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PREPITUS
GeometryAdvanced

Circles

Work with circle equations, arcs, and sectors through one idea: every arc question is a fraction-of-the-circle question.

⏱ 8 min📍 Learn first: Right Triangles and Trigonometry
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Free KC Learning Toolkit

Learn Circles in the format that works for you

Start with a quick audio review, take the deeper podcast lesson, or move directly into the visual guide and knowledge check below.

13 free resources
00:00 / 01:37

DSAT circle theorems and algebraic equations

A short spoken refresher on this concept.

Deep Dive Lesson

A spoken lesson on this concept.

In production — not yet available

Concept Snapshot

Definition
Circle equation: (x − h)² + (y − k)² = r² with center (h, k). Arc length and sector area are proportional slices of 2πr and πr².
What the SAT tests
Reading center/radius from equations (sometimes after completing the square), arc length, sector area, and radian conversion.
Key rule
Slice fraction = central angle/360°. Arc = fraction × 2πr. Sector = fraction × πr². Radians: multiply degrees by π/180.
Most common mistake
Reading (x + 3)² as center x = +3 — the sign flips: h = −3.
Prerequisite
Right Triangles and Trigonometry

Visual Explanation

What to picture: Slice fraction = central angle/360°. Arc = fraction × 2πr. Sector = fraction × πr². Radians: multiply degrees by π/180.

A diagram for this concept is in production. Until it is ready, the worked example below shows the same relationship step by step.

Concept Mind Map

DefinitionCircle equation: (x − h)² + (y − k)² = r² with center (h, k). Arc length and sector area are proportional slices of 2πr and πr².
Key ruleSlice fraction = central angle/360°. Arc = fraction × 2πr. Sector = fraction × πr². Radians: multiply degrees by π/180.

Circles

Geometry

Equation ReadExtract center and radius, minding the sign flips.
Complete the SquareExpanded form x² + y² + 6x − 4y = 12 → standard form first.
Slice ProblemsArc length or sector area via the angle fraction.

How the SAT Tests It

Pattern

Equation Read

Extract center and radius, minding the sign flips.

Clue: "center of the circle"
Pattern

Complete the Square

Expanded form x² + y² + 6x − 4y = 12 → standard form first.

Clue: x² and x terms mixed
Pattern

Slice Problems

Arc length or sector area via the angle fraction.

Clue: "arc", "sector", a central angle

Worked Example

What are the center and radius of the circle x² + y² − 8x + 2y = 8?

The equation is expanded — complete the square in x and y to reach standard form.
x² − 8x → (x − 4)² − 16. Half of −8 is −4; square it.
y² + 2y → (y + 1)² − 1. Combine: (x − 4)² + (y + 1)² = 8 + 16 + 1 = 25.
Center (4, −1) — signs flip from the parentheses — and radius √25 = 5.

Try It Yourself

A circle has radius 9. What is the length of an arc intercepted by a 80° central angle?

Common Traps

What students do

Reading the center's signs straight off the equation.

Why it fails

Standard form subtracts the center coordinates — (x + 3)² means x − (−3), so h = −3.

Do this instead

Force each parenthesis into (x − h) shape before reading: (x + 3)² = (x − (−3))². The flip becomes mechanical.

Interactive Flashcards

Review deckCard 1 of 8

Four-Question Knowledge Check

4 quick questions drawn from this concept's own snapshot, patterns and traps.

Question 1 of 4. Which statement is the key rule for this concept?

Audio Lesson

Circles: equations and slices

Completing the square, sign flips, and fraction-of-circle problems.

The recording for this lesson is in production. Captions and a full transcript will ship with it.

What to notice

Slice fraction = central angle/360°. Arc = fraction × 2πr. Sector = fraction × πr². Radians: multiply degrees by π/180.

What the lesson covers

  • 0:00 Standard form
  • 1:30 Completing the square
  • 3:00 Arcs and sectors
  • 4:40 Practice

Study Guide and Transcript

One-page study guide

The snapshot, the key rule, the patterns and the traps, on one page.

Source-Grounded AI Tutor

Ask the Circles notebook

A tutor grounded in this concept's own sources — ask for a simpler explanation, another example, or why a particular answer is wrong. It is in development and will appear here once its sources pass academic review.

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