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📐 Geometry Formulas

Quick reference for area, perimeter, and volume of triangles, rectangles, circles, and more

Area
78.5398
Perimeter
31.4159
Details
Radius5
Diameter10

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Geometry Formulas Guide

This tool integrates four of the most commonly used plane shape calculation functions: circle, rectangle, triangle, and trapezoid. Select a shape type and enter the necessary side length parameters to automatically calculate the area and perimeter. For triangles, it also supports Heron's formula (calculating area from three side lengths).

Geometry is the branch of mathematics that studies spatial shapes, sizes, and their properties. From ancient Egyptian land surveying to modern architectural CAD design, geometric calculation has always been an indispensable skill in practical work. This tool is especially suitable for students completing homework, engineers making quick estimates, and interior designers calculating material quantities.

Plane Shape Area and Perimeter Formulas

【Circle】Radius r
Area S = πr^2    Perimeter C = 2πr
Diameter d = 2r

【Rectangle】Length a, Width b
Area S = a×b     Perimeter C = 2(a+b)
Diagonal d = sqrt(a^2+b^2)

【Triangle】Base b, Height h
Area S = (1/2)×b×h   Perimeter C = a+b+c
Heron's formula: S = sqrt[s(s-a)(s-b)(s-c)], s=(a+b+c)/2

【Trapezoid】Top base a, Bottom base b, Height h
Area S = (a+b)×h/2
Midsegment m = (a+b)/2

Practical Examples

📋
【Example 1】Circular flower bed Radius r=5m Area = π×5^2 = 78.54 m^2 Perimeter = 2π×5 = 31.42 m Application: Estimate the number of tiles needed 【Example 2】Rectangular room Length 8m, Width 5m Area = 40 m^2, Perimeter = 26m Diagonal = sqrt(64+25) = 9.43m Application: Calculate wallpaper quantity and diagonal distance 【Example 3】Triangular plot Base 12m, Height 8m Area = (1/2)×12×8 = 48 m^2 With sides 9,12,15 (right triangle): Heron's area also = 48 m^2 ✓ 【Example 4】Trapezoidal channel cross-section Top base 4m, Bottom base 8m, Height 3m Area = (4+8)×3/2 = 18 m^2 Midsegment = 6m

Important Notes

💡
- All length parameters must be positive; negative or zero values will cause calculation errors - The triangle area formula (½bh) only requires base and height; three sides are for perimeter and Heron's formula verification - Heron's formula requires the three sides to satisfy the triangle inequality (sum of any two sides greater than the third) - Circle calculation precision depends on the value of π; this tool uses JavaScript's Math.PI - The trapezoid midsegment length equals the average of the top and bottom bases, independent of height

Application Scenarios

  • - Construction: Room area calculation, material budgeting, flooring/paint quantity estimation
  • - Landscaping: Lawn area, fence length, circular fountain/flower bed design
  • - Civil engineering: Road cross-section area, earthwork calculation, foundation bearing capacity analysis
  • - Education: Geometry homework verification, exam review formula reference
  • - DIY projects: Fabric cutting area, frame perimeter, desk film sizing

From Plane to Solid

Mastering plane shape calculations is the foundation for understanding solid geometry. Cylinder volume = base area × height = πr^2 h, cone volume = (1/3)πr^2 h, sphere volume = (4/3)πr^3. These solid formulas are essentially derived by "extruding" or rotating plane shapes along a dimension.

In practical engineering, more complex shape parameterization methods are needed — such as polygon decomposition (breaking complex shapes into basic shapes), integration (curved boundary regions), and numerical methods (finite element meshes). The basic shape calculations covered by this tool are the starting point for these advanced methods.

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