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36 free practice questions a day, shared across all your practice, each with a full explanation. No card needed.

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  • Reading and Writing and Math practice by domain, skill and difficulty
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  • English, Math, Reading and Science practice by category and skill
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Questions, answered

36 practice questions every day, each with a full explanation, shared across your SAT, ACT and AP practice. The SAT and ACT diagnostics, SAT full-length Test A and Challenge Test 1, ACT Practice Test 1, The Daily and all 1,023 SAT vocabulary words are free too. No card needed.

The digital SAT, the ACT and AP exams. Choose a test at the top of this page. The ACT side has English, Math, Reading and Science practice, lessons, a 20-question diagnostic and full-length practice tests. The AP side has practice by unit and topic and full-length practice exams. Every question is original.

ACT and AP practice use the same free daily questions as SAT practice. The 20-question ACT diagnostic and ACT Practice Test 1 are free too. Other ACT practice tests and the AP practice exams come with Pro.

Every day at midnight Central Time.

No. Peakscor is an independent practice tool, not affiliated with College Board or ACT Education Corp. Scores shown are estimates.

Digital SAT® practice

Reach your peak.

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Start here: 20 questions, about 28 minutes. You see your level in every domain and which lessons to take first. Free, no card. You also get 36 practice questions a day: 9 easy, 9 medium, 9 hard and 9 challenging. See what’s free
The DailySTEP 1
  • Three hearts. A miss costs one. Free: one run every 24 hours.
  • Streaks pay. ×2 from three in a row, ×3 from six.
  • Your best run. Most right before three misses.

Full-length tests

Test day, rehearsed.

98 questions across four modules, in the same order and on the same timing as the digital SAT, with a highlighter, a question map and a timer you can see.

A Peakscor full-length test in progress: Reading and Writing, Test A, Module 1 of 4, question 7 of 27, with the passage, four answer choices, a highlighter, a 24:18 timer and Flag, Back and Next buttons.

How it works

A quick run every day. One skill at a time.

  1. 1

    Play the Daily.

    A quick run with three hearts. String right answers together and the questions get harder and the points multiply.

  2. 2

    Miss one, learn why.

    Every question is tagged by domain and skill, and every answer comes with an explanation.

  3. 3

    Drill that skill.

    Practice any single skill at any difficulty, or just the questions you got wrong.

  4. 4

    Beat your best.

    Come back tomorrow and climb. Progress shows your accuracy and your strongest skills.

Built on the digital SAT blueprint.

Every question is tagged by section, domain, skill and difficulty, so you can practice exactly what you need. Full tests run like test day: two Reading & Writing modules, a break, then two Math modules.

Explanations

Every answer, explained.

Not just the right letter: how to get there, and why the tempting answer is wrong.

MathProblem-Solving and Data AnalysisHard

A store raises the price of a jacket by 20%. Later, it lowers the new price by 20%. The final price is what percent of the original price?

A80%
B96%Correct
C100%The trap
D104%

Check with $100: $100then$120then$96

  1. Turn each change into a multiplier.

    Up 20% means × 1.20. Down 20% means × 0.80.

  2. Apply them in order.

    1.20 × 0.80 = 0.96

  3. Read the answer.

    The final price is 96% of the original. That’s B.

  4. Why not 100%?

    The 20% cut is taken from the higher price, so it removes more than the 20% increase added.

Test day

What test day looks like.

Every Peakscor full test follows this order. In the standard tests, the second module of each section adapts to how you did on the first.

  1. Reading & Writing · Module 1

    27 questions32 min

  2. Reading & Writing · Module 2

    27 questions32 min

    Adapts to how you did on Module 1.
  3. Break

    10 min

  4. Math · Module 1

    22 questions35 min

  5. Math · Module 2

    22 questions35 min

    Adapts to how you did on Module 1.

98 questions · 2 hr 14 min of testing, plus the break

Peakscor by the numbers

  • 36free practice questions every day
  • 98questions in every full-length test
  • 1,023SAT vocabulary words
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Free

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  • The Daily, every day
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  • The 20-question Learn diagnostic
  • PEAK Survivor’s first map, plus the opening levels of Peak Hoover and Peak Defense
Start free daily practice

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  • Everything in Pro
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FAQ

Questions, answered.

