📐 Math ResourcesLast updated May 2, 2026
Geometry Guide: Area, Volume & Surface Area Formulas for Every Shape (2026)
Every geometry formula from 2D shapes to 3D solids — with step-by-step examples, real-world applications, and free calculators
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8+
Calculators
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25+
Shapes Covered
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40+
Formulas
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14 min
Read Time
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NIST SI
Standard
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May 2026
Updated
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Key Takeaways
- Rectangle area = l × w, Triangle area = ½bh, Circle area = πr² — these three formulas solve 80% of all real-world area problems.
- Volume formulas all share the same logic: base area × height. Box = l×w×h, Cylinder = πr²h, Cone = (1/3)πr²h, Pyramid = (1/3) × base area × h.
- The Pythagorean theorem (a² + b² = c²) only works for right triangles. Use the Law of Cosines (c² = a² + b² − 2ab·cos C) for all other triangles.
- Surface area is the total area of all outer faces. Sphere: SA = 4πr²; Cylinder: SA = 2πr² + 2πrh; Rectangular box: SA = 2(lw + lh + wh).
- Always ensure unit consistency before calculating — mixing feet and inches is the #1 source of geometry errors. 1 sq ft = 144 sq in; 1 sq m = 10.764 sq ft.
- Heron’s formula finds triangle area from three sides alone: A = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2. No height measurement needed.
- Use π = 3.14159265 (not 3.14) in circle and sphere formulas. Using 3.14 introduces a 0.05% error that compounds in repeated calculations.
- Composite shapes: break into basic shapes, compute each area or volume separately, then add (or subtract for holes and cutouts).
- Coordinate geometry distance formula: d = √[(x₂−x₁)² + (y₂−y₁)²]. Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2).
- Real-world conversions: 1 cubic foot = 7.48 US gallons; 1 acre = 43,560 sq ft; 1 litre = 1,000 cubic centimetres; 1 cubic metre = 1,000 litres.
Geometry is the most visual and practical branch of mathematics. Whether you're calculating the area of a room for flooring, the volume of a water tank, the surface area of a sphere for a paint job, or solving a right triangle for a construction cut, having the right formulas and tools saves time and prevents costly errors. This guide — aligned with NIST SI unit standards and mathematical references from the American Mathematical Society — covers every common shape with step-by-step formulas, worked examples, real-world applications, and free tools including our area calculator, volume calculator, surface area calculator, and right triangle calculator.
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2D Area Formulas: Every Shape with Step-by-Step Examples
The six most common 2D area formulas: Rectangle: A = length × width (l × w). Triangle: A = ½ × base × height (½bh). Circle: A = πr² (radius squared times pi). Trapezoid: A = ½(a+b) × h, where a and b are the parallel sides and h is the perpendicular height. Parallelogram: A = base × height (not side length — the height must be perpendicular). Ellipse: A = π × a × b, where a and b are the semi-major and semi-minor axes. Regular polygon: A = (½) × perimeter × apothem. Sector (pizza-slice): A = (θ/360) × πr², where θ is the angle in degrees. For composite shapes, decompose into basic shapes and sum the areas. For cutouts (e.g., a window in a wall), subtract. NIST reference: NIST SI unit standards. Use our area calculator for instant 2D shape results.
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3D Volume Formulas: Boxes, Cylinders, Spheres, Cones & More
All volume formulas follow one universal concept: Base Area × Height (adjusted by a fraction for tapered shapes). Rectangular box: V = l × w × h. Cube: V = s³. Cylinder: V = πr²h. Sphere: V = (4/3)πr³. Cone: V = (1/3)πr²h (exactly 1/3 of a cylinder with the same base and height). Pyramid: V = (1/3) × base area × h. Triangular prism: V = ½ × b × h₁ × l, where h₁ is the triangle height and l is the prism length. Frustum (truncated cone): V = (1/3)πh(R² + Rr + r²), where R and r are the radii of the two bases. Key conversions: 1 cubic foot = 7.48 US gallons; 1 cubic metre = 1,000 litres; 1 cubic inch = 16.387 cubic centimetres. Use our volume calculator for all shapes.
