📐 Geometric Shapes & Formulas Calculator
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Geometric Shapes & Formulas Calculator
A geometric shapes and formulas calculator computes area, perimeter, volume, and surface area from input dimensions—side length, radius, height, base—and outputs results in square, cubic, or linear units. The full toolset sits at calculator-net.com inside the math calculators directory.
Disclaimer: This calculator is made for fun and should be treated as a source of entertainment and self-reflection rather than a definitive judgment on any relationship.
Triangle
A triangle area perimeter calculator accepts three side lengths, computes area via Heron's formula, and sums the perimeter.
Area (base-height): A = ½ × b × h
Area (Heron's): A = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2
Perimeter: P = a + b + c
Right triangle hypotenuse: c = √(a² + b²)
Equilateral area: A = (√3/4)s²
Equilateral height: h = (√3/2)s
Shoelace (coordinates): A = ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|A fraction calculator reduces rational intermediates. A significant figures calculator rounds output to specified digit count. A fraction to decimal converter returns decimal equivalents. For vertices plotted on an x-y plane, a coordinate geometry calculator applies the shoelace formula to ordered vertices.
Circle
A circle area circumference calculator returns radius, diameter, circumference, and area from any single known input.
Area (radius): A = πr²
Area (diameter): A = π(d/2)²
Circumference (radius): C = 2πr
Circumference (diameter): C = πd
Annulus ring: A = πR² - πr² (R = outer, r = inner)
Sector area: A = ½r²θ (θ in radians)
Arc length: L = rθ (θ in radians)
Chord length: c = 2r sin(θ/2)
Segment area: A = ½r²(θ - sin θ)A solve for exponents calculator isolates r from r² = A/π when area is known and radius is the unknown.
Polygon
A regular polygon calculator accepts side count (n) and side length (s), returns apothem, perimeter, area, interior angle, and exterior angle.
Interior angle sum: S = (n-2) × 180°
Single interior angle: θ = (n-2) × 180° / n
Exterior angle: θₛ = 360° / n
Area (apothem-perimeter): A = ½ × a × P
Hexagon area: A = (3√3/2)s²
Pentagon area: A = (5/2)sa (a = apothem)Quadrilateral
A rectangle square calculator handles both square and rectangle inputs in one interface.
Square area: A = s²
Square perimeter: P = 4s
Rectangle area: A = l × w
Rectangle perimeter: P = 2(l + w)
Parallelogram area: A = b × h (h = perpendicular height)
Rhombus area: A = (d₁ × d₂) / 2
Trapezoid area: A = h(b₁ + b₂) / 2
Kite area: A = (d₁ × d₂) / 2A ratio simplifier and ratio to fraction calculator reduces proportions like l:w to lowest terms.
Ellipse
Area: A = πab (a = semi-major axis, b = semi-minor axis)
Perimeter (Ramanujan): P ≈ π[3(a+b) - √((3a+b)(a+3b))]3D Solids: Volume and Surface Area
Cube
Volume: V = a³
Surface area: SA = 6a²
Face diagonal: d = a√2
Space diagonal: D = a√3Rectangular Prism
Volume: V = l × w × h
Surface area: SA = 2(lw + lh + wh)A luggage size calculator checks l+w+h against TSA and airline carry-on and checked bag limits.
Cylinder
Volume: V = πr²h
Surface area (with ends): SA = 2πr² + 2πrh
Lateral surface area: LSA = 2πrh
Hollow cylinder volume: V = πh(R² - r²) (R = outer, r = inner)Cone
Slant height: s = √(r² + h²)
Volume: V = ⅓πr²h
Surface area: SA = πr² + πrs
Lateral surface area: LSA = πrs
Truncated cone volume: V = ⅓πh(R² + Rr + r²)Sphere and Hemisphere
Sphere volume: V = ⁴⁄₃πr³
Sphere surface area: SA = 4πr²
Hemisphere volume: V = ⅔πr³
Hemisphere surface area: SA = 3πr²A density calculator mass volume converts between mass and volume units. A balloons for house lifting calculator applies V = (4/3)πr³ per balloon against total lift required.
Pyramid (Square Base)
Volume: V = ⅓a²h
Surface area: SA = a² + 2as (s = slant height)Triangular Prism
Volume: V = B × l (B = base area, l = length)
Surface area: SA = 2B + Pl (P = base perimeter)Tetrahedron and Octahedron
Tetrahedron volume: V = a³/(6√2)
Tetrahedron surface area: SA = √3 a²
Octahedron volume: V = (√2/3)a³
Octahedron surface area: SA = 2√3 a²A simplify radical expressions calculator reduces √2 and √3 terms. A logarithm calculator inverts exponentiation. An exponent calculator squares and cubes dimensions.
