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Circles: Area and Circumference

Find the circumference and area of a circle, and work backwards to the radius or diameter.

Practice this skill 30 min Circles: area and circumference Leer en español
Key idea

What you'll learn

In this lesson you will learn the parts of a circle and two key measurements:

  • the circumference — the distance around a circle (like the perimeter), and
  • the area — the space inside a circle.

You will also learn to work backwards: given the circumference or the area, find the radius or the diameter.

Why it matters on the GED: Circle questions appear on nearly every test — pizzas, wheels, round tables, sprinklers, circular gardens and pools. They also come back in composite figures (a semicircle on top of a window) and in 3-D shapes (cylinders and cones use the area of a circle). The single most common mistake on the test is mixing up the radius and the diameter, so we will practice that carefully.

In real life: Circumference tells you how far a wheel rolls in one turn or how much edging goes around a round flower bed. Area tells you how much grass a sprinkler waters or which pizza is the better deal.

Words to know

  • Circle — all the points that are the same distance from one middle point (a clock face, a pizza).
  • Center — the middle point of the circle.
  • Radius (r) — the distance from the center to the edge (a 5 in radius means the edge is 5 in from the middle). Plural: radii.
  • Diameter (d) — the distance straight across the circle through the center. It is two radii: d = 2r (radius 5 in → diameter 10 in).
  • Circumference (C) — the distance around the circle (the length of the crust of a pizza).
  • Area (A) — the space inside the circle, in square units (in², ft²).
  • Pi (\pi) — the number you get when you divide any circle's circumference by its diameter; about 3.14. The TI-30XS has a \pi key.
  • Semicircle — half of a circle (a half-moon window).
  • Chord — a segment joining two points on the circle; a diameter is the longest chord.
  • "In terms of \pi" — leave \pi as a symbol in the answer, like 16\pi, instead of multiplying it out.

All Math words in the glossary

Circle showing the center, a radius r from the center to the edge, a diameter d across through the center (d = 2r), and the circumference C as the distance around center diameter d radius r circumference C = distance around

The radius goes from the center to the edge. The diameter goes all the way across through the center, so it is twice as long as the radius.

Radius and diameter: the first thing to check

Every circle formula on the GED is written with the radius r (or the diameter d for circumference). Before you do anything else, ask: "Was I given the radius or the diameter?"

  • Diameter → radius: divide by 2. A 16 in pizza (the size of a pizza is its diameter) has r = 16 \div 2 = 8 in.
  • Radius → diameter: multiply by 2. A sprinkler that reaches 15 ft from its center waters a circle with d = 30 ft.
Two circles. Left: a 16-inch pizza; the size is the diameter, 16 in (red, straight across through the center), so the radius (blue, center to edge) is 8 in. Right: a sprinkler that reaches 15 ft from its center; that is the radius, 15 ft, so the diameter is 30 ft. d = 16 in r = 8 in 16-inch pizza: d = 16, r = 16 ÷ 2 = 8 in d = 30 ft r = 15 ft Sprinkler reaches 15 ft: r = 15, d = 30 ft

Clues in word problems:

  • "across", "wide", "a 12-inch pizza", "a 26-inch bicycle wheel" → diameter.
  • "from the center", "reaches 15 ft out", "a rope tied to a post" → radius.

Circumference: the distance around

For every circle, the circumference is a little more than 3 times the diameter. That "little more than 3" is \pi \approx 3.14.

C = \pi d \qquad \text{or} \qquad C = 2\pi r

Both formulas are the same because d = 2r.

  • Diameter 26 in: C = 3.14 \times 26 = 81.64 in.
A wheel with diameter 26 in rolled out flat for one full turn. The unrolled blue line is the circumference, a little more than 3 diameters: C = 3.14 × 26 = 81.64 in. d = 26 in 1 d 1 d 1 d 0.14d one turn rolls C = π × 26 ≈ 3.14 × 26 = 81.64 in
  • Radius 7 ft: C = 2 \times 3.14 \times 7 = 43.96 ft.
  • Radius 4 m, in terms of \pi: C = 2\pi(4) = 8\pi m.

Circumference is a length, so it uses plain units (in, ft, m).

Wheels: in one full turn, a wheel rolls forward exactly one circumference. 50 turns of a wheel with C = 75.36 in roll 50 \times 75.36 = 3{,}768 in.

Area: the space inside

A = \pi r^2

Square the radius first, then multiply by \pi. (\pi r^2 means \pi \times r \times r — it does not mean (\pi r)^2 and does not mean \pi \times r \times 2.)

