📐 Math ResourcesLast updated March 14, 2026

Geometry Guide: Area, Volume & Shape Calculations Made Easy

Every geometry formula from triangles to 3D solids — with visual examples

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3+

Calculators

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15+

Shapes Covered

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8 min

Read Time

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Key Takeaways

Rectangle area = l × w, Triangle area = ½bh, Circle area = πr² — these three formulas solve 80% of area problems.
Volume of a box = l × w × h, Cylinder = πr²h, Sphere = (4/3)πr³.
The Pythagorean theorem (a² + b² = c²) only works for right triangles. Use the Law of Cosines for all other triangles.
Surface area = "wrap" the shape in paper; volume = "fill" the shape with water — think physically before computing.
Always ensure consistent units before calculating — mixing feet and inches is the #1 source of geometry errors.

Geometry is the most visual branch of mathematics — and the most practical. Whether you're calculating the area of a room for flooring, the volume of a tank for capacity, or solving a triangle for engineering, having the right formulas and tools saves time and prevents costly errors. This guide — referencing NIST SI unit standards — covers every common shape with formulas, examples, and free tools like our area calculator and volume calculator.

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2D Area Formulas: Every Shape You Need

Rectangle: A = length × width. Triangle: A = ½ × base × height. Circle: A = πr². Trapezoid: A = ½(a + b) × h, where a and b are the parallel sides. Parallelogram: A = base × height. Ellipse: A = π × a × b, where a and b are the semi-axes. For composite shapes, break them into simpler shapes, calculate each area, and sum (or subtract for cutouts). Review Khan Academy's geometry course for interactive lessons. Use our area calculator for instant results with any shape.

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3D Volume & Surface Area Formulas

Rectangular box: V = l × w × h, SA = 2(lw + lh + wh). Cylinder: V = πr²h, SA = 2πrh + 2πr². Sphere: V = (4/3)πr³, SA = 4πr². Cone: V = (1/3)πr²h, SA = πr(r + √(r² + h²)). Pyramid: V = (1/3) × base area × height. For tanks, convert cubic feet to gallons by multiplying by 7.48 (US) or cubic metres to litres by multiplying by 1,000. Use our volume calculator to compute any 3D shape instantly.

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Triangles & the Pythagorean Theorem

The Pythagorean theorem (a² + b² = c²) applies to right triangles only. Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25. For non-right triangles, use the Law of Cosines: c² = a² + b² − 2ab·cos(C), or the Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). To find the area of any triangle given three sides, use Heron's formula: A = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2. Solve any right triangle with our right triangle calculator, or see NIST for reference constants.

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Real-World Geometry Applications

Flooring: measure room area (l × w) and add 10% for waste — use our area calculator. Paint: wall area (perimeter × height) minus windows/doors, divided by coverage rate (350–400 sq ft/gallon) — or try the paint coverage calculator. Landscaping: circular garden bed area = πr² for mulch volume. Pool volume: rectangular = l × w × avg depth × 7.48 for gallons. Roofing: roof area × pitch factor (typically 1.05–1.15 for standard slopes).

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Related Tools & Calculators

3 free tools linked to this guide

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Area Calculator

Calculate the area of any 2D shape: rectangles, triangles, circles, and more.

calculator →

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Volume Calculator

Calculate the volume of 3D shapes including boxes, cylinders, spheres, and cones.

calculator →

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Right Triangle Calculator

Solve right triangles using the Pythagorean theorem and trigonometric ratios.

calculator →

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Frequently Asked Questions

How do I calculate the area of a circle?

Area = πr², where r is the radius. For a circle with a 10-foot diameter (5-foot radius): A = π × 5² = 78.54 square feet. If you only know the circumference, find the radius first: r = C / (2π).

How do I calculate the volume of a cylinder?

Volume = πr²h, where r is the radius of the base and h is the height. For a tank 4 feet wide and 6 feet tall: V = π × 2² × 6 = 75.4 cubic feet = ~564 gallons (1 ft³ = 7.48 gallons).

What is the Pythagorean theorem?

For any right triangle: a² + b² = c², where c is the hypotenuse (longest side opposite the 90° angle). To find a missing side: c = √(a² + b²) or a = √(c² − b²). This only works for right triangles.

How do I find the area of an irregular shape?

Break the shape into regular components (rectangles, triangles, circles), calculate each area separately, then add them together. For curved shapes, approximate using the trapezoidal rule or use our area calculator with composite shape support.

What is the difference between area and surface area?

Area is the space inside a 2D shape (measured in square units). Surface area is the total area of all faces of a 3D shape. For a box: area of one face = l × w; surface area = 2(lw + lh + wh).

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