exit ticket ma.7.gr.1.3 and ma.7.gr.1.4 circles

objective: know the formulas for the area and circumference of a circle and use them to solve problems. (use 3.14 for $\pi$)

1. a circle has a radius of 5 cm. what is its circumference?
2. a circle has a diameter of 12 inches. what is its area?
3. if the circumference of a circle is 31.4 meters, what is the radius?
4. find the area of a semicircle with a radius of 4 ft.
5. what is the ratio of the circumference of a circle to its diameter?

Question

exit ticket ma.7.gr.1.3 and ma.7.gr.1.4 circles

objective: know the formulas for the area and circumference of a circle and use them to solve problems. (use 3.14 for $\pi$)

1. a circle has a radius of 5 cm. what is its circumference?
2. a circle has a diameter of 12 inches. what is its area?
3. if the circumference of a circle is 31.4 meters, what is the radius?
4. find the area of a semicircle with a radius of 4 ft.
5. what is the ratio of the circumference of a circle to its diameter?

Accepted Answer

{ "quality": "clear", "question_count": 5, "skills_matched": [ "step_cot", "explain_then_answer", "tutor" ], "needs_drawing": false, "concepts_used": [ "Linear Equations", "Solving Multi-Step Equations" ], "new_concepts": [ "Circle Circumference", "Circle Area", "Semicircle Area" ], "current_concepts": [ "Circle Circumference", "Circle Area", "Semicircle Area", "Linear Equations", "Solving Multi-Step Equations" ] } ## Calculate circumference of the first circle Using the **Linear Equations** knowledge point to evaluate the formula: $$ \begin{aligned} C &= 2\pi r \ C &= 2 \times 3.14 \times 5 \ C &= 31.4\text{ cm} \end{aligned} $$ ## Calculate area of the second circle Find the radius from the diameter and calculate the area: $$ \begin{aligned} r &= \frac{d}{2} = \frac{12}{2} = 6\text{ inches} \ A &= \pi r^2 \ A &= 3.14 \times 6^2 \ A &= 3.14 \times 36 \ A &= 113.04\text{ square inches} \end{aligned} $$ ## Solve for the radius from circumference Using the **Solving Multi-Step Equations** knowledge point to isolate the radius: $$ \begin{aligned} C &= 2\pi r \ 31.4 &= 2 \times 3.14 \times r \ 31.4 &= 6.28 \times r \ r &= \frac{31.4}{6.28} \ r &= 5\text{ meters} \end{aligned} $$ ## Calculate area of the semicircle Find the area of a full circle and divide by two: $$ \begin{aligned} A_{\text{circle}} &= \pi r^2 \ A_{\text{circle}} &= 3.14 \times 4^2 = 3.14 \times 16 = 50.24\text{ sq ft} \ A_{\text{semicircle}} &= \frac{50.24}{2} \ A_{\text{semicircle}} &= 25.12\text{ sq ft} \end{aligned} $$ ## Determine the ratio of circumference to diameter Evaluate the ratio using the definition of pi: $$ \begin{aligned} \text{Ratio} &= \frac{C}{d} \ \text{Ratio} &= \frac{\pi d}{d} = \pi \ \text{Ratio} &\approx 3.14\text{ (or }\pi\text{)} \end{aligned} $$ ## Question 1 $31.4\text{ cm}$ ## Question 2 $113.04\text{ square inches}$ ## Question 3 $5\text{ meters}$ ## Question 4 $25.12\text{ sq ft}$ ## Question 5 $3.14\text{ (or }\pi\text{)}$ { "subject": "Mathematics", "question_type": "Multi-part", "knowledge_point": [ "Mathematics", "Geometry", "Circle Circumference" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate circumference of the first circle

Calculate area of the second circle

Solve for the radius from circumference

Calculate area of the semicircle

Determine the ratio of circumference to diameter

Answer:

Question 1

Question 2

Question 3

Question 4

Question 5