  • Radius 5 in: A = 3.14 \times 5^2 = 3.14 \times 25 = 78.5 in².
A circle of radius 5 in with a red square of side 5 drawn from the center to the edge; the square has area r squared = 25. The circle area is 3.14 × 25 = 78.5 square inches, about 3.14 of those squares. r = 5 r = 5 r² = 25 A = 3.14 × 25 = 78.5 in² (about 3.14 of the red r × r squares)
  • Diameter 16 in: first r = 8; A = 3.14 \times 8^2 = 3.14 \times 64 = 200.96 in².
  • Radius 4 m, in terms of \pi: A = \pi(4^2) = 16\pi m².

Area uses square units (in², ft²). If your answer is in plain feet, you found the circumference.

Why the radius matters so much: because it is squared, using the diameter by mistake makes the area 4 times too big. With the 16 in pizza, 3.14 \times 16^2 = 803.84 — four times 200.96.

Working backwards: find r or d

From the circumference: divide by \pi to get the diameter.

d = \frac{C}{\pi} \qquad r = \frac{C}{2\pi}
  • C = 62.8 ft: d = 62.8 \div 3.14 = 20 ft, so r = 10 ft.

From the area: divide by \pi, then take the square root.

r^2 = \frac{A}{\pi} \qquad r = \sqrt{\frac{A}{\pi}}
  • A = 113.04 m²: r^2 = 113.04 \div 3.14 = 36, so r = \sqrt{36} = 6 m and d = 12 m.

From area to circumference (two steps): find r from the area, then use C = 2\pi r.

  • A = 50.24 ft² → r^2 = 16 → r = 4 → C = 2(3.14)(4) = 25.12 ft.

When the answer is "in terms of \pi": A = 49\pi means r^2 = 49, so r = 7. C = 18\pi means d = 18, so r = 9.

Formula

Formulas (on the GED formula sheet)

C = 2\pi r = \pi d \qquad A = \pi r^2 \qquad d = 2r \qquad \pi \approx 3.14
  • r = radius (center to edge), d = diameter (all the way across through the center).
  • C = circumference (distance around, plain units); A = area (square units).
  • \pi = pi, about 3.14 (use 3.14 or the calculator's \pi key — whichever the question says).

On the GED formula sheet: yes — the area and circumference of a circle and \pi \approx 3.14 are printed on the sheet. The relationship d = 2r is not printed; you must remember it.

Worked example

Example 1 (easy): both measurements

A circle has a radius of 6 cm. Find its circumference and its area. Use \pi \approx 3.14.

A circle with radius 6 cm drawn from the center to the edge. r = 6 cm
  1. Circumference: C = 2\pi r = 2 \times 3.14 \times 6 = 37.68.
  2. Area: A = \pi r^2 = 3.14 \times 6^2 = 3.14 \times 36 = 113.04.

Answer: C = 37.68 cm; A = 113.04 cm²

Check: C should be a little more than 3 \times 12 = 36 ✓; A should be a little more than 3 \times 36 = 108 ✓.

Worked example

Example 2 (word problem): trim for a round table

A round table is 4 ft across. How much decorative trim is needed to go once around its edge? Use \pi \approx 3.14.

A round table 4 ft across: the red line through the center is the diameter, 4 ft. The thick blue edge is the trim, which is the circumference. d = 4 ft (across) trim = circumference
  1. "4 ft across" is the diameter: d = 4.
  2. Trim goes around the edge, so find the circumference: C = \pi d = 3.14 \times 4 = 12.56.

Answer: 12.56 ft of trim

Check: a little more than 3 diameters: 3 \times 4 = 12 ✓.

Worked example

Example 3 (work backwards): from circumference

A round cake pan has a circumference of 47.1 in. What are its diameter and radius? Use \pi \approx 3.14.

A round cake pan whose circumference (blue edge) is 47.1 in. The diameter (red, through the center) and the radius (green) are unknown. d = ? r = ? C = 47.1 in
  1. C = \pi d, so d = C \div \pi.
  2. d = 47.1 \div 3.14 = 15.
  3. r = d \div 2 = 15 \div 2 = 7.5.

Answer: diameter 15 in; radius 7.5 in

Check: 3.14 \times 15 = 47.1 ✓.

Worked example

Example 4 (work backwards): from area

A lawn sprinkler waters a circle with an area of 314 ft². How far does the water reach from the sprinkler? Use \pi \approx 3.14.

A sprinkler at the center of a watered circle of area 314 square feet. The dashed radius r is unknown. r = ? A = 314 ft² sprinkler
  1. "How far from the sprinkler" is the radius.
  2. A = \pi r^2, so 314 = 3.14 r^2.
  3. Divide by 3.14: r^2 = 314 \div 3.14 = 100.
  4. Square root: r = \sqrt{100} = 10.

