Geometry Calculators
Area, perimeter, diagonal, circumference, arc length, and sector area for circles, rectangles, squares, and triangles — with visual diagrams and verified formulas.
All Geometry Calculators (4)
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Circle Formulas and Quick Reference
All circle properties connect through the radius (r) and the constant π ≈ 3.14159. Knowing any one measurement — radius, diameter, area, or circumference — lets you calculate all the others using the formulas below.
Rectangle Formulas
A rectangle has four right angles with opposite sides equal. Given length (l) and width (w), the three key properties are area, perimeter, and diagonal. The diagonal uses the Pythagorean theorem since the diagonal splits the rectangle into two right triangles.
Square Formulas
A square is a special rectangle where all four sides are equal. This simplifies the formulas — perimeter is just 4 times the side, and the diagonal is the side multiplied by √2 (approximately 1.41421).
Triangle Area — Three Methods
Triangle area can be calculated in different ways depending on what measurements are known. The three most common methods are the base-height formula, Heron's formula (from three sides), and the trigonometric formula (two sides and the included angle).
Shape Comparison — Same Perimeter, Different Areas
Among shapes with the same perimeter, the circle always encloses the most area (isoperimetric inequality). This table compares four shapes all with perimeter = 40 units:
| Shape | Perimeter | Dimensions | Area |
|---|---|---|---|
| Circle | 40 | r = 40÷(2π) = 6.366 | 127.32 sq units |
| Square | 40 | side = 10 | 100 sq units |
| Rectangle | 40 | 12 × 8 | 96 sq units |
| Rectangle | 40 | 15 × 5 | 75 sq units |
The circle (127.32) encloses 27% more area than the square (100) and 33% more than the 12×8 rectangle (96) — all with the same perimeter of 40. Among rectangles, the square always maximizes area for a given perimeter. Source: Math Is Fun — Geometry Reference
5 Geometry Calculation Tips
- Area scales with the square of any linear dimension. Double the radius of a circle and the area quadruples — not doubles. A circle with r=7 has area 153.9380; r=14 gives area 615.7522 (exactly 4×). The same applies to squares and rectangles: double both dimensions and area multiplies by 4. This is why large rooms feel proportionally much bigger than the floor plan dimensions suggest.
- The square diagonal formula (side × √2) is faster than Pythagoras for squares. For a 9×9 square: diagonal = 9 × 1.41421 = 12.7279. Using Pythagoras: √(81+81) = √162 = 12.7279 — same answer, more steps. √2 ≈ 1.41421 is worth memorizing for quick square diagonal estimates.
- Heron's formula works for any triangle — no height needed. When you only know three side lengths (common in real measurements), Heron's formula (Area = √(s(s-a)(s-b)(s-c)) where s is the semi-perimeter) gives exact area without needing to find the height. For sides 5-6-7: s=9, Area = √(9×4×3×2) = √216 = 14.6969.
- Circumference ÷ diameter always equals π, for every circle. This ratio — approximately 3.14159 — is the definition of π. If you measure any circular object and divide its circumference by its diameter, you get π regardless of size. It's a useful field check: if the ratio comes out far from 3.14, either the shape isn't a true circle or one measurement is off.
- For construction and tiling, always add 10% for waste. When using geometry calculators to estimate materials (tiles, flooring, paint), calculate the exact area first, then add 10% for cuts, waste, and irregularities. For a 15×10 m (approx. 49×33 ft) room: area = 150 m² (≈ 1,615 sq ft), with 10% waste allowance order 165 m² (≈ 1,776 sq ft) of material. The construction calculators handle waste factors automatically.