Online Math Calculators

Online Math Calculators

A collection of free online math calculators for quick and accurate computations. Calculate circle area and triangle parameters, solve quadratic equations, convert temperature between scales, and compute volume of geometric solids — all with formulas, visualization, and step-by-step solutions.

txtly.ru math tools help students, teachers, engineers, and anyone who needs a fast calculation without installing software. Every calculator works right in your browser, requires no registration, and is accessible from any device — desktop, tablet, or phone.

Check out other tool categories: business calculators, generators, and converters.

Why Choose txtly.ru Math Calculators

• Free and no registration — no hidden fees, subscriptions, or limits on the number of calculations
• Instant results — all computations happen right in your browser with no server round-trips
• Visual feedback — every calculator includes graphical representation: a circle, triangle, 3D solid, or plot
• Step-by-step solutions — the quadratic equation solver shows every stage of computation, ideal for learning
• Responsive design — calculators work equally well on desktop, tablet, and smartphone
• Precision — double-precision (64-bit floating point) calculations eliminate rounding errors
• Educational value — formulas, tables, and explanations help you understand the math, not just get the answer

How to Use These Calculators

1. Pick a tool from the list below or via the navigation menu
2. Enter your input — radius, triangle sides, equation coefficients, or temperature value
3. Get results instantly — every calculator updates automatically when you change any parameter
4. Explore formulas and examples — each section includes theoretical background and worked problems

Circle Area Calculator

What It Is

The Circle Area Calculator is an online tool for instantly computing the area and circumference of a circle from its radius. Simply enter the radius and you get both results along with a scaled visual representation of the circle.

Formulas

The core formulas used by the calculator:

• Area of a circle: A = π × r², where r is the radius, π ≈ 3.14159
• Circumference: C = 2 × π × r
• Diameter: d = 2 × r

Pi (π) is a mathematical constant representing the ratio of a circle's circumference to its diameter. The calculator uses π with 15 decimal places of precision for accurate results.

When to Use

• Construction and renovation — calculating the area of circular objects: columns, pipes, pools, flower beds
• Manufacturing — material cutting for round shapes, estimating paint coverage for pipes
• Education — checking geometry homework, preparing for math exams
• Design — planning circular interior elements, logo layouts
• Cooking — calculating the area of round baking pans, pizza sizes

Worked Example

Problem: find the area and circumference of a circle with radius 5 cm.

Solution:

• Area: A = π × 5² = 3.14159 × 25 = 78.54 cm²
• Circumference: C = 2 × π × 5 = 31.42 cm
• Diameter: d = 2 × 5 = 10 cm

The calculator instantly provides these values and displays a circle with a 5 cm radius drawn to scale.

Frequently Asked Questions (FAQ)

How do I find the area of a circle using the diameter? Formula: A = π × d² / 4, where d is the diameter. For example, with diameter 10 cm: A = 3.14159 × 100 / 4 = 78.54 cm².

What is the area of a circle with radius 1 meter? A = π × 1² = π ≈ 3.14 m².

How precise are the calculations? The calculator uses π with 15 decimal places, providing practical accuracy for any household or engineering task. Results are rounded to two decimal places.

What happens to the area if I double the radius? The area increases by a factor of 4, since A ∝ r². With radius r, area A = πr²; with radius 2r, area A = π(2r)² = 4πr².

Practical Applications

• 30 cm vs 40 cm pizza: a 30 cm pizza has area 707 cm², while a 40 cm pizza has 1,257 cm² — 78% more. One large pizza is a better deal than two small ones of the same diameter.
• Pipes and plumbing: a pipe's cross-sectional area determines its flow capacity. A 100 mm diameter pipe carries 4 times the flow of a 50 mm pipe at the same pressure.
• Round swimming pool: with a 3 m radius, the water surface area is 28.3 m² — essential for calculating chemical treatment and heating requirements.

