Select Grade Level
Elementary School Shapes
Square
Formula: Area = side × side = a²
Example:
If a square has sides of 5 cm:
Area = 5 cm × 5 cm = 25 cm²
Rectangle
Formula: Area = length × width = a × b
Example:
If a rectangle is 8 cm long and 3 cm wide:
Area = 8 cm × 3 cm = 24 cm²
Triangle
Formula: Area = ½ × base × height = ½ × b × h
Example:
If a triangle has base 6 cm and height 4 cm:
Area = ½ × 6 cm × 4 cm = 12 cm²
Counting Unit Squares
Concept: Area is the number of unit squares that cover a shape
Example:
Count the squares to find the area:
5 squares wide × 3.5 squares tall = 17.5 square units
Middle School Shapes
Circle
Formula: Area = π × radius² = πr²
Example:
If a circle has radius 7 cm (use π ≈ 3.14):
Area = 3.14 × 7 cm × 7 cm ≈ 153.86 cm²
Parallelogram
Formula: Area = base × height = b × h
Example:
If a parallelogram has base 10 cm and height 5 cm:
Area = 10 cm × 5 cm = 50 cm²
Trapezoid
Formula: Area = ½ × (a + b) × h
Example:
If a trapezoid has parallel sides 6 cm and 4 cm, height 5 cm:
Area = ½ × (6 cm + 4 cm) × 5 cm = 25 cm²
Rhombus
Formula: Area = ½ × diagonal₁ × diagonal₂ = ½ × d₁ × d₂
Example:
If a rhombus has diagonals 8 cm and 6 cm:
Area = ½ × 8 cm × 6 cm = 24 cm²
Composite Shapes
Method: Break into simple shapes, calculate each area, then add/subtract
Example:
Shape made of a rectangle (10×4) and a square (5×5):
Area = (10×4) + (5×5) = 40 + 25 = 65 units²
High School Shapes
Ellipse
Formula: Area = π × a × b
Example:
If an ellipse has semi-major axis 8 cm and semi-minor axis 5 cm (π ≈ 3.14):
Area = 3.14 × 8 cm × 5 cm ≈ 125.6 cm²
Regular Pentagon
Formula: Area = ½ × perimeter × apothem = ½ × (5s) × a
Example:
If a regular pentagon has side length 6 cm and apothem 4 cm:
Perimeter = 5 × 6 cm = 30 cm
Area = ½ × 30 cm × 4 cm = 60 cm²
Sector of a Circle
Formula: Area = (θ/360) × πr²
Example:
If a sector has radius 10 cm and central angle 60° (π ≈ 3.14):
Area = (60/360) × 3.14 × 10 cm × 10 cm ≈ 52.33 cm²
Cone Surface Area
Formula: Total Area = πr² + πrl
Example:
If a cone has radius 3 cm, height 4 cm (l=5 cm by Pythagorean theorem):
Base = π×3² ≈ 28.27 cm²
Lateral = π×3×5 ≈ 47.12 cm²
Total ≈ 75.4 cm²
Pyramid Surface Area
Formula: Area = base area + ½ × perimeter × slant height
Example:
Square pyramid with base 6 cm, slant height 5 cm:
Base = 6×6 = 36 cm²
Lateral = ½ × 24 × 5 = 60 cm²
Total = 96 cm²
Circle Segment
Formula: Area = (θ/360)πr² - ½r²sinθ
Example:
Segment with r=10 cm, θ=60° (π≈3.14):
Sector area ≈ 52.33 cm²
Triangle area ≈ 43.30 cm²
Segment ≈ 9.03 cm²