The Daily, plus 36 practice questions every day: 9 easy, 9 medium, 9 hard and 9 challenging, each with a full explanation. Full-length Test A, Challenge Test 1 and all 1,023 vocabulary words are free too, as are the 20-question diagnostic and PEAK Survivor’s first map. No card needed.

Pro is $20 a month or $200 a year: unlimited practice with no daily limit, plus every full-length test and Challenge test, and every game map and level. Peak is $29 a month or $280 a year: everything in Pro, plus the full Learn course and Peak Plan. Cancel anytime from your account.

98 questions across four modules, in the same order as the digital SAT: Reading & Writing Module 1 and Module 2 (27 questions each), a 10-minute break, then Math Module 1 and Module 2 (22 questions each). On official timing that’s 2 hr 14 min of testing. You get estimated section scores and a total at the end.

Yes. In the standard full-length tests, Module 2 of each section is the harder version if you get 60% or more of Module 1 right, and the easier version if you don’t. Challenge tests are built from harder material in every module.

Official timing, extended 1.5× time, or untimed, on every test, including the free ones. Extended time allows 48 minutes for each Reading & Writing module and about 52 for each Math module. Untimed removes the clock so you can focus on accuracy.

A quick run with three hearts. Each right answer builds your streak: the questions get harder as it grows, and your points multiply (×2 from three in a row, ×3 from six). A miss costs a heart and drops you back to easy. Best is your longest streak.

Every day at midnight Central Time. You get a fresh 9 easy, 9 medium, 9 hard and 9 challenging.

No. Peakscor is an independent SAT practice tool, not affiliated with the College Board. Scores shown are estimates.

Your next point starts today.

Free, no card.

Geometry and Trigonometry · Lesson 4

Circles and circle equations

A circle with radius rr has circumference 2πr2\pi r and area πr2\pi r^2. A full circle measures 360∘360^\circ.

Arcs and sectors are fractions of the circle. With a central angle of θ\theta degrees, the fraction is θ360\frac{\theta}{360}: arc length =θ360⋅2πr= \frac{\theta}{360} \cdot 2\pi r and sector area =θ360⋅πr2= \frac{\theta}{360} \cdot \pi r^2. If the angle is in radians, arc length is simply rθr\theta.

Angle facts. An inscribed angle (vertex on the circle) is half the central angle that cuts off the same arc. An angle inscribed in a semicircle is a right angle. A tangent line is perpendicular to the radius drawn to the point of tangency.

The circle equation. In the standard (x, y) coordinate plane, the circle with center (h,k)(h, k) and radius rr is (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2. The signs inside the parentheses are the opposite of the center's coordinates: (x+2)2(x + 2)^2 means h=−2h = -2. The right side is r2r^2, so take its square root to get the radius.

If the equation comes expanded, such as x2+y2−6x+10y+18=0x^2 + y^2 - 6x + 10y + 18 = 0, complete the square: group the x terms and the y terms, add the square of half of each linear coefficient to both sides, and rewrite each group as a squared binomial.

Other conic sections appear less often. A parabola y=a(x−h)2+ky = a(x - h)^2 + k has vertex (h,k)(h, k). An ellipse (x−h)2a2+(y−k)2b2=1\frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 has center (h,k)(h, k), reaches aa units left and right of the center, and bb units up and down. A hyperbola has a minus sign between its squared terms.

Worked example Easy

A circle with a shaded 60° sector660°O
A circle with a shaded 60° sector A circle with center O. A radius of the circle is labelled 6. Two radii form a central angle marked 60°, and the sector between them, a slice of the circle, is shaded.

In the circle shown, O is the center, the radius is 6, and the shaded sector has a central angle of 60∘60^\circ. What is the area of the shaded sector?

  1. π\pi
  2. 2π2\pi
  3. 6π6\piAnswer
  4. 12π12\pi

How to solve it

  1. The sector is 60360=16\frac{60}{360} = \frac{1}{6} of the circle.
  2. The whole circle's area is πr2=π(6)2=36π\pi r^2 = \pi (6)^2 = 36\pi.
  3. Sector area =16⋅36π=6π= \frac{1}{6} \cdot 36\pi = 6\pi.

