The parts of a circle — radius, diameter, chord, tangent, arc, central and inscribed angles — with circumference and area, arc length and sector area as a fraction of the whole, and radians. Chords, tangents and the right triangles they hide; the equation of a circle. Then solids: prisms and cylinders, pyramids and cones and where the ⅓ comes from, spheres, surface area, Cavalieri's principle, density, and how volume scales by k³ — with the numbers that decide which container really holds more. Study cards, hints, a practice quiz at three levels, and four workshop activities that check themselves.
A tank. People ask two completely different questions about it, and they need different formulas.
“Cover” means surface area, in square units. “Fill” means volume, in cubic units. Most points lost in this unit are the right arithmetic done on the wrong question — so underline the word before you pick a formula.
What people get wrong
⚠️People often think…
A circle with diameter 12 has an area of 144π.
The formula wants the radius, so halve the diameter first: r = 6, and the area is 36π. Using 12 gives an answer four times too big. Write r = d ÷ 2 directly on the picture before you touch the formula.
Write r = d ÷ 2 on the picture. Every time.
⚠️People often think…
An inscribed angle equals the arc it sits on.
A CENTRAL angle equals its arc. An INSCRIBED angle is half of it. An 80° arc gives a 40° inscribed angle — so check where the vertex is sitting: at the center, or out on the circle?
Vertex at the center: whole. Vertex on the edge: half.
⚠️People often think…
A cone with the same base and height as a cylinder holds the same amount.
A cone holds exactly one third as much — that is what the ⅓ in the formula is. Cone with r = 5 and h = 12: the cylinder would be 300π, so the cone is 100π. Pyramids get the same ⅓; cylinders and prisms never do.
Pointy top: ⅓. Flat top: no ⅓.
⚠️People often think…
Arc length and sector area are found the same way.
Same fraction, different whole. Arc length is that fraction of the circumference (2πr), and comes out in units. Sector area is the same fraction of the area (πr²), and comes out in square units. The units tell you instantly which one you found.
A crust is a length. A slice is an area.
Worth knowing cold
Which formula, and what it gives you
Check the units on your answer. If they do not match the question, you used the wrong row.
2πrCircumference — around the edge. Units.
πr²Area of the circle — the whole face. Square units.
ArcThat fraction of 2πr. Still a length.
SectorThe same fraction of πr². An area.
Central ∠Equal to its arc. Vertex at the center.
Inscribed ∠Half its arc. Vertex out on the circle.
Prism / cylinderBase area × height. No fraction in front.
Cone / pyramid⅓ × base area × height. Use h, never the slant.
Read the verb, then the picture
1Watch one
A circle has a diameter of 12. Find its area.
The formula wants the radius, not the diameter.
r = 12 ÷ 2 = 6. Write it on the picture.
Square the radius first: 6² = 36.
Area = 36π square units.
2Do one with me
A cone has r = 5 and h = 12. How much does it hold?
A cylinder with that base and height would be πr²h =
A cone holds what fraction of that?
So the cone's volume is
💬One sentence, then you move on
How do you tell from a word problem whether it wants surface area or volume?
3Try one
An inscribed angle sits on an arc of 80°. How many degrees is the angle?
I want a hint first
Inscribed means the vertex is out on the circle, not at the center — and that one puts out half the angle.
💬Last one — then you're done here
Why do cones and pyramids get a ⅓ when cylinders and prisms do not?
Where this goes
Where this lives
Paint for a tank is surface area. Water in it is volume. Buying the wrong one is how a job comes up short — and it is the same mistake on the test.
What this feeds
That closes Geometry. Algebra II picks it up next, with polynomials that bend more than once.
Name one job where covering something and filling it would cost different amounts.
One card at a time — tap “Show me” to check yourself, then Next. Start at Foundation; when those feel easy, climb.
Helpful Hints
🍕 Arc length and sector area — the routine
Write the FRACTION of the circle first: central angle ÷ 360. A 60° piece is 60/360 = 1/6. Then decide what whole you want a fraction of: distance around → circumference 2πr; region inside → area πr². Multiply. Radius 6: arc = 1/6 × 12π = 2π ≈ 6.28; sector = 1/6 × 36π = 6π ≈ 18.85. Leave π for an exact answer; multiply by 3.14159 for a decimal. Put the units on: units for an arc, square units for a sector.