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Surface Area Formulas: How to Wrap Every 3D Shape
Surface area is the total area of all outer faces of a 3D object — imagine flattening the shape into a net. Rectangular box: SA = 2(lw + lh + wh). Cube: SA = 6s². Cylinder: SA = 2πr² + 2πrh (two circular caps + lateral). Sphere: SA = 4πr². Cone (including base): SA = πr² + πrl, where l = √(r² + h²) is the slant height. Cone lateral only: SA = πrl. Square pyramid: SA = b² + 2bl₁, where b is the base side and l₁ is the slant height of each triangular face. Triangular prism: SA = 2 × (triangle base area) + 3 × (rectangle side areas). Practical applications: how much paint covers a wall, how much foil wraps a cylinder, how much concrete forms a retaining wall. Try our surface area calculator for instant results.
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Triangles: Pythagorean Theorem, Law of Cosines & Heron's Formula
Right triangles: Use the Pythagorean theorem (a² + b² = c²). Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25, 9-40-41. Any triangle (with an angle): Law of Cosines: c² = a² + b² − 2ab·cos(C). Any triangle (sides only): Heron’s formula: A = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2. Trigonometry (SOHCAHTOA): sin(θ) = opposite/hypotenuse; cos(θ) = adjacent/hypotenuse; tan(θ) = opposite/adjacent. Triangle angle sum: A + B + C = 180° always. Triangle inequality: each side must be shorter than the sum of the other two. The Khan Academy right triangle module provides interactive lessons. Solve any right triangle with our right triangle calculator.
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Circles, Arcs & Sectors: Complete Formula Reference
Circle fundamentals: Radius (r), Diameter (d = 2r), Circumference (C = 2πr = πd), Area (A = πr²). Arc length: L = (θ/360) × 2πr, where θ is the central angle in degrees. Sector area: A = (θ/360) × πr². Chord length: c = 2r·sin(θ/2). Segment area (region between chord and arc): Aₛₑₜ = sector area − triangle area. Annulus (ring): A = π(R² − r²), where R = outer radius, r = inner radius. Circle equation (coordinate geometry): (x−h)² + (y−k)² = r², centred at (h,k). Practical uses: circular planting beds, pipe cross-sections, swimming pool areas, tank volumes. Use our area calculator with circle and sector modes.
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Coordinate Geometry: Distance, Midpoint, Slope & Area
Coordinate geometry bridges algebra and geometry. Key formulas for points (x₁,y₁) and (x₂,y₂): Distance: d = √[(x₂−x₁)² + (y₂−y₁)²]. Midpoint: M = ((x₁+x₂)/2, (y₁+y₂)/2). Slope: m = (y₂−y₁)/(x₂−x₁). Perpendicular slope: m⊥ = −1/m. Line equation: y = mx + b (slope-intercept) or y−y₁ = m(x−x₁) (point-slope). Area of a polygon given vertices: Shoelace formula = ½|x₁(y₂−yₙ) + x₂(y₃−y₁) + ... + xₙ(y₁−yₙ₋₁)|. Applications: GPS distance calculations, CAD drawing, land survey area computation. Our distance calculator handles all coordinate geometry needs.
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Composite & Irregular Shapes: Step-by-Step Breakdown Method
Real-world geometry problems rarely involve a single perfect shape. The composite method: Step 1 — Identify and outline all basic shapes embedded in the composite figure (rectangles, triangles, circles, semicircles). Step 2 — Assign dimensions to each component from the overall dimensions. Step 3 — Calculate each area or volume separately. Step 4 — Add for combined shapes; subtract for holes or cutouts. Examples: L-shaped room = two rectangles. Circular island in a rectangular yard = rectangle minus circle. Arched doorway = rectangle + semicircle. Hollow cylinder = full cylinder volume minus inner cylinder volume. For land surveys with irregular boundaries, use the trapezoidal rule or divide the polygon into triangles and apply the Shoelace formula. Our area calculator supports composite input.
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Real-World Geometry: Flooring, Painting, Landscaping & Construction
Flooring & tiling: Calculate room area (l × w) and add 10% for cuts and waste. For diagonal tile patterns, add 15–20%. Irregular rooms: use the composite method. Use our area calculator. Paint coverage: wall area = perimeter × height, minus door/window areas. Standard coverage = 350–400 sq ft per gallon. Two coats = double paint needed. Try the paint coverage calculator. Landscaping: circular garden bed = πr²; topsoil volume = area × depth. Convert cubic feet to cubic yards (divide by 27) or litres (multiply by 28.317). Pool volume: rectangular = l × w × avg depth × 7.48 gallons/ft³. Roofing: roof area × pitch factor (1.05 for 4:12 pitch, 1.08 for 6:12, 1.12 for 8:12). Concrete: V = l × w × depth (in ft); divide by 27 for cubic yards. Add 8–10% for waste. Try the concrete calculator.
