a three-dimensional object, measured in cubic units (e.g., cubic inches, cubic centimeters).

Common Formulas for Volume

Example: Volume of a Cylinder

A cylinder with radius 3 in. and height 5 in. is doubled in height. The ratio of the volumes is:

• Small:

• Large:

• Ratio:

The larger cylinder has twice the volume of the smaller one.

Angles in Geometry

Angles of a Triangle

• The sum of the angles in any triangle is .

Example: Solving for Triangle Angles

Given: First angle is three times the second; third is 20° less than the second.

• Let = second angle

• First: ; Third:

• Equation:

• Solve:

• Angles: , ,

Complementary and Supplementary Angles

• Complementary angles: Two angles whose sum is .

• Supplementary angles: Two angles whose sum is .

Algebraic Expressions for Complements and Supplements

• If an angle is :

• Complement:

• Supplement:

Example: Angle and Its Complement

The measure of an angle is twice its complement. Let be the angle, its complement:

• Equation:

• Solve:

• Complement: