Geometry & Trigonometry Examples & Practice Problems

Work through examples and practice problems for geometry & trigonometry symbols.

Geometry & Trigonometry Examples

Work through these examples to understand geometric and trigonometric symbols.

Here are common usage examples for each symbol:

Pi ( $\pi$ ): The ratio of a circle's circumference to its diameter ( $\pi \approx 3.14159$ )
Degree ( $°$ ): Measure of angle ( $90°$ )
Angle ( $\angle$ ): Denotes an angle ( $\angle ABC$ )
Perpendicular ( $\perp$ ): Lines meeting at a $90°$ angle ( $AB \perp CD$ )
Parallel ( $\parallel$ ): Lines that never intersect ( $AB \parallel CD$ )
Triangle ( $\triangle$ ): Represents a triangle shape ( $\triangle ABC$ )
Congruent ( $\cong$ ): Same shape and size (geometric equality) ( $\triangle ABC \cong \triangle DEF$ )
Similar ( $\sim$ ): Same shape but different size ( $\triangle ABC \sim \triangle DEF$ )
Theta ( $\theta$ ): Common variable for an unknown angle ( $\theta = 45°$ )

Circle: The circumference of a circle with radius $r$ is $C = 2\pi r$ .
Right Angle: If $AB \perp CD$ , then $\angle ABC = 90°$ .
Similar Triangles: If $\triangle ABC \sim \triangle DEF$ , their corresponding angles are equal and sides are proportional.

Problem 1: Find the circumference of a circle with radius $r = 5$ units.

Solution: Using the formula $C = 2\pi r$ :

Answer: $C = 10\pi$ units (or approximately $31.42$ units)

Problem 2: In triangle $\triangle ABC$ , if $\angle ABC = 90°$ and line $AB \perp BC$ , what can you say about the triangle?

Solution:

Answer: $\triangle ABC$ is a right triangle with the right angle at $B$ .

Problem 3: If $\triangle ABC \sim \triangle DEF$ and $AB = 3$ , $DE = 6$ , find the scale factor.

Solution:

Answer: Scale factor $k = 2$

Geometric and trigonometric concepts appear everywhere in daily life, from construction to navigation to design.

Right Angles ( $\perp$ ): Ensure walls are perpendicular. When hanging pictures or installing shelves, use a level to check that surfaces are at $90°$ angles ( $\perp$ ).
Parallel Lines ( $\parallel$ ): Install parallel elements. When laying floor tiles or installing fence posts, ensure they're parallel ( $\parallel$ ) for a professional look.
Area Calculations: Calculate paint needed. For a rectangular room $12$ feet by $10$ feet, area = $12 \times 10 = 120$ square feet. For a circular table with radius $2$ feet, area = $\pi \times 2^2 = 4\pi \approx 12.57$ square feet.

Angles ( $\angle$ , $°$ ): Use angles for navigation. When turning, a $90°$ turn means you're going perpendicular to your original direction. A $180°$ turn means you're going back the way you came.
Distance Calculations: Use the Pythagorean theorem. If you walk $3$ blocks north and $4$ blocks east, you're $\sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$ blocks from your starting point.

Circles ( $\pi$ ): Calculate serving sizes. If a pizza has radius $6$ inches, its area is $\pi \times 6^2 = 36\pi \approx 113$ square inches. This helps determine how many people it can serve.
Angles: Cut food at specific angles. Cutting a sandwich diagonally creates $45°$ angles, giving you triangular pieces.

Angles: Understand ball trajectories. When shooting a basketball, the angle ( $\theta$ ) affects the arc. A $45°$ angle often provides optimal range.
Triangles ( $\triangle$ ): Set up equipment. When positioning a camera tripod, the three legs form a triangle for stability.

Parallel Lines ( $\parallel$ ): Create visual harmony. In graphic design, parallel lines create rhythm and organization.
Right Angles ( $\perp$ ): Establish structure. Most buildings use perpendicular lines ( $\perp$ ) for stability and aesthetic appeal.
Similar Shapes ( $\sim$ ): Scale designs. If a logo design is similar ( $\sim$ ) but needs to be larger, maintain proportions.

Angles: Optimize viewing angles. For computer monitors, a $90°$ viewing angle ( $\perp$ to your line of sight) reduces eye strain.
Circles: Understand screen sizes. A $24$ -inch monitor refers to the diagonal, but the actual viewing area depends on the circle's area calculations.

When working with shapes and angles:

Geometry and trigonometry aren't just academic subjects—they're essential tools for understanding and interacting with the physical world!