Here are the solutions to fill in the missing blanks on your worksheet.
Part 1: Area of Sectors
Top Left Box (Quarter Circle, radius 8 cm)
*
Formula setup: The angle is $90^\circ$ (right angle).
* Fraction: $\frac{90}{360}$
* Radius squared ($r^2$): $8^2 = 64$
*
Calculation: $\frac{90}{360} \times \pi \times 64$ simplifies to $\frac{1}{4} \times 64\pi$.
*
Final Answer: $16\pi$
Top Right Box (Sector, angle $60^\circ$, radius 12 cm)
*
Formula setup:
* Fraction: $\frac{60}{360}$
* Radius squared ($r^2$): $12^2 = 144$
*
Calculation: $\frac{60}{360} \times \pi \times 144$ simplifies to $\frac{1}{6} \times 144\pi$.
*
Final Answer: $24\pi$
Bottom Left Box (General Sector, radius 10 cm)
*
Formula setup:
* Fraction: $\frac{\theta}{360}$
* Area of full circle: $\pi r^2$ or specifically $\pi(10)^2 = 100\pi$
*
Blanks: The first box is $\theta$, second is $360$. The large box is $100\pi$ (or $\pi \times 10^2$).
Bottom Right Box (General Formula)
*
Blanks:
* Fraction: $\frac{\theta}{360}$
* Large box: $\pi r^2$
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Part 2: Arc Lengths
Top Left Box (Quarter Circle, radius 8 cm)
*
Formula setup: Angle is $90^\circ$. Circumference is $2\pi r = 16\pi$.
*
Calculation: $\frac{90}{360} \times 16\pi$ simplifies to $\frac{1}{4} \times 16\pi$.
*
Final Answer: $4\pi$
Top Right Box (Sector, angle $36^\circ$, radius 7 cm)
*
Formula setup: Circumference is $2\pi r = 14\pi$.
*
Calculation: $\frac{36}{360} \times 14\pi$ simplifies to $\frac{1}{10} \times 14\pi$.
*
Final Answer: $1.4\pi$ (or $\frac{7\pi}{5}$)
Bottom Left Box (General Sector, radius 22 cm)
*
Formula setup:
* Fraction: $\frac{\theta}{360}$
* Circumference: $2\pi r = 44\pi$
*
Blanks: First box $\theta$, second $360$. Large box $44\pi$.
Bottom Right Box (General Formula)
*
Blanks:
* Fraction: $\frac{\theta}{360}$
* Large box: $2\pi r$ (Circumference)
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Final Answer:
Area of Sectors Section:
1.
Top Left: $\frac{90}{360} \times \pi \times 64 = \mathbf{16\pi}$
2.
Top Right: $\frac{60}{360} \times \pi \times 144 = \mathbf{24\pi}$
3.
Bottom Left: $\frac{\theta}{360} \times \mathbf{100\pi}$
4.
Bottom Right: $\frac{\theta}{360} \times \mathbf{\pi r^2}$
Arc Lengths Section:
1.
Top Left: $\frac{90}{360} \times 16\pi = \mathbf{4\pi}$
2.
Top Right: $\frac{36}{360} \times 14\pi = \mathbf{1.4\pi}$
3.
Bottom Left: $\frac{\theta}{360} \times \mathbf{44\pi}$
4.
Bottom Right: $\frac{\theta}{360} \times \mathbf{2\pi r}$
Parent Tip: Review the logic above to help your child master the concept of arc length and sector area worksheet.