Answer: 10 ft

Check: 3.14 \times 10^2 = 3.14 \times 100 = 314 ✓.

Worked example

Example 5 (GED-hard): buying grass seed

A sprinkler sprays water 15 ft in every direction. Grass seed is sold in bags that each cover 200 ft². How many bags are needed to seed the whole watered circle? Use \pi \approx 3.14.

  1. 15 ft "in every direction" from the center is the radius: r = 15.
  2. Area: A = 3.14 \times 15^2 = 3.14 \times 225 = 706.5 ft².
  3. Bags: 706.5 \div 200 = 3.5325.
  4. You can't buy part of a bag, and 3 bags cover only 600 ft², so round up.
Left: a watered circle with radius 15 ft and area 3.14 × 225 = 706.5 square feet. Right: four bags of seed, 200 square feet each, laid end to end up to 800; the lawn bar of 706.5 goes past 3 bags (600), so 4 bags are needed. r = 15 ft A = 706.5 ft² bag 1 bag 2 bag 3 bag 4 lawn 706.5 ft² 0 800 600 3 bags = 600 < 706.5; 4 bags = 800 ✓

Answer: 4 bags

Check: 4 \times 200 = 800 \ge 706.5 ✓ and 3 \times 200 = 600 < 706.5 ✓.

Calculator tip

Calculator tip: TI-30XS MultiView

Circle questions almost always allow the calculator.

  • The \pi key: press π (it is on its own key on the TI-30XS). π × 6 x² enter gives 36\pi; press toggle to see 113.0973355. With 3.14 you get 113.04. The question will say which to use; if it says "use 3.14", type 3.14, because answer choices are often rounded from 3.14.
  • Area: type 3.14 × 8 x² enter → 200.96. The calculator squares the 8 before multiplying, which is exactly what \pi r^2 means.
  • Radius from area: 2nd x² gives √. Type 113.04 ÷ 3.14 under the root, press the right arrow to leave it, then enter → 6. The whole division must be under the root.
  • Fraction answers: if a result shows as a fraction, the toggle key switches it to a decimal.

When not to use it: if the answers are "in terms of \pi" (like 16\pi), don't multiply by 3.14 at all — just work with the radius.

Common mistake

Common mistakes

  • Using the diameter as the radius. A 16 in pizza has r = 8, so A = 3.14 \times 64 = 200.96. Using 16 gives 803.84 — 4 times too big.
  • Using the radius in C = \pi d. With r = 7, C = 2\pi(7) = 43.96, not \pi(7) = 21.98 (that is only half way around).
  • Mixing up area and circumference. "Fence/edging/trim/how far it rolls" → circumference. "Cover/seed/paint/how much pizza" → area.
  • "Doubling" instead of squaring. r^2 means r \times r. For r = 5: 5^2 = 25, not 5 \times 2 = 10.
  • Forgetting the square root when working backwards. If r^2 = 36, then r = 6, not 36.
  • Stopping at the radius when the diameter is asked (or the reverse). Re-read the question's last line before you choose.
Test strategy

GED test strategy

  • Circle the word "radius" or "diameter" in the question and in the answer choices — the test-makers know this is the most common trap.
  • Estimate with 3. C is a little more than 3d; A is a little more than 3r^2. A 10 ft-radius circle has area just over 300 ft², so 31.4 or 3,140 are out.
  • Check units. Square units → area; plain units → circumference, radius or diameter.
  • Plug answers back in on work-backwards questions: if a choice says r = 6, check 3.14 \times 36 = 113.04.
  • Follow the instructions on \pi: use 3.14 if told; use the \pi key if told; leave \pi in the answer if the choices show \pi.

Check yourself

Circles: area and circumference

Find the area of the circle. Use \pi \approx 3.14.

Circle with center marked and a radius of 5 in drawn to the right 5 in
Answer choices
Hint

The area formula uses the radius. What do you do to the radius before multiplying by 3.14?

Check yourself

Circles: area and circumferenceFill in

A bicycle wheel has a diameter of 26 inches. What is the circumference of the wheel? Use \pi \approx 3.14.

inches

Type your answer in the box. Fractions like 3/4 are fine.

Hint

You were given the diameter. Which circumference formula uses the diameter directly?

Summary

  • Radius = center to edge; diameter = all the way across = 2r. Check which one you were given first.
  • Circumference (distance around): C = \pi d = 2\pi r, in plain units.
  • Area (inside): A = \pi r^2 — square the radius, then multiply by \pi; answer in square units.
  • Backwards: d = C \div \pi; r = \sqrt{A \div \pi}.
  • One wheel turn = one circumference. Use 3.14, the \pi key, or "in terms of \pi" exactly as the question says.
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