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MATH · GRADE 7Measurement

Solving Circle Problems

Fencing and sod for the same round garden — why do they need different formulas?

Grade 7
the fencethe sod?why two formulas…
In this lesson

What a circle problem asks

Every circle problem comes down to one decision: are you measuring the edge of the circle, or the inside of it? That single choice picks the formula. The arithmetic is the easy part — choosing right is the whole skill.

rAround the edgea length · metres (m)Across the insidea surface · square metres (m²)
  • Circumference when the question is about the edge: a border, fencing, a track, or the distance around. It is a length, so the answer lands in metres or centimetres.
  • Area when the question is about the inside: a surface, a covering, planting, painting, or material. It is a surface, so the answer lands in square metres or square centimetres.

Use π≈3.14\pi \approx 3.14 unless the question tells you otherwise.

Try it both ways

Before you compute anything, decide which formula reaches the answer in one step. The three formulas sit side by side — the same choice you'll make on every circle problem.

Solve both ways

Pick the formula that gets you there in one step, enter the answer, then check. Watch the units — a length or a surface?

Problem 1 of 6

Pick the circle formula that fits the given and the asked quantity, enter the answer, then check. Problems run forward (given a length, find circumference or area) and backward (given circumference or area, find a length).
Given
radius = 5 cm
Find
area
Formula you used

Work backward

Sometimes the problem gives the final measurement first.

If a circle has area 78.5 cm², work backward:

78.5=3.14r278.5 = 3.14r^2 r2=25r^2 = 25 r=5 cmr = 5\text{ cm}

If a circle has circumference 31.4 m, work backward from

C=πDC = \pi D:

31.4=3.14D31.4 = 3.14D D=10 mD = 10\text{ m}

The units help check your choice. Radius and diameter are lengths. Area is square units. Circumference is a length around the edge.

Two measurements in one problem

A circular garden has radius 3 m. The gardener needs edging around the outside and mulch over the inside.

The edging is circumference:

C=2(3.14)(3)=18.84 mC = 2(3.14)(3) = 18.84\text{ m}

The mulch is area:

A=3.14(32)=28.26 m2A = 3.14(3^2) = 28.26\text{ m}^2

Same circle, two different questions. The edge is measured in metres; the inside is measured in square metres.

On a prairie farm

Drive past a farm near Lethbridge or Camrose and you will pass rows of steel grain bins — each one a cylinder standing on a round concrete pad.

Pouring a new pad takes both formulas. The crew fills the inside of the pad with concrete, so that amount is area. Then a steel collar wraps the outside edge to hold the pour, so that length is circumference.

For a pad of radius 2.5 m:

A=3.14(2.52)=19.625 m2 of concreteA = 3.14(2.5^2) = 19.625\text{ m}^2 \text{ of concrete} C=2(3.14)(2.5)=15.7 m of collarC = 2(3.14)(2.5) = 15.7\text{ m of collar}

One pad, two orders to phone in — one measured in square metres, one in metres. Naming the attribute before reaching for a formula is what keeps them straight.

Misconception probe

Worksheet

These aren't graded. For each one, decide first what's being measured — the edge or the inside — then pick the formula and solve. When a problem hands you the finished measurement and asks for the radius or diameter, work backward.

Practice · Not graded

MA.7.MEA.1

Practice the idea

01 / 10

A circle has circumference 31.4 cm. Without calculating, which is closest to its area?

Multiple choice: a circle has circumference 31.4 cm. Without calculating, which value is closest to its area?
Show common mistakes

Student says

“The circle has area 78.5 cm², so the radius is 78.5 ÷ 3.14 = 25 cm.”

What it reveals

Stopped halfway through the inverse area problem. Dividing by π gives r², not r.

Targeted response

After 78.5 ÷ 3.14, you have r² = 25. The radius is the number whose square is 25, so r = 5 cm.

Student says

“The edging and mulch both use the same answer because it's the same circle.”

What it reveals

Confused the circle with the attribute being measured. Same object can have both an edge length and an inside area.

Targeted response

Ask what is being measured. Edging follows the outside path, so it is circumference in metres. Mulch covers the inside surface, so it is area in square metres.

Student says

“The radius is 7 m, so I used A = 3.14 × 14² because I needed the diameter.”

What it reveals

Replaced the radius with the diameter in the area formula. A = πr² always takes the radius directly.

Targeted response

The formula is A = πr². If the radius is 7 m, use 7 — not 14. Diameter appears only in the circumference formula C = πD. For area, use the radius you were given.

Going further

The same formula-choice habit carries straight into surface area and volume. Before calculating, name the attribute: edge length, surface area, or volume. The formula follows the attribute — just as it does here, where naming "edge" or "inside" decides everything.