Open Circle Area Calculator →

Temperature Converter

What It Is

The Temperature Converter is an online tool for converting values between the three main temperature scales: Celsius (°C), Fahrenheit (°F), and Kelvin (K). An interactive slider lets you smoothly adjust the temperature while watching all three scales update simultaneously.

Conversion Formulas

Conversions between scales use the following formulas:

Temperature Reference Points

When to Use

• Travel — converting weather forecasts from °F to °C when visiting the United States and other Fahrenheit-using countries
• Cooking — adapting American recipes: 350°F = 175°C (oven), 400°F ≈ 200°C
• Science and engineering — converting to the absolute Kelvin scale for physics and chemistry calculations
• Medicine — interpreting thermometer readings in different units
• Education — learning about temperature scales in physics class

Worked Example

Problem: convert 98.6°F to Celsius and Kelvin.

Solution:

• To Celsius: °C = (98.6 − 32) × 5/9 = 66.6 × 5/9 = 37°C
• To Kelvin: K = 37 + 273.15 = 310.15 K

This is normal human body temperature — 37°C.

Frequently Asked Questions (FAQ)

Why does the United States use Fahrenheit? The Fahrenheit scale was proposed by German physicist Daniel Gabriel Fahrenheit in 1724 and historically became standard in the United States, Liberia, and the Cayman Islands. Most other countries use Celsius.

What is absolute zero? Absolute zero (0 K = −273.15°C) is the theoretical minimum possible temperature, at which all thermal motion of particles ceases. It is the starting point of the Kelvin scale.

How can I quickly convert °F to °C mentally? Approximate formula: °C ≈ (°F − 30) / 2. For example, 86°F → (86 − 30) / 2 = 28°C (exact: 30°C). The error is about 2°C — acceptable for a quick weather estimate.

Open Temperature Converter →

Triangle Calculator

What It Is

The Triangle Calculator is a tool for computing all parameters of a triangle from a minimal set of input data. It supports two modes: right triangle calculation from two legs, and arbitrary triangle calculation from three sides using Heron's formula.

The triangle is one of the fundamental geometric shapes, and this calculator quickly finds its area, perimeter, angles, heights, and other properties. The tool is especially useful for geometry homework and exam preparation.

Formulas

Right triangle (from legs a and b):

• Hypotenuse: c = √(a² + b²) (Pythagorean theorem)
• Area: A = (a × b) / 2
• Perimeter: P = a + b + c
• Angle α = arctan(a / b), angle β = arctan(b / a), angle γ = 90°

Arbitrary triangle (from three sides a, b, c):

• Semiperimeter: s = (a + b + c) / 2
• Area (Heron's formula): A = √(s × (s − a) × (s − b) × (s − c))
• Angles via the law of cosines: cos α = (b² + c² − a²) / (2bc)
• Perimeter: P = a + b + c

Triangle inequality check: the sum of any two sides must be greater than the third: a + b > c, b + c > a, a + c > b.

Types of Triangles

When to Use

• Construction — calculating the area of triangular plots, roof gables, truss bracing
• Surveying — measuring areas of non-rectangular land parcels
• Education — solving geometry problems, exam preparation
• Design and graphics — calculating triangular elements, polygonal modeling
• Navigation — triangulation, distance determination

Worked Example

Problem (right triangle): legs a = 3 cm, b = 4 cm. Find the hypotenuse, area, and angles.

Solution:

• Hypotenuse: c = √(3² + 4²) = √25 = 5 cm
• Area: A = (3 × 4) / 2 = 6 cm²
• Perimeter: P = 3 + 4 + 5 = 12 cm
• Angle α = arctan(3/4) ≈ 36.87°, angle β = arctan(4/3) ≈ 53.13°, angle γ = 90°

Problem (Heron's formula): triangle sides: 13, 14, 15 cm. Find the area.

