Circle explorer

What the explorer shows

The radius is the distance from the center of a circle to its edge. The diameter goes all the way across through the center, so it is twice the radius. The circumference is the distance around the circle. On this page, drag the purple handle or use the stepper to set the radius in steps of half a unit, and the readout gives the diameter, the circumference and the area with the formulas filled in.

Unroll the circumference peels the edge of the circle off and lays it flat as a red line. When it is fully unrolled, three blocks the length of the diameter appear underneath. The red line is always a little longer than the three blocks: about 3.14 diameters. That ratio, circumference ÷ diameter, is the number π (pi).

Why unroll it

Students often meet C = 2πr as a rule with no reason behind it. Seeing the circumference laid out next to the diameter shows that π is not a magic number but a measurement: every circle's circumference is the same multiple of its diameter, so C = πd, which is the same as 2πr. Common Core standard 7.G.B.4 asks students to know the formulas for circumference and area and to give an informal argument for how they are related. Changing the radius and comparing results also shows that doubling the radius doubles the circumference but multiplies the area by 4.

Common mix-ups

• Using the diameter in πr². Area uses the radius; if you know the diameter, halve it first.
• Thinking π is exactly 3.14 or 22/7. Both are approximations; π goes on forever without repeating.
• Expecting the area to double when the radius doubles. It grows four times, because the radius is squared.
• Writing area in plain units. Circumference is a length, in units; area is in square units.

A 10-minute circle lesson

• Set the radius to 2 and unroll the circumference. Ask how many diameters fit.
• Repeat with a radius of 7. Ask whether the answer is still a bit more than 3.
• Introduce π as that ratio and write C = π × d.
• Double the radius from 3 to 6 and compare the circumference and area in the readout.
• Press Challenge and let the student find the radius from a circumference or area before pressing Check.

Where it fits by grade

• Grades 5 and 6: radius and diameter as part of describing shapes, and practice with decimals.
• Grade 7: formulas for the circumference and area of a circle and the relationship between them (7.G.B.4).
• Grade 8: circles return in the volume of cylinders, cones and spheres (8.G.C.9).
• England: Year 6 names the radius, diameter and circumference and knows that the diameter is twice the radius. Key Stage 3 calculates the circumference and area of circles.

Learning more

Area and perimeter are on Eurekami's skill map, and Daily Practice brings them back when they start to fade. For older students, Exam Prep includes GCSE Maths practice papers, and the AI tutor asks guiding questions when a circle problem is tricky.