Calculate the area of 9 standard geometric shapes with live-updating SVG diagrams, units, and worked steps.
Measurement Unit
Radius (r)10 cm
Calculated Area (circle)
314.1593 cm²
The total 2D planar space enclosed inside the boundary lines of the circle.
How is this calculated?
Formula: A = π × r²
Substitute: A = π × 10²
Calculate: A = π × 100
Result: A ≈ 314.159265 cm²
About the Area Calculator
Area is a fundamental mathematical quantity that measures the two-dimensional space enclosed within a boundary or geometric shape. Calculated in square units (such as square meters, square centimeters, or square feet), area is critical across diverse disciplines including architecture, civil engineering, agriculture, and graphic design. The mathematical approach to calculating area varies based on the geometry of the shape: linear polygons like rectangles and triangles use simple products of perpendicular base and height dimensions; regular curves like circles and ellipses require transcendental constants (π) to account for round boundaries; and complex polygons utilize integration or triangulation methods. Understanding these formulas allows for accurate space planning, material cost estimation, and structural analysis.
Mathematical Formula & Logic
Area formulas represent the mathematical relationship between the boundary dimensions of specific geometric shapes:
1. Rectangle:
Area = Length × Width
2. Circle:
Area = π × Radius²
3. Triangle (Base-Height):
Area = 0.5 × Base × Height
4. Triangle (Heron's):
Area = √[s(s - a)(s - b)(s - c)] where s = (a + b + c) / 2
5. Trapezoid:
Area = 0.5 × (Base_A + Base_B) × Height
6. Ellipse:
Area = π × Semi-major_axis × Semi-minor_axis
7. Circle Sector:
Area = π × Radius² × (Angle / 360°)
8. Parallelogram:
Area = Base × Height
9. Rhombus:
Area = 0.5 × Diagonal_1 × Diagonal_2
10. Regular Polygon:
Area = 0.25 × n × Side_Length² / tan(π / n)
Step-by-Step Example
Calculate the area of a trapezoid with parallel bases of 12 cm and 16 cm, and a vertical height of 8 cm:
1. Identify the variables:
- Base A = 12 cm
- Base B = 16 cm
- Height = 8 cm
2. Apply the trapezoid area formula:
Area = 0.5 × (Base_A + Base_B) × Height
Area = 0.5 × (12 + 16) × 8
3. Simplify the expression:
Area = 0.5 × 28 × 8
Area = 14 × 8 = 112
4. The area of the trapezoid is 112 square centimeters (112 cm²).
Reference Data & Values
shape
key variables
area formula
example calculation
Circle
Radius (r)
πr²
r=5 cm → Area ≈ 78.54 cm²
Rectangle
Length (l), Width (w)
l × w
l=10, w=5 → Area = 50
Triangle (Base-Height)
Base (b), Height (h)
0.5 × b × h
b=6, h=4 → Area = 12
Triangle (Heron's)
Sides (a, b, c)
√[s(s-a)(s-b)(s-c)]
a=3, b=4, c=5 → Area = 6
Trapezoid
Bases (a, b), Height (h)
0.5(a+b)h
a=12, b=16, h=8 → Area = 112
Ellipse
Axes (a, b)
π × a × b
a=6, b=4 → Area ≈ 75.40
Circle Sector
Radius (r), Angle (θ)
πr²(θ/360)
r=10, θ=90 → Area ≈ 78.54
Parallelogram
Base (b), Height (h)
b × h
b=10, h=6 → Area = 60
Rhombus
Diagonals (d1, d2)
0.5 × d1 × d2
d1=12, d2=16 → Area = 96
Regular Polygon
Sides (n), Length (s)
0.25 × n × s² / tan(π/n)
n=5, s=10 → Area ≈ 172.05
Frequently Asked Questions
Area measures the two-dimensional space inside a shape's boundary, whereas perimeter measures the total length of the outer boundary itself. For instance, a fence around a yard represents its perimeter, while the grass inside represents its area.
Area measures two-dimensional space, which is defined by multiplying two perpendicular linear measurements together (e.g., length in meters × width in meters). This multiplication of units results in units squared (e.g., m², ft²).
Vertical height (perpendicular height) is the straight-line distance from the highest point of a shape to its base at a 90-degree angle. Slant height is the length of the slanted outer edge. Standard area formulas (triangle, trapezoid, parallelogram) require the vertical height, not the slant height.
Irregular shapes can be calculated using decomposition (triangulation). You break the complex shape down into smaller standard shapes (rectangles, triangles, circles), calculate the area of each, and sum them to find the total.
Heron's formula calculates the area of any triangle given only its three side lengths (a, b, c). First, find the semi-perimeter s = (a + b + c) / 2. The area is then computed as A = √[s × (s - a) × (s - b) × (s - c)].
A sector is a pie-slice section of a circle. Its area is a fraction of the circle's total area, calculated as A = π × r² × (θ / 360) where θ is the central angle in degrees, or A = 0.5 × r² × θ if the angle is in radians.
The area of a regular polygon with n sides of length s is calculated using the formula A = 0.25 × n × s² / tan(π / n). Alternatively, it can be computed as A = 0.5 × perimeter × apothem, where the apothem is the distance from the center to the midpoint of any side.
The area of an ellipse is A = π × a × b, where a is the semi-major axis (half the width) and b is the semi-minor axis (half the height). If a = b = r, the formula simplifies to A = π × r², showing that a circle is a special case of an ellipse.