Solution:

• Semiperimeter: s = (13 + 14 + 15) / 2 = 21 cm
• Area: A = √(21 × 8 × 7 × 6) = √7056 = 84 cm²

This is the famous "Heronian triangle" — a triangle with integer sides and an integer area.

Frequently Asked Questions (FAQ)

Can I calculate a triangle from two sides and the included angle? Yes, the area can be found using: A = ½ × a × b × sin(C), where a and b are sides and C is the angle between them. The current calculator version supports three-side input and right triangle by two legs.

What if the triangle doesn't exist? The calculator checks the triangle inequality: the sum of any two sides must exceed the third. If this condition is not met, an error message is displayed.

What is the precision of the calculations? All computations use double-precision (64-bit) floating-point numbers, providing accuracy to about 10⁻¹² for most operations.

Open Triangle Calculator →

Volume Calculator

What It Is

The Volume Calculator is an online tool for calculating the volume and surface area of four basic geometric solids: cylinder, sphere, cube, and cone. Each shape includes a 3D visualization for intuitive understanding.

Formulas

Cylinder (r — base radius, h — height):

• Volume: V = π × r² × h
• Lateral surface area: S_lat = 2 × π × r × h
• Total surface area: S_tot = 2 × π × r × (r + h)

Sphere (r — radius):

• Volume: V = 4/3 × π × r³
• Surface area: S = 4 × π × r²

Cube (a — edge length):

• Volume: V = a³
• Surface area: S = 6 × a²
• Face diagonal: d = a × √2
• Space diagonal: D = a × √3

Cone (r — base radius, h — height):

• Volume: V = 1/3 × π × r² × h
• Slant height: l = √(r² + h²)
• Lateral surface area: S_lat = π × r × l
• Total surface area: S_tot = π × r × (r + l)

Formula Summary Table

When to Use

• Construction and renovation — calculating concrete volume for cylindrical columns, round pool capacity
• Logistics — determining capacity of cylindrical tanks, cubic containers
• Manufacturing — estimating material for pipes, tanks, spherical vessels
• Cooking — calculating the volume of round and rectangular baking pans
• Education — solving solid geometry problems, SAT/ACT math preparation

Worked Example

Problem (cylinder): base radius 10 cm, height 30 cm.

Solution:

• Volume: V = π × 10² × 30 = 3.14159 × 100 × 30 = 9,424.78 cm³ ≈ 9.42 L
• Surface area: S = 2 × π × 10 × (10 + 30) = 2 × 3.14159 × 10 × 40 = 2,513.27 cm²

Problem (sphere): radius 5 cm.

Solution:

• Volume: V = 4/3 × π × 5³ = 4/3 × 3.14159 × 125 = 523.60 cm³
• Surface area: S = 4 × π × 5² = 4 × 3.14159 × 25 = 314.16 cm²

Frequently Asked Questions (FAQ)

Why is the cone volume exactly 1/3 of the cylinder volume? A cone with the same base and height as a cylinder has exactly one-third the volume. This can be proven through integration or demonstrated experimentally — three full cones fill a cylinder of the same radius and height.

How do I convert cubic centimeters to liters? 1 liter = 1,000 cm³. Divide the value in cm³ by 1,000. For example, 9,424.78 cm³ = 9.42 L.

Which shape has the largest volume for a given surface area? The sphere. Among all solids with the same surface area, the sphere encloses the maximum volume — this is the isoperimetric property in three dimensions.

Practical Volume Applications

• Concrete column: a cylindrical column with radius 20 cm and height 3 m requires V = π × 0.2² × 3 = 0.377 m³ of concrete (about 9 fifty-kilogram bags).
• Hot air balloon: the volume of a spherical hot air balloon with radius 5 m is V = 4/3 × π × 125 ≈ 523.6 m³ — that's how much helium is needed to fill it.
• Moving box: a standard 40×30×30 cm box has a volume of 36,000 cm³ = 36 liters — useful for planning a move and ordering transport.

