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98 questions across four modules, in the digital SAT’s order and on its timing, 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.

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    Play the Daily.

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

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    Miss one, learn why.

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

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    Beat your best.

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

Built around the real 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.

Free daily practice

12 easy, 12 medium and 12 hard questions free every day, each with a full explanation. Filter by section, domain, skill and difficulty. Resets at midnight.

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Take any test on official timing, extended 1.5× time, or untimed to focus purely on accuracy.

Adaptive full-length tests

Four modules in official order. The second module of each section adapts to how you did on the first.

Vocabulary

1,023 essential SAT words. Browse the list, flip flashcards, or quiz yourself.

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Games

PEAK Survivor, Hole and Peak Defense: three games powered by SAT practice questions.

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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

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FAQ

Questions, answered.

The Daily, plus 36 practice questions every day: 12 easy, 12 medium and 12 hard, each with a full explanation. Full-length Test A, Challenge Test 1 and all 1,023 vocabulary words are free too. 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. 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 digital SAT’s order: 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 use the hardest material throughout.

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. You get a fresh 12 easy, 12 medium and 12 hard.

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

Your next point starts today.

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Right triangles and trigonometry · Lesson 2

Special right triangles: 45-45-90 and 30-60-90

Two right triangles have side lengths that always follow a fixed pattern, and both patterns are on the reference sheet. Knowing them lets you skip the Pythagorean theorem and gives exact answers with square roots.

45-45-90: the two legs are equal, and the hypotenuse is a leg times 2\sqrt{2}: ss, ss, s2s\sqrt{2}. Legs of 8 give a hypotenuse of 828\sqrt{2}. Going from the hypotenuse back to a leg, divide by 2\sqrt{2}. A square's diagonal cuts it into two 45-45-90 triangles, so a square with side ss has a diagonal of s2s\sqrt{2}.

30-60-90: call the shortest side xx. It is across from the 30° angle. The longer leg, across from the 60° angle, is x3x\sqrt{3}, and the hypotenuse is 2x2x. A short leg of 2 gives a long leg of 232\sqrt{3} and a hypotenuse of 4. From any one side, find xx first, then the others.

To divide by a square root, multiply the top and bottom by that root: a3=a33\frac{a}{\sqrt{3}} = \frac{a\sqrt{3}}{3}. Answer choices are usually written this way, with no root on the bottom.

An equilateral triangle cut by its height splits into two 30-60-90 triangles: half of a side is xx, the height is x3x\sqrt{3}, and a full side is 2x2x.

The 45-45-90 triangle45°45°sss√2
The 45-45-90 triangle An isosceles right triangle with the right angle at the bottom left, marked with a square. Each of the other two angles is marked 45°. Both legs are labelled s and the hypotenuse is labelled s√2.
The 30-60-90 triangle30°60°xx√32x
The 30-60-90 triangle A right triangle with the right angle at the bottom left, marked with a square. The angle at the right end of the horizontal leg is marked 30° and the top angle is marked 60°. The vertical leg, across from the 30° angle, is labelled x; the horizontal leg, across from the 60° angle, is labelled x√3; the hypotenuse is labelled 2x.

Worked example Easy

An isosceles right triangle45°x12
An isosceles right triangle A right triangle with the right angle at the bottom left, marked with a square. The two legs each have one tick mark, showing they are equal. The angle at the right end of the horizontal leg is marked 45°. The horizontal leg is labelled x and the hypotenuse is labelled 12.

In the right triangle shown, the two legs have equal length. What is the value of xx?

  1. 66
  2. 434\sqrt{3}
  3. 626\sqrt{2}Answer
  4. 12212\sqrt{2}

How to solve it

  1. Equal legs and a right angle make this a 45-45-90 triangle, so the hypotenuse is a leg times 2\sqrt{2}: x2=12x\sqrt{2} = 12.
  2. Divide by 2\sqrt{2}: x=122x = \frac{12}{\sqrt{2}}.
  3. Multiply the top and bottom by 2\sqrt{2}: x=1222=62x = \frac{12\sqrt{2}}{2} = 6\sqrt{2}.
  4. Check: 62⋅2=6⋅2=126\sqrt{2} \cdot \sqrt{2} = 6 \cdot 2 = 12.

