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Area e perimetro delle figure piane

Calcola area e perimetro di 15 figure: rettangoli, triangoli, trapezi, cerchi, ellissi, poligoni regolari e corone circolari, con disegni in scala.

Aggiornato Esempi verificati: 16

Prova
Area
cm²
Area: 40 cm²
Massimo di cifre decimali: 4; Al più vicino; a parità lontano da zero
Perimetro
26cm
Diagonal
9.434cm

The rectangle covers 40 cm² and its boundary is 26 cm long.

The rectangle, to scale

l = 8 cmw = 5 cm
Come si calcola S

  1. Area

    A=l×w=8×5=40 cm2A = l \times w = 8 \times 5 = 40\,\text{cm}^{2}

  2. Perimetro

    P=2(l+w)=2(8+5)=26 cmP = 2(l + w) = 2(8 + 5) = 26\,\text{cm}

  3. Diagonal

    d=l2+w2=82+52=9.433981 cmd = \sqrt{l^2 + w^2} = \sqrt{8^2 + 5^2} = 9.433981\,\text{cm}

Informazioni su Area e perimetro delle figure piane

Choose one of 15 plane shapes and enter the dimensions that define it. The area and perimeter come back with the other lengths that shape has: diagonals, heights, arc and chord lengths, a regular polygon's apothem and circumradius, or a ring's width. A triangle from three sides uses Heron's formula, A = √(s(s − a)(s − b)(s − c)) with s half the perimeter, and a regular polygon uses A = ns²/(4 tan(180°/n)).

The default rectangle, 8 cm by 5 cm, has an area of 40 cm², a perimeter of 26 cm and a diagonal of 9.434 cm. Area sizes flooring, turf and paint; perimeter sizes fencing, edging and skirting board. The drawing is to scale, so a mistyped dimension shows up as the wrong shape.

Base and height do not fix a triangle's slanted sides, and a trapezoid needs its legs, so those two inputs give area only. An ellipse's perimeter has no closed form; Ramanujan's second approximation is used, accurate to 0.04% or better.

Esempi svolti

Rectangle 8 × 5 cm

Shape
Rectangle
Lunghezza
8 cm
Larghezza
5 cm
Unità dei risultati
Centimetri (cm)
Area
40 cm²
Perimetro
26 cm
Diagonal
9.433981 cm

Fonte di verifica: 8 × 5, 2(8 + 5); Python 3.8 math.sqrt(89)

3-4-5 triangle by Heron's formula

Shape
Triangle (three sides)
Side a
3 cm
Side b
4 cm
Side c
5 cm
Unità dei risultati
Centimetri (cm)
Area
6 cm²
Perimetro
12 cm
Altezza
4 cm

Fonte di verifica: Right triangle with legs 3 and 4: ½·3·4 = 6; height onto side 3 is the other leg, 4

Regular hexagon, side 2 m

Shape
Regular polygon
Side length
2 m
Number of sides
6
Unità dei risultati
Metri (m)
Area
10.392305 m²
Perimetro
12 m
Apothem (inradius)
1.732051 m
Circumradius
2 m

Fonte di verifica: Python 3.8 math: 6*4/(4*tan(pi/6)) = 6√3, 2/(2*tan(pi/6)) = √3; a hexagon's circumradius equals its side

Square as a regular 4-gon (exact tan 45°)

Shape
Regular polygon
Side length
3 cm
Number of sides
4
Unità dei risultati
Centimetri (cm)
Area
9 cm²
Apothem (inradius)
1.5 cm
Circumradius
2.12132 cm

Fonte di verifica: Square of side 3: area 9, apothem 3/2; Python 3.8 math: 3/sqrt(2)

Domande

How do you find the area of a trapezoid?

Average the two parallel sides and multiply by the perpendicular height: A = (a + b)/2 × h. Bases of 10 cm and 4 cm with a height of 4 cm give 7 × 4 = 28 cm². British English calls this shape a trapezium; the formula is the same. The perimeter also needs the two slanted legs: legs of 5 cm each make it 24 cm.

How do you find the area of a triangle from three sides?

Use Heron's formula. Take the semi-perimeter s = (a + b + c)/2, then A = √(s(s − a)(s − b)(s − c)). Sides of 5, 6 and 7 cm give s = 9 and A = √(9 × 4 × 3 × 2) = √216 ≈ 14.697 cm². Three lengths only make a triangle if each is shorter than the other two combined, so 3, 4 and 8 have no area.

What is the area of a regular hexagon?

A = (3√3/2)s², about 2.598 times the side squared. A hexagon with 5 cm sides covers 64.952 cm², and one with 2 m sides covers 10.392 m². Any regular polygon with n sides of length s has A = ns²/(4 tan(180°/n)). A hexagon's circumradius equals its side, and its apothem is s√3/2 ≈ 0.866s.

Is there a formula for the perimeter of an ellipse?

Not an exact one in elementary functions: the exact perimeter is a complete elliptic integral. Ramanujan's 1914 approximation, P ≈ π(a + b)(1 + 3h/(10 + √(4 − 3h))) with h = (a − b)²/(a + b)², is off by at most 0.04%, reached only as the ellipse flattens into a line. For semi-axes of 10 and 6 it gives 51.0539977, within 1.2 × 10⁻⁹ of the exact value.

How do you convert square feet to square metres?

Multiply by 0.09290304, the square of the international foot (0.3048 m exactly, fixed in 1959). One square metre is about 10.764 ft². A 12 ft × 10 ft room is 120 ft², or 11.148 m². Area scales with the square of the length unit, so converting each side first and then multiplying gives the same result.

Quanto è preciso «Area e perimetro delle figure piane»?

La precisione dipende dai dati inseriti e dalle ipotesi del metodo. Il calcolo decimale usa 50 cifre significative, ma stime, metodi numerici e dati di origine possono essere meno precisi; l’arrotondamento visualizzato non elimina questi limiti. Esempi svolti verificati con fonti indipendenti: 16. Per esempio, «Rectangle 8 × 5 cm» viene verificato con 8 × 5, 2(8 + 5); Python 3.8 math.sqrt(89).

Da dove proviene il metodo?

Weisstein, E. W. “Heron's Formula”, “Regular Polygon”, “Ellipse” — MathWorld; Ramanujan, S. (1914) “Modular equations and approximations to π”, Quarterly Journal of Mathematics 45, 350–372 (perimeter approximation II); NIST Handbook 44, Appendix C — exact inch–centimetre relation (1 in = 2.54 cm).

Informazioni su questa calcolatrice

A△=s(s−a)(s−b)(s−c),An-gon=ns24tan⁡(π/n)Pellipse≈π(a+b)(1+3h10+4−3h)\begin{gathered} A_\triangle = \sqrt{s(s-a)(s-b)(s-c)},\quad A_{n\text{-gon}} = \frac{n s^2}{4\tan(\pi/n)} \\ P_\text{ellipse} \approx \pi(a+b)\left(1 + \frac{3h}{10 + \sqrt{4-3h}}\right) \end{gathered}

Fonti

  1. Weisstein, E. W. “Heron's Formula”, “Regular Polygon”, “Ellipse” — MathWorld
  2. Ramanujan, S. (1914) “Modular equations and approximations to π”, Quarterly Journal of Mathematics 45, 350–372 (perimeter approximation II)
  3. NIST Handbook 44, Appendix C — exact inch–centimetre relation (1 in = 2.54 cm)

Verificato con le fonti

Questa calcolatrice include 16 esempi svolti con risposte da fonti indipendenti. Fanno parte della suite di test e puoi eseguirli anche qui.

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