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Composite figures and shaded areas
An L-shaped room, a picture frame, a square with a circle cut out: the test builds odd shapes from simple ones. There are only two moves, add the pieces or subtract the hole, and one rule for perimeter.
Add the pieces, or subtract the hole
A composite figure is made of shapes you already know. Either cut it into pieces and add their areas, or see it as a big shape with a piece missing and subtract. Both give the same answer; pick the one with fewer steps.
Perimeter is only the outside edge
When you cut a shape into pieces, the cut lines are inside. They count for area but never for perimeter: walk around the outside and add only the edges you walk along.
How to solve them
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1
Split it into rectangles and add
Draw one line to cut the shape into rectangles. Work out any side that is not labeled from the sides across from it.
Next step: Subtract the holeExampleThe same L-shape, cut with a vertical line
- Left piece: 6 m wide (10 − 4) and the full 8 m tall: 6 × 8 = 48.
- Right piece: 4 m wide and 5 m tall (8 − 3): 4 × 5 = 20.
- 48 + 20 = 68, the same as 80 − 12.
Answer 68 m²
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2
Subtract the hole: frames, borders and paths
For a frame, a border or a path around something, take the area of the whole outer shape and subtract the area of what is inside.
Next step: Shaded regionsExampleA 14 × 10 photo sits in a frame that is 20 × 16 overall. What is the area of the frame itself?
- Outside: 20 × 16 = 320.
- Photo: 14 × 10 = 140.
- Frame: 320 − 140 = 180.
Answer 180 square units
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3
Shaded regions: outer area minus inner area
When a circle sits inside a square (touching all four sides), the circle’s diameter equals the side of the square. The shaded corners are the square minus the circle.
Next step: PerimeterExampleA circle fits exactly inside a square of side 10. What is the area of the four corners outside the circle? (π ≈ 3.14)
- Square: 10 × 10 = 100.
- The circle’s radius is 10 ÷ 2 = 5: 3.14 × 25 = 78.5.
- Corners: 100 − 78.5 = 21.5.
Answer 21.5 square units
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4
The perimeter of an L-shape
Cutting a rectangle out of a corner does not change the perimeter: the two edges you remove are replaced by two edges of the same lengths. Check by walking the edge.
Next step: Half circlesExampleThe perimeter of the 10 × 8 L-shape with a 4 × 3 corner removed
- Walk around from the top-left corner: top 6, step down 3, step across 4, right side 5, bottom 10, left side 8.
- 6 + 3 + 4 + 5 + 10 + 8 = 36.
- The same as the full rectangle: 2 × 10 + 2 × 8 = 36.
Answer 36 m
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5
Rectangles with half circles
A half circle on the end of a rectangle adds half of a circle’s area. Its diameter is the side it sits on, so its radius is half of that side.
Next step: Missing sidesExampleA 12 × 6 rectangle has a half circle on one 6-unit end. What is the total area? (π ≈ 3.14)
- Rectangle: 12 × 6 = 72.
- Half circle: radius 3, so ½ × 3.14 × 9 = 14.13.
- 72 + 14.13 = 86.13.
Answer 86.13 square units
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6
Missing sides come from the opposite sides
In shapes made of rectangles, the total of the horizontal edges on the top equals the total on the bottom, and the same for the vertical edges on the left and right.
HorizontalBottom 10, top pieces 6 and ?
10 − 6 = 4
VerticalLeft side 8, right pieces 3 and ?
8 − 3 = 5
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Counting the cut line in the perimeter
An L-shape cut into two rectangles:
Wrongadd both rectangles’ perimeters
Rightadd only the outside edges
Adding the hole instead of subtracting
A 20 × 16 frame around a 14 × 10 photo:
Wrong320 + 140 = 460
Right320 − 140 = 180
A whole circle instead of half
Half circle of radius 3:
Wrong3.14 × 9 = 28.26
Right½ × 28.26 = 14.13
The diameter as the radius
A half circle sitting on a 6-unit side:
Wrongradius 6
Rightradius 3
5 questions at test level
Pick an answer, then read why. No timer, no account.
An L-shaped room is a 12 ft × 10 ft rectangle with a 4 ft × 3 ft corner removed. What is its area?
Show the answer
B. Whole rectangle: 12 × 10 = 120. Missing corner: 4 × 3 = 12. Area: 120 − 12 = 108. 120 forgets the missing corner, and 132 adds it. The area is 108 square feet.
A 9-inch by 12-inch photo sits in a frame that is 13 inches by 16 inches overall. What is the area of the frame itself, not counting the photo?
Show the answer
B. Outside: 13 × 16 = 208. Photo: 9 × 12 = 108. Frame: 208 − 108 = 100. 208 is the whole thing and 108 is the photo. The frame is 100 square inches.
A circle fits exactly inside a square with sides of 8. What is the area inside the square but outside the circle? (use π ≈ 3.14)
Show the answer
A. Square: 8 × 8 = 64. The circle’s radius is 4: 3.14 × 16 = 50.24. Outside the circle: 64 − 50.24 = 13.76. 50.24 is the circle itself. The answer is 13.76.
An L-shaped room is a 12 ft × 10 ft rectangle with a 4 ft × 3 ft corner cut away. What is its perimeter?
Show the answer
C. Cutting out a corner replaces two edges with two edges of the same lengths, so the perimeter equals the full rectangle’s: 2 × 12 + 2 × 10 = 44. 51 counts the 4 and the 3 twice, and 37 leaves them out. The perimeter is 44 feet.
A rectangle 10 m by 4 m has a half circle attached to one of its 4 m sides. What is the total area? (use π ≈ 3.14)
Show the answer
A. Rectangle: 10 × 4 = 40. The half circle has radius 2: ½ × 3.14 × 4 = 6.28. Total: 40 + 6.28 = 46.28. 52.56 adds a whole circle, and 65.12 uses 4 as the radius. The total is 46.28 m².
9 flashcards
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