Why each choice is right or wrong

  • A. Incorrect. This halves the hypotenuse, which is the rule for the short leg of a 30-60-90 triangle. With legs of 6, this triangle's hypotenuse would be 626\sqrt{2}, not 12.
  • B. Incorrect. This divides 12 by 3\sqrt{3} instead of 2\sqrt{2}: 123=1233=43\frac{12}{\sqrt{3}} = \frac{12\sqrt{3}}{3} = 4\sqrt{3}. The factor 3\sqrt{3} belongs to the 30-60-90 pattern.
  • C. Correct. x2=12x\sqrt{2} = 12, so x=122=62x = \frac{12}{\sqrt{2}} = 6\sqrt{2}.
  • D. Incorrect. 12212\sqrt{2} multiplies the hypotenuse by 2\sqrt{2}, as if 12 were a leg. The hypotenuse is the longest side, so each leg is 1212 divided by 2\sqrt{2}, not multiplied by it.

Worked example Medium

A right triangle with a 30° angle30°9h
A right triangle with a 30° angle A right triangle with the right angle at the bottom left, marked with a square. The angle at the right end of the horizontal leg is marked 30°. The horizontal leg is labelled 9 and the hypotenuse is labelled h. The vertical leg is not labelled.

In the right triangle shown, what is the value of hh?

  1. 333\sqrt{3}
  2. 932\frac{9\sqrt{3}}{2}
  3. 636\sqrt{3}Answer
  4. 929\sqrt{2}

How to solve it

  1. The angles are 30°, 60° and 90°. The side of length 9 touches the 30° angle, so it is across from the 60° angle. That makes it the long leg.
  2. The long leg is the short leg times 3\sqrt{3}, so the short leg is 93=933=33\frac{9}{\sqrt{3}} = \frac{9\sqrt{3}}{3} = 3\sqrt{3}.
  3. The hypotenuse is twice the short leg: h=2⋅33=63h = 2 \cdot 3\sqrt{3} = 6\sqrt{3}.
  4. Check: 63≈10.396\sqrt{3} \approx 10.39, which is longer than 9, as a hypotenuse must be.

Why each choice is right or wrong

  • A. Incorrect. 333\sqrt{3} is the short leg, the side across from the 30° angle. The hypotenuse is twice that.
  • B. Incorrect. 932\frac{9\sqrt{3}}{2} multiplies 9 by 32\frac{\sqrt{3}}{2} instead of dividing. The long leg is 32\frac{\sqrt{3}}{2} of the hypotenuse, so h=9÷32=63h = 9 \div \frac{\sqrt{3}}{2} = 6\sqrt{3}. Also, 932≈7.79\frac{9\sqrt{3}}{2} \approx 7.79 is shorter than the leg of length 9, and a hypotenuse can't be.
  • C. Correct. The long leg is 9, so the short leg is 93=33\frac{9}{\sqrt{3}} = 3\sqrt{3} and the hypotenuse is 2⋅33=632 \cdot 3\sqrt{3} = 6\sqrt{3}.
  • D. Incorrect. This uses the 45-45-90 pattern, multiplying a leg by 2\sqrt{2}. This triangle has a 30° angle, so its legs are not equal.

Trap Putting √3 on the wrong side

In a 30-60-90 triangle, the short leg is across from the 30° angle, and the side across from the 60° angle is the short leg times 3\sqrt{3}. Match each side to the angle across from it, not the angle it touches. Since 3≈1.73\sqrt{3} \approx 1.73, the order from shortest to longest is xx, x3x\sqrt{3}, 2x2x.

Trap Using the wrong pattern

Halving the hypotenuse works only in a 30-60-90 triangle, and only for the short leg. Check the angles first: equal legs or a 45° angle means 45-45-90 and the factor 2\sqrt{2}; a 30° or 60° angle means 30-60-90 and the factors 3\sqrt{3} and 2.

Trap Multiplying when you should divide

From a leg to the hypotenuse of a 45-45-90 triangle, multiply by 2\sqrt{2}. From the hypotenuse to a leg, divide. If a leg comes out longer than the hypotenuse, you multiplied.

Desmos When Desmos isn’t faster

These answers are often exact, with 2\sqrt{2} or 3\sqrt{3} in them, and the patterns give them in one or two steps. That's faster than any calculator.

Desmos helps you match a radical to a decimal choice or check a size. Type 6sqrt(3) and Desmos shows about 10.392, longer than the leg of 9, as a hypotenuse should be.

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Full-length tests

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

A little harder than the real SAT. These tests can run slightly tougher than test day, which makes them good practice: if you can handle these, the real one should feel easier. No question appears in more than one test, Challenge Tests included.

Challenge Tests Hardest of the hardest

Same 98-question format, but every question is drawn from the toughest SAT material: Challenging-tier math throughout and the hardest reading.

Vocab

Study the classic SAT vocabulary list. Browse, flip flashcards, or quiz yourself.

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