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Geometry Unit Conversions: Area, Volume & Length Reference
Length: 1 inch = 2.54 cm; 1 foot = 30.48 cm; 1 yard = 0.9144 m; 1 mile = 1.6093 km; 1 metre = 3.2808 ft. Area: 1 sq ft = 144 sq in = 0.0929 sq m; 1 sq m = 10.764 sq ft; 1 acre = 43,560 sq ft = 4,047 sq m; 1 hectare = 10,000 sq m = 2.471 acres; 1 sq mile = 640 acres = 2.59 sq km. Volume: 1 cubic ft = 1,728 cubic in = 7.481 US gal = 0.0283 cubic m; 1 cubic m = 1,000 L = 264.2 US gal = 35.315 cubic ft; 1 US gallon = 231 cubic in = 3.785 L; 1 UK gallon = 4.546 L; 1 litre = 0.264 US gal = 61.02 cubic in. These conversions are defined by NIST in NIST SP 811. Use our unit converter for all geometry conversions.
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Geometry Formulas Cheat Sheet: Quick Reference for All Shapes
2D Shapes — Area & Perimeter: Square (s²; 4s). Rectangle (lw; 2(l+w)). Triangle (½bh; a+b+c). Circle (πr²; 2πr). Trapezoid (½(a+b)h; a+b+c+d). Parallelogram (bh; 2(a+b)). Ellipse (πab; approx 2π√(½(a²+b²))). Regular n-gon (½n·s·a; ns). 3D Shapes — Volume & Surface Area: Cube (s³; 6s²). Box (lwh; 2(lw+lh+wh)). Cylinder (πr²h; 2πr(r+h)). Sphere ((4/3)πr³; 4πr²). Cone ((1/3)πr²h; πr(r+√(r²+h²))). Square pyramid ((1/3)b²h; b²+2b√(h²+(b/2)²)). Triangular prism (Bh where B=triangle area; 2B+lateral). Frustum ((1/3)πh(R²+Rr+r²); π(R+r)√(h²+(R−r)²)+π(R²+r²)). Coordinate Geometry: Distance √(Δx²+Δy²); Midpoint ((x₁+x₂)/2,(y₁+y₂)/2); Slope (y₂−y₁)/(x₂−x₁). Print or save this reference and use our area calculator or volume calculator for instant computation.
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Related Tools & Calculators
14 free tools linked to this guide
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Area Calculator
Calculate the area of any 2D shape: rectangles, triangles, circles, trapezoids, ellipses, and composite shapes.
calculator →Volume Calculator
Calculate the volume of 3D shapes: boxes, cylinders, spheres, cones, pyramids, and frustums.
calculator →Surface Area Calculator
Compute the surface area of every 3D shape including cubes, cylinders, cones, and spheres.
calculator →Right Triangle Calculator
Solve right triangles for missing sides and angles using the Pythagorean theorem and SOHCAHTOA.
calculator →Perimeter Calculator
Calculate the perimeter or circumference of rectangles, triangles, circles, and polygons.
calculator →Distance Calculator
Find the distance between two points using the coordinate geometry distance formula.
calculator →Unit Converter
Convert area (sq ft to sq m), volume (gallons to litres), and length units instantly.
calculator →Percentage Calculator
Calculate percentage of a number, percentage increase/decrease, and proportion problems.
calculator →Area & Volume Formulas Cheat Sheet
Every 2D area and 3D volume formula on one printable page — rectangles to frustums.
blog →Percentage Math Guide
Master percentages, ratios, proportions, and percent change for math, finance, and science.
guide →Engineering Calculators Guide
Structural, civil, and mechanical engineering formulas and free calculators for professional use.
guide →Construction Calculators Guide
Concrete, flooring, roofing, and material estimation calculators for construction projects.
guide →Paint Coverage Calculator
Estimate gallons of paint needed based on wall area, ceiling height, and number of coats.
calculator →Concrete Calculator
Calculate cubic yards of concrete for slabs, footings, columns, and walls.
calculator →❓
Frequently Asked Questions
How do I calculate the area of a circle?
