How To Solve “Shaded Region” Questions

Relax – no one else knows the formula for that oddly shaped shaded region either!  Shaded region questions nearly always involve subtraction.  Usually, all you have to do is subtract the white area from the area of the whole figure are to get the gray.

Suppose you’re told that the length of a side of the larger square is 25, and the smaller square has sides of length 10.  What is the area of the shaded region?

Simply use these 4 steps:  English, Formulas, Substitute, Solve.

Step 1 (English)  large square – small square = shaded region

Step 2 (Formulas)  S2 – s2 = answer

Step 3 (Substitute) 252 – 102 = answer

Step 4 (Solve) 625 – 100 = 525

🙂

If you use these steps, most of these problems will be fairly easy.  Need to subtract a semicircle?  No problem; just use (πr2)/2.

Let’s do a trickier one:

Okay, the large and medium squares still have sides of 25 and 10, respectively.  Suppose the small square has sides of length 3.  What is the area of the unshaded region?

Step 1 (English)  large square – medium square + small square = unshaded region

Step 2 (Formulas)  l2 – m2 + s2 = answer

Step 3 (Substitute) 252 – 102 + 32 = answer

Step 4 (Solve) 625 – 100 + 9 = 534

If you didn’t see Step 1 right away, that’s okay.  It’s easier if you break things down into simpler steps anyhow.  Just find the area of the shaded region first (medium square  – small square), and then subtract that from the large square (whole – gray = white).

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8 comments on “How To Solve “Shaded Region” Questions

  1. Katelyn deuter says:

    What if the square is the same length on both sides? Do you still use the same formula?

  2. Hi Katelyn,

    I’m not sure what you’re asking. By definition, all four sides of a square have the same length.

    • Jules says:

      HELP! How do I solve the following:
      A square has sides ABCD.
      There is a triangle that shares the base of the square, which is AD.
      The height of the triangle is 6cm.
      The area of the triangle is 21 cm squared.
      What is the area of the shaded region?

  3. Hi Jules,

    First of all, if this is a multiple choice question, I would draw the figure (if it isn’t already shown), and just estimate the area.

    Otherwise, use the four steps from this article.

    (note – s^2 means s squared)

    1) (English) square – triangle = shaded region

    2) (Formulas) s^2 – 1/2bh = shaded region

    3) (Substitute) First you need to find the base of the triangle. 1/2bh = 21, and h = 6, so 3h = 21, and h = 7 (this is b and s, since they’re shared). So 7^2 – 21 = shaded region.

    4) (Solve) 49 – 21 = 28.

  4. Four circles of radius 10 are arranged as shown. The centers are the vertices of a square. What is the area of the shaded region?

  5. Hi Gabrielle,

    “As shown”? I can only guess that the four circles are tangent, and that the shaded region is the portion of the square that’s not part of the circles. So just solve square – 4 quarter circles, which is the same as square – one circle. Note that the side of the square is 20.

  6. Kevin Mcalister says:

    HELP NEEDED
    A trapezium has sides ABCD with AD ||BC.POD is a diameter of the semi circle with centre O ,Given that AB = 21 cm BC=34cm AP= 5 cm PD=18cm calculate the area of the shaded region take pi = 3.142 and give your answer correct to 2 decimal places

  7. Hi Kevin,

    Sorry, but without a diagram, there is no way to solve this, since I can’t see the shaded region.

    Note also that the word “trapezium” will not appear on the SAT.

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