Open Volume Calculator →

Quadratic Equation Solver

What It Is

The Quadratic Equation Solver is an online calculator for equations of the form ax² + bx + c = 0 with full step-by-step solution output. The solver computes the discriminant, finds roots (including complex ones), and shows every stage of the calculation — from substituting coefficients to the final answer.

Formulas and Theory

Quadratic equation: ax² + bx + c = 0, where a ≠ 0.

Discriminant: D = b² − 4ac

The discriminant determines the number and nature of roots:

Vieta's formulas (for verification):

• Sum of roots: x₁ + x₂ = −b/a
• Product of roots: x₁ × x₂ = c/a

Incomplete quadratic equations:

• If b = 0: ax² + c = 0 → x² = −c/a
• If c = 0: ax² + bx = 0 → x(ax + b) = 0 → x₁ = 0, x₂ = −b/a
• If b = 0 and c = 0: ax² = 0 → x = 0

When to Use

• Education — solving algebra problems, preparing for SAT Math, ACT, GCSE, and other standardized tests
• Physics — computing trajectories, fall time, kinematics problems (equations with t²)
• Economics — finding break-even points, profit optimization (quadratic functions)
• Engineering — beam deflection calculations, resonant circuit analysis
• Programming — solving quadratic equations in algorithms, computer graphics (ray-sphere intersection)

Worked Examples

Example 1 (D > 0): 2x² + 5x − 3 = 0

Step-by-step solution:

1. Coefficients: a = 2, b = 5, c = −3
2. Discriminant: D = 5² − 4 × 2 × (−3) = 25 + 24 = 49
3. √D = √49 = 7
4. Roots:
   • x₁ = (−5 + 7) / (2 × 2) = 2 / 4 = 0.5
   • x₂ = (−5 − 7) / (2 × 2) = −12 / 4 = −3
5. Verification via Vieta: x₁ + x₂ = 0.5 + (−3) = −2.5 = −b/a = −5/2 ✓

Example 2 (D = 0): x² − 6x + 9 = 0

Step-by-step solution:

1. Coefficients: a = 1, b = −6, c = 9
2. Discriminant: D = (−6)² − 4 × 1 × 9 = 36 − 36 = 0
3. Root: x = −(−6) / (2 × 1) = 6 / 2 = 3
4. This is a perfect square: (x − 3)² = 0

Example 3 (D < 0): x² + 2x + 5 = 0

Step-by-step solution:

1. Coefficients: a = 1, b = 2, c = 5
2. Discriminant: D = 2² − 4 × 1 × 5 = 4 − 20 = −16
3. √|D| = √16 = 4
4. Complex roots:
   • x₁ = (−2 + 4i) / 2 = −1 + 2i
   • x₂ = (−2 − 4i) / 2 = −1 − 2i

Frequently Asked Questions (FAQ)

What if coefficient a equals zero? Then it's not a quadratic equation but a linear one: bx + c = 0. Solution: x = −c/b (if b ≠ 0). The calculator will warn you that a cannot be zero and prompt you to enter a valid value.

Can I solve a quadratic equation without the discriminant? Yes, there are several methods: completing the square, Vieta's formulas (guess-and-check), and graphing (parabola). However, the discriminant method is the most universal and reliable.

Why do we need complex roots? Complex numbers are widely used in electrical engineering (AC circuit analysis), quantum mechanics, signal processing, aerodynamics, and many other fields. The fact that an equation has no real solutions doesn't make it useless.

How can I verify my solution? Plug each root back into the original equation — you should get zero. For a quick check, use Vieta's formulas: the sum of roots should equal −b/a, and the product should equal c/a.

Other Useful Tools

After mastering these math calculators, check out the unit converters — unit conversion is often needed alongside mathematical calculations. For automating repetitive computations, use the generators — such as the random number generator or password generator.

Open Quadratic Equation Solver →