Area = πr², where r is the radius (half the diameter). For a circle with a 10-foot diameter (5-foot radius): A = π × 5² = 78.54 square feet. If you only know the circumference C, find radius first: r = C / (2π), then apply A = πr². If you know the diameter d: A = π(d/2)² = πd²/4.
How do I calculate the volume of a cylinder?
Volume = πr²h, where r is the base radius and h is the height. For a tank 4 ft wide (2 ft radius) and 6 ft tall: V = π × 2² × 6 = 75.4 cubic feet. To convert to US gallons, multiply by 7.48: 75.4 × 7.48 = 564 gallons. Surface area of same cylinder: SA = 2πr² + 2πrh = 2π(4) + 2π(2)(6) = 100.5 sq ft.
What is the Pythagorean theorem and when can I use it?
For any right triangle: a² + b² = c², where c is the hypotenuse (longest side opposite the 90° angle). Common Pythagorean triples: 3-4-5, 5-12-13, 8-15-17. To find a missing side: c = √(a² + b²); a = √(c² − b²). IMPORTANT: This only works for right triangles. For any other triangle, use the Law of Cosines: c² = a² + b² − 2ab·cos(C).
How do I find the area of an irregular or composite shape?
Step 1: Divide the shape into recognisable basic shapes (rectangles, triangles, semicircles). Step 2: Calculate the area of each component separately. Step 3: Add all areas together (or subtract for cutouts/holes). Example: an L-shaped room = two rectangles. A room with a circular table cutout = rectangle area − circle area. Our area calculator supports composite shape input.
What is the difference between area and surface area?
Area measures the space inside a 2D (flat) shape in square units (sq ft, sq m). Surface area is the total area of all outer faces of a 3D shape — imagine unfolding the shape and measuring every face. For a box (l=5, w=3, h=2): floor/ceiling = 2×5×3 = 30; front/back = 2×5×2 = 20; sides = 2×3×2 = 12; total SA = 62 sq units.
How do I calculate the volume and surface area of a sphere?
Volume = (4/3)πr³. For a sphere with 10-inch radius: V = (4/3)π(10)³ = 4,189 cubic inches. Surface area = 4πr². Same sphere: SA = 4π(100) = 1,257 sq inches. Note: doubling the radius increases volume by 8× (cubic relationship) but surface area by only 4× (square relationship).
How do I calculate the perimeter of different shapes?
Rectangle: P = 2(l + w). Square: P = 4s. Triangle: P = a + b + c. Circle (circumference): C = 2πr = πd. Regular polygon: P = n × side length, where n = number of sides. For an irregular polygon, simply sum all side lengths. Perimeter is always measured in linear units (ft, m, cm) — not square units.
What is the distance formula in coordinate geometry?
The distance between points (x₁, y₁) and (x₂, y₂) is: d = √[(x₂−x₁)² + (y₂−y₁)²]. This is the Pythagorean theorem applied to a coordinate plane. Example: distance from (1, 2) to (4, 6): d = √[(4−1)² + (6−2)²] = √[9+16] = √25 = 5 units.
How do I calculate the volume of a cone or pyramid?
Cone: V = (1/3)πr²h. For a cone with 3-inch radius and 9-inch height: V = (1/3)π(9)(9) = 84.8 cubic inches. Pyramid: V = (1/3) × base area × height. For a square pyramid (4×4 base, 6 height): V = (1/3)(16)(6) = 32 cubic units. Both the cone and pyramid are exactly 1/3 the volume of their corresponding cylinder/prism.
How do I convert between area units like square feet and square meters?
1 square foot = 0.0929 square meters. 1 square meter = 10.764 square feet. 1 acre = 43,560 sq ft = 4,047 sq meters. 1 hectare = 10,000 sq meters = 2.471 acres. 1 square mile = 640 acres = 2.59 sq km. To convert sq ft to sq m: multiply by 0.0929. To convert sq m to sq ft: multiply by 10.764. Our unit conversion calculator handles all area, volume, and length conversions instantly.
What is Heron's formula and when should I use it?
Heron’s formula calculates triangle area when you know all three side lengths but NOT the height: A = √[s(s−a)(s−b)(s−c)], where s = (a+b+c)/2 (the semi-perimeter). Example: triangle with sides 5, 7, 8: s = (5+7+8)/2 = 10; A = √[10(10−5)(10−7)(10−8)] = √[10×5×3×2] = √300 = 17.32 sq units. Use this when measuring an irregular land plot or any triangle without a right angle.
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