Shaded Regions Blueprint Worksheet
GEOMETRY STUDIO Module 8.4 • Area Decomposition
SHADED REGIONS BLUEPRINT
Calculate composite areas involving circles, sectors, and geometric cutouts.
Name:
Date:
Period:
Score: _____ / 70
FORMULAS:
\(A_{\text{circle}} = \pi r^2\)
\(C = 2\pi r\)
\(A_{\text{sector}} = \frac{\theta}{360^\circ}\pi r^2\)
\(A_{\text{triangle}} = \frac{1}{2}bh\)
Use \(\pi \approx 3.1416\)
01
Square with Inscribed Circle Residuals
10 Points
r = 7 cm side s = 14 cm
Circle is tangent to all 4 sides
A square metal plate has a side length of \(14\text{ cm}\). A circular hole of radius \(r = 7\text{ cm}\) is punched through the center, touching all four edges. Calculate the total area of the four shaded corner remnants.
Answer:
02
Concentric Annular Scoring Ring
10 Points
r = 5 m R = 9 m
Concentric circles sharing center O
An outdoor plaza target consists of two concentric circles. The inner unshaded circle has a radius of \(r = 5\text{ m}\), and the outer circular boundary has a radius of \(R = 9\text{ m}\). Calculate the area of the shaded outer ring (annulus).
Answer:
SHADED REGIONS BLUEPRINT • PRACTICE SERIES PAGE 1 OF 3 INCLUDE APPROPRIATE SQUARED UNITS
SECTION B: INTERMEDIATE COMPOSITE DECOMPOSITIONS
Calculate composite areas using sectors, polygons, and symmetric cutouts.
CANDIDATE ID: _________
03
Circular Segment (Quadrant Minus Triangle)
10 Points
r = 12 cm r = 12 cm O
A quarter circle of radius \(r = 12\text{ cm}\) contains an unshaded right isosceles triangle formed by the chord connecting its arc endpoints. Calculate the area of the shaded circular segment.
Answer:
04
Rectangular Plate with Semicircular End Notches
10 Points
L = 20 in d = 10 in
A steel bracket is machined from a \(20\text{ in} \times 10\text{ in}\) rectangular sheet. Two semicircular notches of diameter \(d = 10\text{ in}\) are cut inward from both shorter ends. Calculate the remaining shaded area.
Answer:
05
The Four-Corner Star (Quadrant Inversions)
10 Points
s = 16 m (each r = 8 m)
A square courtyard of side length \(16\text{ m}\) contains four unshaded quarter-circular regions of radius \(r = 8\text{ m}\) at each corner meeting tangentially. Calculate the area of the remaining central shaded star.
Answer:
SHADED REGIONS BLUEPRINT • PRACTICE SERIES PAGE 2 OF 3 FOUR QUARTERS FORM ONE COMPLETE CIRCLE
SECTION C: ADVANCED COMPOSITE MASTERY
Solve classic geometric configurations: Archimedean Arbelos & Overlapping Lenses.
FINAL LEVEL
06
The Classical Arbelos (Three Tangent Semicircles)
10 Points
A C B AC = 12 cm CB = 8 cm
Entire baseline AB = 20 cm
A large semicircle has diameter \(AB = 20\text{ cm}\). Point \(C\) lies on \(AB\) such that \(AC = 12\text{ cm}\) and \(CB = 8\text{ cm}\). Two smaller semicircles are drawn on diameters \(AC\) and \(CB\) inside the large semicircle. Calculate the shaded area between the large and small semicircles.
Answer:
07
The Overlapping Foliate Lens (Dual Quadrant Petal)
10 Points
side s = 10 mm 10 mm
Centers at opposite square vertices
Inside a square of side length \(s = 10\text{ mm}\), two quarter circles of radius \(r = 10\text{ mm}\) are drawn centered at opposite vertices. Calculate the total area of the shaded overlapping lens region.
Answer:
Geometric Audit & Self-Check Checklist
1. Radius vs. Diameter: Did you divide given diameters by 2 before calculating \(\pi r^2\)?
2. Decomposition: Did you correctly subtract unshaded cutouts from the bounding figure?
3. Square Units: Are appropriate squared units (\(\text{cm}^2, \text{m}^2, \text{in}^2, \text{mm}^2\)) stated?
SHADED REGIONS BLUEPRINT • PRACTICE SERIES PAGE 3 OF 3 COMPLETED BLUEPRINT SUBMISSION
