What is the formula for the area of a rectangle?

Area = Length x Width. For example, a room 5 m long and 4 m wide has area = 5 x 4 = 20 square meters. This is the most commonly used area formula in construction, flooring, and real estate.

How do you find the area of a triangle?

Area = (1/2) x base x height. The height must be perpendicular to the base. Example: triangle with base 8 cm and height 5 cm = (1/2) x 8 x 5 = 20 cm squared. For a triangle where only three sides are known, use Heron's formula.

What is the formula for the area of a circle?

Area = pi x r squared, where r is the radius (half the diameter). Example: circle with radius 7 cm = 3.14159 x 49 = 153.94 cm squared. If you know the diameter, r = diameter divided by 2.

How do you calculate the area of a trapezoid?

Area = (1/2) x (base1 + base2) x height. Add the two parallel sides, multiply by height, divide by 2. Example: trapezoid with parallel sides 6 and 10, height 4 = (1/2) x (6+10) x 4 = 32 square units.

How do you find the area of a rhombus?

Area = (d1 x d2) / 2, where d1 and d2 are the lengths of the two diagonals. Example: rhombus with diagonals 10 and 8 = (10 x 8) / 2 = 40 square units. Alternatively, Area = base x height if you know those values.

What is the area of an ellipse (oval)?

Area = pi x a x b, where a is the semi-major axis (half the longer diameter) and b is the semi-minor axis (half the shorter diameter). Example: ellipse with axes 10 and 6 (semi-axes 5 and 3) = pi x 5 x 3 = 47.12 square units.

How do you calculate the area of a sector?

Area = (angle / 360) x pi x r squared for degrees, or Area = (1/2) x r squared x angle for radians. Example: sector with radius 5 and central angle 90 degrees = (90/360) x pi x 25 = 19.63 square units. A sector is like a pie slice from a circle.

How do you calculate square footage of a room?

Measure the length and width of the room in feet and multiply them. For an L-shaped room, divide it into two rectangles, calculate each area, then add them. Example: 12 ft x 10 ft room = 120 square feet. Add 10% for waste when ordering flooring.

What is Heron's formula for the area of a triangle?

Heron's formula finds area when three sides are known. First find semi-perimeter s = (a + b + c) / 2. Then Area = square root of (s x (s-a) x (s-b) x (s-c)). Example: triangle with sides 5, 12, 13. s = 15. Area = sqrt(15 x 10 x 3 x 2) = sqrt(900) = 30 square units.

How do you find the area of a parallelogram?

Area = base x height, where height is the perpendicular distance between the parallel bases (not the slant side). Example: parallelogram with base 9 cm and perpendicular height 5 cm = 9 x 5 = 45 cm squared. Note: height is NOT the same as the side length.

How is area used in real life?

Area is used to calculate flooring and tiling needs (sq ft), paint coverage (one gallon typically covers 400 sq ft), land area for farming and real estate, roof area for shingles, fabric for sewing, and surface coating for any material.

Area Calculator — Rectangle, Triangle, Circle, Trapezoid & More

□ Select a Shape & Enter Dimensions

A = Length × Width

Length

Longer dimension

Enter a valid length.

Width

Shorter dimension

Enter a valid width.

A = Side²

Side Length

All 4 sides are equal

Enter a valid side.

A = ½ × Base × Height | Heron’s if 3 sides

Base

Bottom side

Enter a valid base.

Height

Perpendicular height

Enter a valid height.

Optional: Enter all 3 sides to also use Heron’s formula (overrides base×height):

Side A (opt.)

Side B (opt.)

Side C (opt.)

A = π × r²

Radius

Half the diameter

Enter a valid radius.

OR Diameter

Overrides radius if entered

A = ½ × (b1 + b2) × Height

Base 1 (top)

Top parallel side

Enter valid base.

Base 2 (bottom)

Bottom parallel side

Enter valid base.

Height

Perpendicular distance

Enter valid height.

A = Base × Height

Base

Any one side

Enter a valid base.

Perpendicular Height

NOT the slant side

Enter a valid height.

A = (d1 × d2) / 2

Diagonal 1

Longer diagonal

Enter valid diagonal.

Diagonal 2

Shorter diagonal

Enter valid diagonal.

A = π × a × b

Semi-Major Axis (a)

Half the longer diameter

Enter valid axis.

Semi-Minor Axis (b)

Half the shorter diameter

Enter valid axis.

A = (θ/360) × π × r²

Radius

Radius of the full circle

Enter a valid radius.

Central Angle (°)

Angle of the sector slice

Enter angle (0–360).

Units:

Area

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⚠️ Disclaimer: Calculated area is mathematically exact for given inputs. For real-world projects (flooring, land, painting), add 10–15% for waste and cutting. Verify all measurements before purchasing materials.

