Find the area and the circumference (or perimeter) of each object. (a) A penny, (b) a nickel, (c) a dime, (d) a quarter, (e) a half-dollar, (f) a silver dollar, (g) a Susan B. Anthony dollar, (h) a dollar bill, and (i) one face of the pyramid on the back of a $ 1 bill.
The correct answer and explanation is:
To find the area and circumference (or perimeter) of each object, we need their dimensions. Coins are circular, so we use:
- Area of a circle: A = πr²
- Circumference of a circle: C = 2πr
A dollar bill and the pyramid are not circular, so we use:
- Area of a rectangle: A = length × width
- Area of a triangle (for pyramid face): A = ½ × base × height
- Perimeter depends on shape (sum of all sides)
(a) Penny
Diameter = 0.75 in → Radius = 0.375 in
Area = π × (0.375)² ≈ 0.4418 in²
Circumference = 2 × π × 0.375 ≈ 2.356 in
(b) Nickel
Diameter = 0.835 in → Radius = 0.4175 in
Area ≈ π × (0.4175)² ≈ 0.5475 in²
Circumference ≈ 2 × π × 0.4175 ≈ 2.623 in
(c) Dime
Diameter = 0.705 in → Radius = 0.3525 in
Area ≈ π × (0.3525)² ≈ 0.3901 in²
Circumference ≈ 2 × π × 0.3525 ≈ 2.214 in
(d) Quarter
Diameter = 0.955 in → Radius = 0.4775 in
Area ≈ π × (0.4775)² ≈ 0.716 in²
Circumference ≈ 2 × π × 0.4775 ≈ 2.998 in
(e) Half-Dollar
Diameter = 1.205 in → Radius = 0.6025 in
Area ≈ π × (0.6025)² ≈ 1.139 in²
Circumference ≈ 2 × π × 0.6025 ≈ 3.783 in
(f) Silver Dollar
Diameter = 1.5 in → Radius = 0.75 in
Area ≈ π × (0.75)² ≈ 1.767 in²
Circumference ≈ 2 × π × 0.75 ≈ 4.712 in
(g) Susan B. Anthony Dollar
Diameter = 1.043 in → Radius = 0.5215 in
Area ≈ π × (0.5215)² ≈ 0.855 in²
Circumference ≈ 2 × π × 0.5215 ≈ 3.276 in
(h) Dollar Bill
Length = 6.14 in, Width = 2.61 in
Area = 6.14 × 2.61 ≈ 16.0254 in²
Perimeter = 2 × (6.14 + 2.61) ≈ 17.5 in
(i) One Face of the Pyramid
Estimate base = 1 in, height = 0.85 in (approx. visual from the bill)
Area ≈ ½ × 1 × 0.85 = 0.425 in²
Perimeter = 2 sides + base. If each slant side is ~0.9 in → 0.9 + 0.9 + 1 = 2.8 in
These measurements illustrate real-world applications of geometry in currency. Coins are modeled as perfect circles, and paper money and designs use basic shapes like rectangles and triangles. This helps in teaching and understanding geometry through familiar everyday items.