Why each choice is right or wrong

  • A. Incorrect. π\pi is 16⋅π⋅6\frac{1}{6} \cdot \pi \cdot 6, which forgets to square the radius. The full circle's area is π(6)2=36π\pi (6)^2 = 36\pi.
  • B. Incorrect. 2π2\pi is the length of the arc, 16⋅2π(6)\frac{1}{6} \cdot 2\pi (6). The question asks for the area of the sector, which uses πr2\pi r^2.
  • C. Correct. A 60∘60^\circ sector is 16\frac{1}{6} of the circle, and 16⋅36π=6π\frac{1}{6} \cdot 36\pi = 6\pi.
  • D. Incorrect. 12π12\pi divides by 180 instead of 360: 60180⋅36π\frac{60}{180} \cdot 36\pi. A full circle measures 360∘360^\circ.

Worked example Medium

In the standard (x, y) coordinate plane, what are the center and radius of the circle with equation x2+y2−6x+10y+18=0x^2 + y^2 - 6x + 10y + 18 = 0?

  1. Center (3,−5)(3, -5), radius 4Answer
  2. Center (3,−5)(3, -5), radius 16
  3. Center (−3,5)(-3, 5), radius 4
  4. Center (−3,5)(-3, 5), radius 16

How to solve it

  1. Group the x terms and the y terms and move the constant: (x2−6x)+(y2+10y)=−18(x^2 - 6x) + (y^2 + 10y) = -18.
  2. Complete each square: half of −6-6 is −3-3, and (−3)2=9(-3)^2 = 9; half of 10 is 5, and 52=255^2 = 25. Add 9 and 25 to both sides.
  3. (x−3)2+(y+5)2=−18+9+25=16(x - 3)^2 + (y + 5)^2 = -18 + 9 + 25 = 16.
  4. Compare with (x−h)2+(y−k)2=r2(x - h)^2 + (y - k)^2 = r^2: the center is (3,−5)(3, -5) and r2=16r^2 = 16, so r=4r = 4.

Why each choice is right or wrong

  • A. Correct. Completing the square gives (x−3)2+(y+5)2=16(x - 3)^2 + (y + 5)^2 = 16, so the center is (3,−5)(3, -5) and the radius is 16=4\sqrt{16} = 4.
  • B. Incorrect. A radius of 16 uses r2r^2 as the radius. The center (3,−5)(3, -5) is right, but the radius is 16=4\sqrt{16} = 4.
  • C. Incorrect. The center (−3,5)(-3, 5) copies the signs of −6x-6x and +10y+10y the wrong way. In (x−3)2+(y+5)2(x - 3)^2 + (y + 5)^2, the center is (3,−5)(3, -5).
  • D. Incorrect. Center (−3,5)(-3, 5) with radius 16 makes both slips: it flips the signs of the center and uses r2=16r^2 = 16 as the radius.

Trap Arc length or sector area

Arc length is a fraction of the circumference, 2πr2\pi r; sector area is a fraction of the area, πr2\pi r^2. Both use the same fraction θ360\frac{\theta}{360}, so a choice built from the wrong one looks just as clean as the key. Read whether the question asks for a length or an area.

Trap Signs and squares in the circle equation

(x−3)2+(y+5)2=16(x - 3)^2 + (y + 5)^2 = 16 has center (3,−5)(3, -5), not (−3,5)(-3, 5), and radius 4, not 16. Expect choices that flip the signs of the center, that report r2r^2 as the radius, or that do both.

Calculator When a calculator is faster

For a circle equation, Desmos can be faster than completing the square. Type the equation exactly as given, such as x^2+y^2-6x+10y+18=0, and Desmos graphs the circle. Read the center and radius from the grid (or add a point such as (3,-5) to check it) and check them against the choices.

For arcs, sectors, and inscribed angles, the work is choosing the right fraction of the circle, so a calculator adds little. On paper, a graphing calculator may need the equation solved for y first, which makes completing the square the quicker route there.

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Each test mirrors the digital SAT®: 98 questions across four modules, 2 hr 14 min on official timing.

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Same 98-question format, built from harder material: a hard Module 1, then a Module 2 made mostly of the hardest questions.

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