Cut the picture into shapes you have formulas for, then add — or subtract.
What is the area of the shaded region?
A circular hole of diameter 10 inches is cut from a
rectangular sheet of metal that is 16 inches long and
12 inches wide. What is the area, in square inches, of the
remaining sheet?
Sheet minus hole. The rectangle is 16 × 12 = 192, and the hole has radius 10 ÷ 2 = 5, so its area is π(5)² = 25π. The answer is 192 − 25π ≈ 113.5 square inches. A subtracts the wrong way round and is negative, since 25π ≈ 78.5. B uses πr instead of πr², which is not an area at all. D squares the diameter instead of the radius — the commonest slip in the whole topic, because the stem gives you the diameter on purpose.
1
Five formulas cover everything the SAT draws.
The rectangle, the triangle and the circle are on the reference sheet; the parallelogram and the trapezoid are not there at all, and no perimeter formula is. Learn all five — looking anything up costs time you do not have.
Shape
Area
Around the edge
rectangle
A = bh
P = 2(b + h)
triangle
A = ½bh
add the three sides
parallelogram
A = bh
add the four sides
trapezoid
A = ½(a + b)h
add the four sides
circle
A = πr²
C = 2πr = πd
trapezoid: A =
a + b2
× h
The trapezoid is a rectangle whose width changes steadily, so the average of the two
parallel sides does the job. The two slanted sides never enter the area formula.
h means the perpendicular height
Not the slanted side. A parallelogram with base 20 and slant 10 whose vertical
height is 8 has area 20 × 8 = 160, not 200. For a circle, the other trap is the
same one twice: A = πr² and C = 2πr both want the radius, so halve any
diameter the stem hands you before you square anything.
2
A composite area is the pieces added together.
A window is a rectangle 10 feet wide and 8 feet tall with a semicircle on top. What is its area?
rectangle: 10 × 8 = 80
square feet
radius = 10 ÷ 2 = 5
the semicircle sits on the 10-foot side, so that side is its diameter
semicircle: ½π(5)² = 12.5π
half a circle, so half the area
total = 80 + 12.5π
≈ 119.3 square feet
80 + 12.5π ≈ 119.3 square feet
Draw the cut line yourself. Once the picture is one rectangle and one half-circle,
there is no “composite figure” left — only two formulas you already know.
3
A shaded region is almost always big shape minus small shape.
The opening problem: a 16-by-12 sheet with a circular hole of diameter 10 cut out of it.
sheet: 16 × 12 = 192
the whole rectangle, hole included
r = 10 ÷ 2 = 5
the stem gave the diameter
hole: π(5)² = 25π
not π(10)² — that would be four times too big
shaded = 192 − 25π
≈ 113.5 square inches
192 − 25π ≈ 113.5 square inches
Subtract in the right order
Outer minus inner. Writing 25π − 192 gives −113.5, and no region has a negative
area — which makes it the fastest choice to cross out. If the parts you have added
come to more than the whole, you have double-counted an overlap somewhere.
4
Leave π in the answer.
The choices are usually written in terms of π, so decimalising early only makes matching harder.
exact
192 − 25π
matches a choice on sight
rounded
113.5
only when the stem says “closest to”
Treat π as a unit you carry along, like a variable: 25π + 12.5π = 37.5π, and
192 − 25π cannot be simplified further because 192 is not a multiple of π. Round only
at the very end, and only if the question asks you to.
Desmos does earn its place here. Type 192 − 25π and it returns 113.4601…,
so you can check an exact answer against a “closest to” choice in one line — but do
the algebra in exact form first and let the calculator confirm it.
5
Perimeter is a length, and internal edges are not on it.
Two rectangles joined along a shared edge. Walk around the outside: the join is never part of the walk.
the combined shape is 17 by 6
12 + 5 across
P = 2(17 + 6) = 46
units, not square units
the two perimeters separately: 36 + 22 = 58
12 too many
58 − 2(6) = 46 ✓
the shared edge was counted twice, once from each side
Areas do add straight across — 72 + 30 = 102 square units — because no area is
shared. Only the boundary changes when you glue two shapes together.
Check the units the question wants
Square feet means area; feet means perimeter. The two answers are both in the
choices, and the word “perimeter” in a stem full of areas is easy to skim past.
6
Cutting a corner out changes the area but not the perimeter.
A 16-by-9 rectangle with a 4-by-4 square removed from one corner.
area = 16 × 9 − 4 × 4 = 144 − 16 = 128
square units — the area really does drop
the cut removes 4 from the top and 4 from the right side
− 8 from the boundary
and adds the two new edges, 4 and 4
+ 8 back, because they are now on the outside
P = 2(16 + 9) = 50
exactly the original rectangle's perimeter
area 128 square units · perimeter 50 units
Walk the outline and check: 16 + 5 + 4 + 4 + 12 + 9 = 50. The two new edges count
because the walk goes along them — unlike the shared edge in the last step, which the
walk never touches. That is the whole distinction.
7
A curved edge contributes its arc, not its whole circle.
Back to the window: rectangle 10 by 8 with a semicircle of radius 5 on top. What is the perimeter?
bottom + two sides = 10 + 8 + 8 = 26
the straight part of the walk
whole circle: C = 2π(5) = 10π
if it were a full circle
semicircle arc = ½(10π) = 5π
half the circle, half the circumference
P = 26 + 5π
≈ 41.7 feet
26 + 5π ≈ 41.7 feet
The top of the rectangle is gone
That 10-foot edge is the semicircle's diameter, and it sits inside the window, so
it is not on the perimeter — adding it gives 36 + 5π, a distractor built for
exactly this slip. A quarter circle works the same way: its arc is ¼ of 2πr, which
is πr2.
✓
Check: is the size believable?
the hole25π ≈ 78.5 of 192about 41% of the sheet ✓
shaded left192 − 78.5 = 113.5positive and smaller than 192 ✓
unitssquare inches vs inchesarea squared, perimeter not ✓
Remember the flow
Cut it into known shapes · Add or subtract · Keep π
Halve the diameter first, and never walk an internal edge.
Now you try
Practice problem
What is the perimeter?
A rectangular garden measures 16 feet by
9 feet. A square section measuring
4 feet on each side is removed from one corner of the
garden. What is the perimeter, in feet, of the remaining region?
Walking the outline gives 16 + 5 + 4 + 4 + 12 + 9 = 50 feet — the same as the original rectangle's 2(16 + 9) = 50, because the cut takes 4 + 4 off the outside and puts 4 + 4 back as the two new edges. The area drops from 144 to 128 square feet; the perimeter does not move. B subtracts the 8 that was removed and forgets to add the new edges. D adds the new edges without removing the old ones. A does both subtractions, 50 − 8 − 8.
Hint: do not reach for a formula. Trace the boundary with your finger, writing down
each edge as you pass it, and stop when you are back where you started. The answer
that “feels” right here is usually the one that forgot half of the notch.
