Minds On
Relating the area of two-dimensional shapes
Investigate a circle that that fits exactly inside a square. The side length of the square is 3 cm.
Press ‘Hint’ to reveal questions that will guide your thinking.
If we know that the area of the square is 9 cm², what does this information tell you about the area of the circle?
- What is an estimate for the area of the circle that is too low?
- What is an estimate for the area of the circle that is too high?
- What is your estimate for the area of the circle?
- Is the area of the square, including the circle, larger, equal to, or smaller than the area of the circle? Why?
- How might you use the area of the square to help you estimate the area of the circle?
Action
Investigating the area of a circle
We have explored the relationships between the radius, diameter, and circumference of a circle using the formula C = πd (circumference equals pi multiplied by diameter). We know that pi equals approximately 3.14. We also discovered that the diameter is twice (2 times) the radius or d = 2r.
Answer the following questions and record your responses using a method of your choice.
- What is the diameter and circumference of a circle with a radius of 4 cm?
- What is the circumference of a circle with a diameter of 12 cm?
Press ‘Answer’ to reveal the solutions.
If I know that the radius is 4 centimetres, then:
d = 2r
d = 2 × 4
d = 8
The diameter of the circle is 8 centimetres.
In order to calculate the circumference of the circle, I can use the formula C = 𝝅d.
C = 𝝅d
C = 3.14 × 8
C ≈ 25.12 cm
The circumference of the circle is approximately 25.12 centimetres.
In the second question, I know that the diameter of the circle is 12 centimetres.
To calculate the circumference of the circle, I can use the formula C = πd.
C = πd
C = 3.14 × 12
C ≈ 37.68 cm
The circumference of the circle is approximately 37.68 centimetres.
How could you rearrange parts of the circle so they resemble a parallelogram? Use a math manipulative such as 1/8 fraction pieces, or pieces cut from cardstock, or record a detailed audio/written description of what you would do.
Compare the shape you have created to a parallelogram. What do you notice? What do you wonder?
Given that the height of the parallelogram is equal to the radius of the circle, and half the circumference is equal to πr, how might you find the area of this new parallelogram created from wedges of the circle?
Record all your answers using a method of your choice.
Diameter: d = 2r
Circumference: C= πd
An image of a “curvy” parallelogram is constructed from 8 parts of a circle. The parts of the original circle are marked as radius and half the circumference (pi × r), or πr. For comparison, the following is an image of a parallelogram with the base and height marked.
Calculating area of a circle
Formula: Area of a circle = πr²
If in the curved parallelogram, the "base" is πr and the "height" is r, then the formula to find the area of the shape is A = πr²
Calculate and measure the area of the following circles using the formula:
- a circle with a radius of 6 mm
- a circle with a diameter of 12 mm
- a circle with a circumference of 37.68 cm
What do you notice is the relationship between the three circles?
Press ‘Solution’ to reveal the solution for this task.
Solution:
1) If I know that the radius of the circle is 6 mm, I can use the formula:
A = 3.14 × 6²
A = 3.14 × 36
A = 113.04
The area of the circle is approximately 113.04 mm².
2) If the diameter of the circle is 12 mm, I know that half of the diameter is the radius which equals 6 mm.
A = πr²
A = π × 6²
A = 3.14 × 6²
A = 3.14 × 36
A = 113.04 mm²
3) If I know that the circumference of a circle is 37.68 cm , I know the formula for circumference is
C = πd
37.68 = 3.14 × d
37.68 ÷ 3.14 = d
12 cm = d
Therefore, r = 6 cm
A = πr²
A = π × 6²
A = 3.14 × 6²
A = 3.14 × 36
A = 113.04 cm²
All three circles share the same circumference and area.
Consolidation
Solving problems with circles
Problem one
Apply your knowledge of area formulas to determine the area of the square and the area of the circle.
A circle inside a square. The circle touches all four sides of the square. There is a line through the centre of the circle labelled 6 cm.
Determine:
- the area of the square (A = l × w)
- the area of the circle (A = πr²)
How do the areas of the square and circle compare? Does this seem reasonable?
Problem two
Apply your knowledge of area formulas to determine the area of the white ring only.
Calculate the area of both circles. What should you do with these two areas to find the area of the ring between the circles? The “ring” is the area outside the inner circle, but inside the larger circle.
There is a large circle with a smaller circle inside it. Both have the centre marked O. There is a line from O to a point (D) on the rim of the inner circle, which measures 4.2 cm. There is a line from O to a point (B) on the rim of the outer circle, which measures 4.8 cm.
Reflection
As you read the following descriptions, select the one that best describes your current understanding of the learning in this activity. Press the corresponding button once you have made your choice.
I feel...
Now, expand on your ideas by recording your thoughts using a voice recorder, speech-to-text, or writing tool.
When you review your notes on this learning activity later, reflect on whether you would select a different description based on your further review of the material in this learning activity.
Press ‘Discover More’ to extend your skills.
Discover MoreA circular pool at the local park is 6 metres deep, and has a diameter of 20 metres. Determine how much water the pool can hold.
You may wish to apply your understanding of the formula for the area of a circle A = πr ² and the formula for the volume of a cylinder (which is the area of the circular base x height, A = 𝜋r ² × h).
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