Free ATI TEAS 7 Mathematics guide
October 2026 - Circumference & Area of a Circle Study Guide for nursing students in
Circumference is 2πr and area is πr². The commonest slip is being handed a diameter and using it as the radius, which doubles the circumference and quadruples the area — so the first thing to do with any circle question is decide which one you have been given.
The formulas, and both forms
Each can be written in terms of the radius or the diameter. Knowing both saves a conversion step.
| Quantity | With radius | With diameter | Units |
|---|---|---|---|
| Circumference | C = 2πr | C = πd | Ordinary (cm, m) |
| Area | A = πr² | A = π(d/2)² | Squared (cm², m²) |
Radius, diameter, and the trap
- The radius runs from the centre to the edge. The diameter runs all the way across, through the centre, so
d = 2r. - Questions describe a circle by whichever is more natural in context — "a wound 6 cm across" is a diameter; "a catheter with a 2 mm radius" is a radius.
- Halve a diameter before it enters
πr². Using the diameter as the radius gives four times the correct area, and that value is always among the choices.
A circle described by its diameter
Worked example
A circular dressing is 14 cm across. Find its circumference and its area, using π ≈ 3.14.
- 14 cm is the diameter, so the radius is 7 cm.
- Circumference:
πd = 3.14 × 14 = 43.96cm. (Or2πr = 2 × 3.14 × 7, the same thing.) - Area:
πr² = 3.14 × 7² = 3.14 × 49 = 153.86cm². - Check the units: circumference in cm, area in cm².
Circumference ≈ 43.96 cm, area ≈ 153.86 cm². Using 14 as the radius would have given an area of 615.44 cm² — four times too large.
Square the radius, not the whole expression. πr² means π times (r squared), so for a radius of 5 it is 3.14 × 25, not (3.14 × 5)². The order of operations does this automatically if you follow it.
Working backwards
Given a circumference and asked for the radius, divide by 2π: a circumference of 31.4 cm gives 31.4 ÷ 6.28 = 5 cm. Given an area, divide by π and take the square root: an area of 78.5 cm² gives 78.5 ÷ 3.14 = 25, and the square root of 25 is 5 cm.
Semicircles follow directly. Half a circle has half the area — πr² ÷ 2 — but its perimeter is half the circumference plus the diameter, because the straight edge is part of the boundary.
Terms to know
- Radius
- The distance from the centre of a circle to its edge.
- Diameter
- The distance across a circle through its centre. Twice the radius.
- Circumference
- The distance around a circle.
- Pi (π)
- The ratio of circumference to diameter, approximately 3.14.
- Chord
- A straight line between two points on a circle. The diameter is the longest chord.
- Arc
- Part of the circumference.
- Sector
- A slice of a circle, bounded by two radii and an arc.
What the TEAS asks most
The same core ideas appear in different wording. If you can answer these without stopping to think, you have what this section requires.
Find the circumference of a circle with this radius or diameter.
2πr or πd. Identify which measurement you have been given first.
Find the area.
πr², halving the diameter first if that is what the question gave you.
The circumference is X. What is the radius?
Divide by 2π.
What is the perimeter of this semicircle?
Half the circumference plus the diameter — the straight edge counts.
Key points
- Circumference
2πrorπd; areaπr². - Diameter is twice the radius. Halve it before using
πr². - Using a diameter as a radius gives four times the correct area.
- Square the radius before multiplying by π.
- A semicircle's perimeter includes its straight edge.
Review quiz
4 questions on this topic, each with the reasoning worked through. Pick an answer to see how it went, or open the explanation straight away.
-
Question 1 of 4
A circle has a radius of 6 cm. What is its area, using
π ≈ 3.14?Show the answer and explanation
Correct answer: B.
113.04cm²3.14 × 6² = 3.14 × 36 = 113.04cm².37.68is the circumference rather than the area,452.16treats 6 as the diameter and squares 12, and18.84is3.14 × 6— π times the radius, with no squaring at all. -
Question 2 of 4
A circular wound measures 10 mm across. What is its circumference, using
π ≈ 3.14?Show the answer and explanation
Correct answer: A.
31.4mm"Across" means diameter, so
C = πd = 3.14 × 10 = 31.4mm.62.8treats 10 as the radius,78.5is the area of a circle of radius 5, and15.7halves the correct answer. -
Question 3 of 4
A circle has a circumference of
25.12cm. What is its radius, usingπ ≈ 3.14?Show the answer and explanation
Correct answer: B.
4cmDividing by
2πgives25.12 ÷ 6.28 = 4cm, and checking forwards confirms it:2 × 3.14 × 4 = 25.12. 8 cm is the diameter, and12.56comes from halving the circumference without dividing by π. -
Question 4 of 4
What is the perimeter of a semicircle with a diameter of 8 cm, using
π ≈ 3.14?Show the answer and explanation
Correct answer: B.
20.56cmHalf the circumference is
(3.14 × 8) ÷ 2 = 12.56, and the straight edge adds the 8 cm diameter, giving20.56cm. Choice A stops at the curve and forgets the flat edge — the standard error on semicircle perimeters.
Put this TEAS topic into practice
Build a set on circumference and area of a circle and use the rationale on every question to reinforce what you just reviewed.
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