Measuring Space: Perimeter and Area: common misconceptions

If your child says… “The same perimeter means the same area.”

Ask together

Do these shapes contain the same number of squares?

What’s true

Changing the outline can change the enclosed area without changing the perimeter.

If your child says… “Doubling a circle’s width makes its border four times longer.”

Ask together

Does dividing each travelled distance by its diameter give very different results?

What’s true

Circumference grows by the same multiplier as diameter, which is twice the radius.

If your child says… “Enough matching digits make a fraction exactly pi.”

Ask together

Do the third decimal digits match?

What’s true

Both 3.14 and twenty-two sevenths approximate pi, and closer fractions always exist.

If your child says… “The same angle means the same curved distance.”

Ask together

If the radius doubles, what happens to the circumference you take a fraction of?

What’s true

Equal angles select equal fractions of circumferences, but larger radii give longer arcs.

If your child says… “Multiply the base by the sloping side.”

Ask together

Which distance becomes the rectangle’s width when you move the triangle?

What’s true

The rectangle’s width matches the perpendicular height, rather than the sloping side.

If your child says… “Those smaller triangles have different shapes, so their areas differ.”

Ask together

Which changed between the smaller triangles: base length or perpendicular height?

What’s true

Equal bases and equal perpendicular heights give equal areas, even when shapes differ.

If your child says… “Four side lengths always tell us a shape’s area.”

Ask together

A square leans into a rhombus, keeping its side lengths. Must its area stay unchanged?

What’s true

Brahmagupta’s rule requires all four corners on one circle, and a zero-length fourth side gives Heron’s.

If your child says… “A quarter-circle’s area is a quarter of its curved border’s length.”

Ask together

Does the shaded quarter cover part of the surface or just part of the border?

What’s true

A sector takes its angle’s fraction of the circle’s area, rather than its circumference.