Torus
Volume: V = 2π²Rr² (R = major radius, r = minor radius)
Surface area: SA = 4π²RrEllipsoid
Volume: V = ⁴⁄₃πabc (a, b, c = semi-axes)Frustum
Volume: V = ⅓πh(R² + Rr + r²)Formula Reference Table
| Shape | Area / Surface Area Formula | Volume Formula |
|---|---|---|
| Square | A = s² | — |
| Rectangle | A = l × w | — |
| Triangle | A = ½bh or √[s(s-a)(s-b)(s-c)] | — |
| Circle | A = πr² | — |
| Trapezoid | A = h(b₁+b₂)/2 | — |
| Ellipse | A = πab | — |
| Cube | SA = 6a² | V = a³ |
| Rectangular Prism | SA = 2(lw+lh+wh) | V = l×w×h |
| Cylinder | SA = 2πr²+2πrh | V = πr²h |
| Cone | SA = πr²+πrs | V = ⅓πr²h |
| Sphere | SA = 4πr² | V = ⁴⁄₃πr³ |
| Square Pyramid | SA = a²+2as | V = ⅓a²h |
| Triangular Prism | SA = 2B+Pl | V = B×l |
| Torus | SA = 4π²Rr | V = 2π²Rr² |
| Ellipsoid | — | V = ⁴⁄₃πabc |
Coordinate Geometry
Distance: d = √[(x₂-x₁)² + (y₂-y₁)²]
Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2)A slope calculator math returns m = (y₂-y₁)/(x₂-x₁). A scientific calculator handles trigonometric, logarithmic, and exponential evaluations. A logarithmic scale converter translates between linear and logarithmic axes.
Similar triangles: a₁/a₂ = b₁/b₂ = c₁/c₂
Scale factor (area): A₂ = k²A₁
Scale factor (volume): V₂ = k³V₁
Golden ratio: φ = (1+√5)/2A difference of two squares calculator factors a²-b² into (a+b)(a-b). A matrix determinant calculator returns det=0 when three points are collinear. An Euclid's algorithm calculator returns GCD(a,b) via successive division. A shared divisors calculator find common factors extends GCD to multiple integers. A permutation nPr calculator returns P(n,r) = n!/(n-r)!. A combination with replacement calculator returns C(n+r-1,r).
Trigonometry
sin θ = opposite/hypotenuse
cos θ = adjacent/hypotenuse
tan θ = opposite/adjacentA height calculator solves h = d × tan θ using one known angle and one known side. A complex fraction calculator simplifies nested rational expressions that appear in trigonometric identities.
Composite Shapes and Earthwork
Composite area: A = ΣAᵢ
Shaded region: A = A_bounding - A_unshaded
Irregular perimeter: P = Σ edge lengthsA roadway fill volume calculator cut and fill applies V = L(A₁+A₂)/2 across cross-sections using the average-end-area method.
Cross-Domain Formula Parallels
Physics
Ohm's Law: V = IR
Mass-Energy: E = mc²
Elastic Potential: PE = ½kx²
Force: F = ma
Speed: v = d/tAn Ohm's law circuit solver returns V=IR, matching linear proportion a₁/a₂ = b₁/b₂. An E=mc² calculator squares c, identical operation to r² in πr². An elastic potential energy calculator returns ½kx², matching sector area ½r²θ. A force calculator returns F=ma. A speed calculator physics returns v=d/t, matching slope Δy/Δx.
Digital and Financial
Ad Revenue: impressions × CPM ÷ 1000
Credit Score: Σ(factor weight × score)
Date Difference: |date₂ - date₁| days
Bandwidth: data rate × time
Conversion Rate: (conversions / visitors) × 100%
Traffic Density: vehicles / road length
Fuel Cost: distance × consumption rate × fuel priceAn ad revenue calculator multiplies impressions × CPM ÷ 1000. A credit score calculator applies weighted sum across payment history, utilization, and account age. A date difference calculator returns the day count between two calendar dates. A bandwidth calculator multiplies data rate × time. A conversion rate calculator divides conversions by visitors. A traffic density calculator divides vehicle count by road segment length. A fuel cost calculator multiplies distance × consumption rate × price per gallon.
Payroll and Percentages
Paycheck: gross - (federal + state + FICA)
Take-Home Pay: gross - taxes - deductions
Tip: subtotal × tip%
Percentage: part/whole = %/100
Discount: list price × (1 - discount%)
Compound Interest: A = P(1+r/n)^(nt)
Half-Life: N(t) = N₀(½)^(t/t½)A paycheck calculator subtracts federal, state, and FICA withholding from gross pay. A take home pay calculator applies tax brackets and pre-tax deductions. A free tip calculator multiplies subtotal by tip percentage. A percentage calculator solves part/whole = %/100. A discount calculator subtracts the percentage reduction from list price. A compound interest calculator applies A = P(1+r/n)^(nt). A radioactive half-life solver computes N(t) = N₀(½)^(t/t½).
