Geometry Basics
Master shapes, angles, and spatial reasoning for the ParaPro test
What You'll Learn
Basic 2D Shapes
Two-dimensional figures
Polygons (Closed Shapes)
Triangle
3 sides, 3 angles
Sum of angles = 180°
Square
4 equal sides, 4 right angles
All angles = 90°
Rectangle
4 sides, 4 right angles
Opposite sides equal
Rhombus
4 equal sides
Like a tilted square
Pentagon
5 sides, 5 angles
Sum of angles = 540°
Hexagon
6 sides, 6 angles
Sum of angles = 720°
Types of Triangles
By Sides:
Equilateral
All 3 sides equal
Isosceles
2 sides equal
Scalene
No sides equal
By Angles:
Right
One 90° angle
Acute
All angles < 90°
Obtuse
One angle > 90°
Understanding Angles
Measuring rotation
Types of Angles
Acute Angle
Less than 90°
Example: 45°, 60°, 30°
Right Angle
Exactly 90°
Forms an "L" shape
Obtuse Angle
More than 90°, less than 180°
Example: 120°, 135°, 150°
Straight Angle
Exactly 180°
Forms a straight line
Angle Relationships
Complementary Angles
Two angles that add up to 90°
Example: 30° + 60° = 90°
Supplementary Angles
Two angles that add up to 180°
Example: 110° + 70° = 180°
Perimeter
The distance around a shape
Perimeter Formulas
Square
P = 4s
s = side length
Rectangle
P = 2l + 2w
l = length, w = width
Triangle
P = a + b + c
Add all three sides
Circle (Circumference)
C = 2πr or πd
r = radius, d = diameter
Example: Finding Perimeter
Find the perimeter of a rectangle with length 8 cm and width 5 cm
Step 1: Use the formula P = 2l + 2w
Step 2: P = 2(8) + 2(5)
Step 3: P = 16 + 10 = 26 cm
Area
The space inside a shape
Area Formulas
Square
A = s²
s = side length
Rectangle
A = l × w
l = length, w = width
Triangle
A = ½ × b × h
b = base, h = height
Circle
A = πr²
r = radius (π ≈ 3.14)
Parallelogram
A = b × h
b = base, h = height
Trapezoid
A = ½(b₁ + b₂) × h
b₁, b₂ = parallel bases
Example: Finding Area
Triangle: base = 10 cm, height = 6 cm
A = ½ × b × h
A = ½ × 10 × 6
A = 30 cm²
Circle: radius = 5 cm
A = πr²
A = 3.14 × 5²
A ≈ 78.5 cm²
3D Shapes and Volume
Three-dimensional figures
Common 3D Shapes
Cube
6 square faces
V = s³
Rectangular Prism
6 rectangular faces
V = l × w × h
Sphere
Perfectly round ball
V = ⁴⁄₃πr³
Cylinder
Circular top and bottom
V = πr²h
Cone
Circular base, point top
V = ⅓πr²h
Pyramid
Polygon base, point top
V = ⅓ × base × h
Example: Finding Volume
Find the volume of a rectangular box: length = 4 in, width = 3 in, height = 5 in
Step 1: Use the formula V = l × w × h
Step 2: V = 4 × 3 × 5
Step 3: V = 60 cubic inches (in³)
Common Mistakes to Avoid
Confusing perimeter and area
Perimeter is distance around, area is space inside
✓ Perimeter = linear units (cm), Area = square units (cm²)
Forgetting to use height for triangles
Using a slant side instead of the perpendicular height
✓ Height must be perpendicular to the base
Using diameter instead of radius
Most circle formulas use radius, not diameter
✓ Remember: radius = diameter ÷ 2
Wrong units in answers
Forgetting to include proper units
✓ Perimeter: cm, m | Area: cm², m² | Volume: cm³, m³
Practice Problems
Test your understanding
Perimeter
Find the perimeter of a square with sides of 7 inches.
Area
What is the area of a triangle with base 12 cm and height 8 cm?
Angles
If two angles are supplementary and one is 65°, what is the other?
Volume
Find the volume of a cube with sides of 4 cm.
Circle
What is the area of a circle with diameter 10 cm? (Use π = 3.14)
Show Answers
1. P = 4s = 4 × 7 = 28 inches
2. A = ½ × 12 × 8 = 48 cm²
3. 180° - 65° = 115°
4. V = s³ = 4³ = 64 cm³
5. r = 5 cm, A = πr² = 3.14 × 25 = 78.5 cm²
Key Takeaways
Shapes: Know properties of triangles, squares, rectangles, circles
Angles: Acute < 90°, Right = 90°, Obtuse > 90°
Perimeter: Add all sides (or use formulas)
Area: Measure space inside (square units)
Volume: Measure space inside 3D shapes (cubic units)
Continue Learning
Ready to Practice?
Test your geometry knowledge with practice questions