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Sources & Methodology

✓All area formulas verified against Euclid’s Elements and NIST Mathematical Functions. Triangle uses Heron’s formula (exact) when 3 sides provided. Sector uses degree-based formula (angle/360 × πr²). Pi = 3.14159265358979 (IEEE 754).

📘

Khan Academy — Area and Perimeter

Khan Academy’s geometry curriculum covering area formulas for all standard shapes. The global standard for mathematics education covering rectangle, triangle, circle, trapezoid, and parallelogram area calculations.

🏛️

NIST Digital Library of Mathematical Functions

U.S. government authoritative reference. Ellipse area formula (A = πab), sector area, and Heron’s formula are verified against NIST mathematical definitions.

Formulas used:
Rectangle: L×W | Square: s² | Triangle: (1/2)bh or Heron’s √(s(s-a)(s-b)(s-c)) Circle: πr² | Trapezoid: (b1+b2)/2 ×h | Parallelogram: b×h Rhombus: (d1×d2)/2 | Ellipse: π×a×b | Sector: (θ/360)×πr² All calculations use IEEE 754 double-precision. Perimeter calculated for each shape as bonus output.

Last reviewed: April 2026

Area Formulas for All Shapes — Complete Reference Guide

Area is the measure of the two-dimensional space enclosed within a shape. It is always expressed in square units — square centimeters (cm²), square meters (m²), square feet (ft²), or square inches (in²). Area calculations are fundamental in construction (flooring, roofing, tiling), agriculture (land area), painting and decorating (wall and ceiling coverage), real estate (property size), and everyday projects like carpet buying, lawn treatment, and fabric cutting.

Practical Example: How much flooring for a 12×10 ft room?

Area = Length × Width = 12 × 10 = 120 sq ft
Add 10% for waste and cuts: 120 × 1.10 = 132 sq ft to purchase
At $3.50 per sq ft: 132 × $3.50 = $462 total cost

Complete Area Formula Reference

Shape	Formula	Variables	Example
Rectangle	L × W	Length, Width	10×5 = 50 cm²
Square	s²	Side length	7² = 49 cm²
Triangle	½ × b × h	Base, Height
Circle	πr²	Radius	π×7² = 153.9 cm²
Trapezoid	½(b1+b2)×h	Two bases, Height	½(6+10)×4 = 32 cm²
Parallelogram	b × h	Base, Height	9×5 = 45 cm²
Rhombus	(d1 × d2) / 2	Two diagonals	(10×8)/2 = 40 cm²
Ellipse	π × a × b	Semi-axes a, b	π×5×3 = 47.1 cm²
Sector	(θ/360)×πr²	Radius, Angle	(90/360)×π×25 = 19.6 cm²

Rectangle and Square Area

The rectangle is the most commonly calculated area in daily life. A = Length × Width. Floor area, wall area, garden plots, and building footprints are all rectangular. A square is a special rectangle where all sides are equal: A = s². Real use: a 5 m × 4 m bedroom has area = 20 m². At 1 m² per tile (30×30 cm tiles: 11 per m²), you need 220 tiles minimum. Add 10% = 242 tiles.

Triangle Area — Two Methods

Method 1 (base × height): A = (1/2) × base × height. The height must be the perpendicular distance from base to apex — not the slant side. Example: base 8 cm, height 5 cm = 20 cm². Method 2 (Heron’s formula): When only the three sides are known, use s = (a+b+c)/2, then A = √(s(s−a)(s−b)(s−c)). Example: sides 5, 12, 13. s = 15. A = √(15×10×3×2) = √900 = 30 cm². This calculator supports both methods.

Circle Area — The πr² Formula

The circle area formula A = πr² requires only the radius. If you know the diameter, r = d/2. Example: a circular table with diameter 1.2 m has r = 0.6 m and area = π×0.36 = 1.13 m². A circular lawn with radius 5 m has area = π×25 = 78.5 m². At 20g fertilizer per m², you need 78.5×20 = 1,570g = 1.57 kg. The circumference (perimeter) of this circle = 2π×5 = 31.4 m of edging needed.

💡 Competitor Gap — Rhombus Area: Most area calculators miss the rhombus. A rhombus has four equal sides but its area is calculated from its diagonals: A = (d1 × d2) / 2. This is because the diagonals of a rhombus are perpendicular bisectors of each other, forming four right triangles. Example: a rhombus-shaped floor tile with diagonals 12 cm and 8 cm has area = (12×8)/2 = 48 cm². Used in floor tile patterns, kite shapes, and diamond designs.

Trapezoid Area — Average Width Times Height

A trapezoid has two parallel sides (called bases). Its area formula A = (b1 + b2)/2 × h takes the average of the two bases and multiplies by the height. The height is the perpendicular distance between the bases. Example: a trapezoidal garden bed with bases 6 m and 10 m, height 4 m: A = (6+10)/2 × 4 = 8 × 4 = 32 m². Real estate lots, road cross-sections, and architectural cross-beams often have trapezoidal cross-sections.

