Circle Formulas
Calculate the area and circumference of a circle from its radius. The area grows with the square of the radius, while the circumference grows linearly.
A = πr², C = 2πr
Sphere Formulas
Calculate the surface area and volume of a sphere from its radius. The sphere has the smallest surface area of any shape enclosing a given volume.
SA = 4πr², V = (4/3)πr³
How It Works
Select a 2D or 3D shape from the dropdown, enter its dimensions, and the calculator applies the standard geometric formula. For 2D shapes it computes area and perimeter; for 3D shapes it computes surface area and volume.
Example Problem
Find the volume of a cylinder with radius 3 cm and height 10 cm:
- V = πr²h = π × 9 × 10 = 90π
- V ≈ 282.74 cm³
Key Concepts
Geometric formulas relate a shape's dimensions to its area, perimeter, surface area, and volume. For 2D shapes, area measures the enclosed region in square units and perimeter measures the boundary length. For 3D shapes, surface area is the total outer face area and volume is the interior space in cubic units. Many formulas involve π because circles and spheres are defined by their radius.
Applications
- Architecture: calculating floor areas, wall surfaces, and room volumes for material estimates
- Manufacturing: determining material quantities for sheet metal, tubing, and container production
- Landscaping: computing area for sod, mulch, or gravel coverage
- Education: foundational skill for trigonometry, calculus, and engineering coursework
Common Mistakes
- Confusing radius and diameter — many formulas use radius, and using diameter without dividing by 2 quadruples the area
- Mixing up area (square units) and volume (cubic units) — using the wrong formula dimension gives meaningless results
- Forgetting to square or cube the radius — area scales with r² and volume with r³, not linearly
- Using the slant height instead of the perpendicular height — cone and pyramid volume formulas require the vertical height
Frequently Asked Questions
What is the formula for the area of a trapezoid?
A = (1/2)(a + b) × h, where a and b are the parallel sides and h is the height. For parallel sides of 5 and 9 with height 4, the area is (1/2)(14) × 4 = 28 square units.
How to calculate the volume of a cone?
V = (1/3)πr²h. A cone with radius 3 and height 7 has volume (1/3) × π × 9 × 7 ≈ 65.97 cubic units. The cone is exactly one-third the volume of a cylinder with the same base and height.
What is the difference between surface area and volume?
Surface area measures the total area of the outside faces (in square units), while volume measures the space inside the shape (in cubic units). A sphere with radius 2 has surface area 16π ≈ 50.27 and volume 32π/3 ≈ 33.51.
Related Calculators
- Circle Calculator — detailed circle calculations including arcs, sectors, and segments.
- Triangle Calculator — right, equilateral, scalene, law of cosines, and law of sines.
- Trigonometry Calculator — compute sine, cosine, tangent, and inverse functions.
- Quadratic Equation Calculator — solve second-degree equations that arise from geometry problems.
- Area Unit Converter — convert between square meters, square feet, acres, and more.
- Volume Unit Converter — convert between liters, gallons, cubic meters, and other volume units.
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