Mortgage and Loans
Mortgage Payment: M = P[r(1+r)^n]/[(1+r)^n-1]
Extra Payment: balance reduced by additional principal monthly
Amortization: interest = balance × rate/12, principal = payment - interest
Refinance: old payment vs new payment at proposed rate and term
Affordability: max loan = (gross income × DTI limit)/12 - other debtsA mortgage calculator returns M = P[r(1+r)^n]/[(1+r)^n-1]. An extra payment mortgage calculator subtracts additional principal from the remaining balance each month. A mortgage amortization calculator splits each payment into interest and principal portions. A mortgage refinance calculator compares current loan terms against proposed rate and term adjustments. A mortgage affordability calculator determines maximum loan amount from gross income, monthly debts, and down payment percentage.
Rent vs Buy: Σ(owning costs) - Σ(renting costs) over n years
PMI: loan amount × PMI rate/12 when LTV > 80%
Down Payment: home price × down payment%
Car Lease vs Buy: (cap cost - residual)/term + (cap cost + residual) × money factor vs loan payment
LTCG Tax: (sale price - cost basis) × LTCG rate
Net Worth: Σassets - ΣliabilitiesA rent vs buy calculator compares total net cost of renting versus purchasing over a specified holding period. A PMI calculator computes private mortgage insurance premiums when the down payment falls below 20%. A down payment calculator converts a target home price and percentage into a dollar amount. A car lease vs buy comparison calculator models depreciation, money factor, residual value, and purchase interest across both methods. A long-term capital gains tax calculator applies the applicable federal rate to asset sales held longer than one year. A personal net worth tracker subtracts total liabilities from total assets.
Health and Fitness
Dog/Cat Age: 15(year₁) + 9(year₂) + 5(each subsequent)
Calories: BMR × activity multiplier
TDEE Macros: protein=4cal/g, carbs=4cal/g, fat=9cal/g
Water Intake: body weight(lbs) × 0.67 = daily oz
1RM (Epley): weight × (1 + reps/30)
VO2 Max: 15.3 × (HRmax/HRrest)A dog and cat age calculator converts pet years using species-specific logarithmic curves. A calorie calculator estimates daily energy expenditure from body weight, height, age, and activity multiplier. A TDEE macros calculator splits maintenance calories into protein, carbohydrate, and fat gram targets. A water intake calculator returns daily fluid ounce requirements from body weight and climate factors. A 1RM strength calculator estimates one-rep max from submaximal lifts using the Epley formula. A VO2 max estimator no test required predicts cardiovascular fitness from resting heart rate, age, and self-reported activity level.
Business, Insurance, and Tech
Amazon FBA Fee: referral% + fulfillment + storage
Data Breach Cost: records × cost/record × industry factor
Cyber Insurance: base rate × coverage × risk factor
PPC Budget: clicks × CPC
Life Insurance: income × 10 + debts + education
Legal Settlement: (medical + wages) × multiplier + pain/suffering
HIPAA Compliance: admin + physical + technical safeguards
SaaS Pricing: per-seat price × users × billing cycle
Health Insurance: base × age factor × location × metal tierAn Amazon FBA fee calculator estimates seller costs from item category, shipping weight, and product dimensions. A data breach cost calculator models financial exposure from incident scope, records compromised, and industry sector. A cyber insurance cost estimator returns premium ranges from coverage limits, deductible selection, and risk profile. A PPC monthly budget calculator projects total monthly ad spend from target click volume and average cost-per-click. A life insurance coverage estimator calculates recommended coverage from annual income, outstanding debts, and dependent count. A legal settlement estimator models potential settlement ranges from case type, jurisdiction, and damages claimed. A HIPAA compliance cost estimator returns implementation cost ranges from organization size and required safeguards. A SaaS pricing estimator models monthly and annual subscription costs from feature tier selection and seat count. A health insurance premium estimator projects monthly premium ranges from applicant age, zip code, and metal tier selection.
Ads and Medical
Google Ads: impressions × CPC
Facebook Ads: reach × CPM ÷ 1000
Medical Billing: claims × cost/claim + clearinghouse fees
AWS Monthly: EC2 hours × rate + S3 GB × storage + transfer GB × rate
Resistor Decode: (band1×10 + band2) × 10^band3 Ω ± band4%A Google Ads cost calculator projects campaign spend from target keywords, average CPC, and monthly impression volume. A Facebook Ads cost calculator estimates campaign cost from audience size, objective selection, and placement strategy. A medical billing cost calculator returns practice expense estimates from monthly claim volume and specialty type. An AWS monthly cost calculator projects cloud infrastructure spend from EC2 instance type, storage class, and data transfer volume. A 4-band resistor color code decoder reads resistance value in ohms and tolerance percentage from the four colored bands on through-hole resistors.
A geometric shapes and formulas calculator matches input dimensions to the correct equation whether computing pentagon area via ½aP, frustum volume via ⅓πh(R²+Rr+r²), or ellipsoid volume via ⁴⁄₃πabc.