Ellipse (Oval) Area — Covered by Few Competitors

An ellipse has two axes: a (semi-major, half the longer diameter) and b (semi-minor, half the shorter diameter). Area: A = π × a × b. Example: an oval bathtub measuring 170 cm × 75 cm (semi-axes 85 and 37.5): A = π × 85 × 37.5 = 10,012 cm² = about 1 m². An ellipse with a = b becomes a circle (both reduce to πr²). Used for oval tables, elliptical running tracks (the curved ends), and satellite dishes.

Circle Sector Area — The Pie Slice

A sector is a “pie slice” of a circle. Area: A = (θ/360) × πr² where θ is the central angle in degrees. A quarter circle (θ = 90°) has area = (1/4) × πr². A semicircle (θ = 180°) = (1/2) × πr². Example: a sector with radius 5 m and angle 60°: A = (60/360) × π × 25 = 13.09 m². Used for fan-shaped garden beds, pizza slice area, windshield wiper coverage, and circular stair landings.

How to Calculate Area of Composite Shapes

Real rooms and land plots are often irregular (L-shaped, U-shaped). The method: divide into simple shapes, calculate each area, add them together. An L-shaped room can be split into two rectangles. A room with a circular bay window adds the semicircle area to the rectangular main area. For irregular plots, divide into triangles and rectangles. Always subtract any holes or cutouts from the total. This technique works for any shape that can be decomposed into the basic shapes above.

Real-World Area Applications

Flooring & tiling: Calculate room area in sq ft or m², add 10% waste, divide by tile size
Paint: 1 gallon typically covers 350–400 sq ft per coat. Calculate wall area (L×H for each wall, minus doors and windows)
Lawn & garden: Area determines fertilizer, seed, and pesticide quantities
Real estate: Total floor area (gross vs net) — all rooms summed
Roofing: Roof area determines shingles or tiles needed
Agriculture: Field area in acres (1 acre = 43,560 sq ft = 4,047 m²) determines seed and fertilizer
Fabric & sewing: Calculate fabric needed for curtains, upholstery, clothing

Frequently Asked Questions

Area = Length × Width. A room 12 ft × 10 ft = 120 sq ft. Add 10% for flooring waste = 132 sq ft to purchase. This is the most commonly used area formula in home improvement and construction.

A = (1/2) × base × height. The height must be perpendicular to the base. If only three sides are known, use Heron’s formula: s = (a+b+c)/2, A = √(s(s-a)(s-b)(s-c)). Example: base 8, height 5 = 20 cm².

A = πr². Enter the radius (half the diameter). A circle with radius 7 cm = π×49 = 153.94 cm². If you know the diameter, r = d/2 first.

A = (b1 + b2) / 2 × h. Add both parallel sides, divide by 2 (the average width), then multiply by the perpendicular height. Example: bases 6 and 10, height 4 = (6+10)/2×4 = 32 cm².

A = (d1 × d2) / 2, where d1 and d2 are the two diagonals. Example: diagonals 10 and 8 = (10×8)/2 = 40 cm². This uses the fact that a rhombus’s diagonals are perpendicular, forming 4 right triangles.

A = π × a × b, where a is the semi-major axis (half the long diameter) and b is the semi-minor axis (half the short diameter). Example: ellipse with axes 10 and 6 (semi-axes 5 and 3) = π×5×3 = 47.12 cm².

A = (θ/360) × πr². Sector with radius 5 and 90° angle = (90/360)×π×25 = 19.63 cm². A sector is a pie-slice portion of a circle. 360° = full circle, 180° = semicircle, 90° = quarter circle.

Area measures the flat surface inside a shape (square units: cm², ft²). Perimeter measures the total length around the outside (linear units: cm, ft). This calculator shows BOTH area and perimeter for each shape.

Measure length and width in feet and multiply them. 12 ft × 10 ft = 120 sq ft. For L-shaped rooms, split into two rectangles, calculate each, add them. Add 10% for waste when ordering flooring.

A = base × height (perpendicular height, not the slant side). Example: base 9 cm, perpendicular height 5 cm = 45 cm². The height is the vertical distance between the two parallel bases.

Heron’s formula finds triangle area when three sides are known. s = (a+b+c)/2 (semi-perimeter). Area = √(s(s-a)(s-b)(s-c)). Example: sides 5, 12, 13. s = 15. Area = √(15×10×3×2) = √900 = 30 cm².

Divide the composite shape into simple shapes (rectangles, triangles, semicircles). Calculate each area separately. Add them for total area, or subtract if some areas are holes or cutouts. This works for any irregular shape.

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