Fraction, whole, multiply, units.
🔑 The whole unit in one idea
Everything round is built from π and the radius. Around a circle: 2πr. Inside a circle: πr². A piece of either: multiply by the fraction of the turn. Stack a base to make a solid: V = Bh. Bring it to a point: ⅓Bh. Round it into a sphere: (4/3)πr³. Cover a solid instead of filling it: add up its faces — surface area. Scale everything by k and lengths go ×k, areas ×k², volumes ×k³. Every formula in this unit is one of those moves.
Around, inside, a piece, a stack, a point, a ball, a skin.
⚠️ Traps the test loves
Using the diameter as the radius. Diameter 12 → r = 6 → area 36π, not 144π. Write r = d ÷ 2 on the picture first.
Mixing up arc length (2πr, units) and sector area (πr², square units). Same fraction, different whole.
Treating an inscribed angle like a central angle. Central = the arc; inscribed = HALF the arc. An 80° arc gives a 40° inscribed angle.
Forgetting the ⅓ on cones and pyramids — or applying it to a cylinder. Cone r = 5, h = 12 is ⅓ × 300π = 100π.
Using the slant height for volume or the height for lateral area. Volume wants h; πrℓ wants the slant ℓ = √(r² + h²).
Answering the wrong question: paint for a tank is SURFACE AREA (78π), water in it is VOLUME (90π). Read “cover” vs “fill.”
📐 The formulas — the chart
Figure
Formula
Worked example
Circle — circumference
C = 2πr = πd
r = 6: 12π ≈ 37.70
Circle — area
A = πr²
r = 6: 36π ≈ 113.10
Arc length
(θ/360) × 2πr
60°, r = 6: 2π ≈ 6.28
Sector area
(θ/360) × πr²
60°, r = 6: 6π ≈ 18.85
Inscribed angle
½ × intercepted arc
arc 80° → 40°; semicircle → 90°
Prism / cylinder
V = Bh = πr²h
r = 3, h = 10: 90π ≈ 282.74
Pyramid / cone
V = ⅓Bh = ⅓πr²h
r = 5, h = 12: 100π ≈ 314.16
Sphere
V = (4/3)πr³ · S = 4πr²
r = 3: V = 36π ≈ 113.10, S = 36π ≈ 113.10
Cylinder — surface
S = 2πr² + 2πrh
r = 3, h = 10: 78π ≈ 245.04
Cone — surface
S = πr² + πrℓ, ℓ = √(r² + h²)
r = 5, h = 12: ℓ = 13, S = 90π ≈ 282.74
Scaling by k
length ×k · area ×k² · volume ×k³
k = 2: area ×4, volume ×8
🎯 How the test will ask
A circle with a radius or diameter — “find the circumference / area, in terms of π.” Halve a diameter first; keep π unless told to round.
A shaded sector or an arc with a central angle — set up the fraction θ/360, then choose 2πr or πr².
An angle with its vertex on the circle — halve the arc. Vertex at the center — equal to the arc. Two chords crossing inside — half the sum.
A tangent and a radius — a right angle, then Pythagoras. A chord and its distance from the center — half the chord, then Pythagoras.
“Which container holds more?” — compute BOTH volumes; never trust the shape.
A word problem about paint, wrapping, filling or fencing — decide surface area, volume or perimeter before touching a formula.
An equation (x − h)² + (y − k)² = r² — read the center with the signs flipped and take the square root for r.
✅ Before the test, can you…
Name radius, diameter, chord, secant, tangent, arc, central angle and inscribed angle on a diagram?
Find circumference and area from a radius OR a diameter, exactly and to two decimals?
Compute an arc length and a sector area from a central angle, and say which is which by its units?
Use the inscribed angle theorem, including the 90° in a semicircle?
Find the volume of a cylinder, cone, sphere and prism, and explain where the ⅓ comes from?
Tell surface area from volume in a word problem, and compute a cylinder's surface area?
Write the equation of a circle from its center and radius, and read them back from an equation?
Pick your level
Look back at anything you missed — the hint that appeared is exactly what to reread tonight.
How sure did you feel?
Workshop
Work like an engineer: name the parts of a circle, put a sector-area solution in order, argue which container really holds more, then sort real jobs into area, surface area and volume. Every activity checks itself, and hints are free.
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Your practice record — saved on this device
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