Mathematics Year 6: Circle: circumference, π and area
Mathematics, Year 6 (Portugal): See that the perimeter of a circle is directly proportional to its diameter, with constant π, and use P = π × d and A = π × r² in problems.
π: the number that links the edge to the diameter
The perimeter P of a circle (the length of its edge, its circumference) is always a little more than three times the diameter d. That number, the same for every circle, is called π (say “pi”). So P = π × d; since the diameter is twice the radius r, also P = 2 × π × r. The area of the circle is A = π × r × r = π × r². The value π ≈ 3.14 is only an approximation: a result calculated with it is “about”.
Why P and d are proportional
With a piece of string, measure the edge and the diameter of three round objects, in cm: d = 6.0 and P = 18.8; d = 9.5 and P = 29.8; d = 14.0 and P = 44.0. Divide P by d: 3.13; 3.14; 3.14. It is (almost) always the same number, and the small differences come from measuring. If P ÷ d is constant, P and d are directly proportional, and π is the constant of proportionality: P = π × d. For the area, a square of side r has area r × r, and the circle covers π of those squares: A = π × r × r.
A case: a circle with radius 6 cm
A circle has radius r = 6 cm. Diameter: d = 2 × 6 = 12 cm. Perimeter: P = π × 12 ≈ 3.14 × 12 = 37.68 cm. Area: A = π × 6² = π × 36 ≈ 3.14 × 36 = 113.04 cm². Is it reasonable? The perimeter must lie between 3 and 4 diameters, that is, between 36 cm and 48 cm: 37.68 does. And the circle fits inside a square of side 12 cm, with area 144 cm²: 113.04 is less. Look at the units: cm for the perimeter, cm² for the area.
The trap: radius or diameter, r² or 2 × r
Three common slips. One: using the radius where the formula wants the diameter. With r = 6 cm, P = π × 12, not π × 6. Two: mixing up r² with 2 × r: r² = r × r = 36, but 2 × r = 12. Three: giving a perimeter in cm² or an area in cm: the perimeter is a length (cm), the area is a surface (cm²). And remember 3.14 is approximate: 37.68 cm is “about”, not the exact value.
Where you see this outside school
A wheel with a diameter of 60 cm turns once and moves forward about 3.14 × 60 = 188.4 cm: that is its perimeter. To know the ribbon around a round cake, you use the perimeter; to know the paint for a round table, the cloth for a lid or the cheese covering a pizza, you use the area. Rule of thumb: edge → perimeter (cm); surface → area (cm²).
Test what you learned
You divide the perimeter of any circle by its diameter. What number do you get?
- a) Always 2
- b) Always π, about 3.14
- c) Always 4, as in a square
- d) A different number for each circle, bigger for bigger circles
Correct answer: b) Always π, about 3.14 — Yes: P ÷ d is the same for every circle. That constant of proportionality is π, and P = π × d.
A circular track has a radius of 20 m. How far do you run in one lap along its edge? Use π ≈ 3.14.
- a) 62.8 m
- b) 1256 m
- c) 43.14 m
- d) 125.6 m
Correct answer: d) 125.6 m — Yes: d = 2 × 20 = 40 m, and P = 3.14 × 40 = 125.6 m, about.
A circle has a radius of 4 cm. What is its area? Use π ≈ 3.14.
- a) 50.24 cm²
- b) 200.96 cm²
- c) 25.12 cm²
- d) 12.56 cm²
Correct answer: a) 50.24 cm² — Yes: A = π × r² = 3.14 × 4 × 4 = 3.14 × 16 = 50.24 cm², about.
Keep exploring
Other languages
Loading MyLeoNes™…