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AQA GCSE · Question 26 · Geometry and Measures
A large circle and a small circle are shown.
The radius of the large circle is 12 cm.
radius of large circle : radius of small circle = 4 : 1
Work out the shaded area. Give your answer in terms of π.
A large circle and a small circle are shown.
The radius of the large circle is 12 cm.
radius of large circle : radius of small circle = 4 : 1
Work out the shaded area. Give your answer in terms of π.
How to approach this question
1. **Find the radius of the small circle:**
The ratio is 4 : 1 (large : small).
The radius of the large circle is 12 cm, which corresponds to the 4 parts.
Value of one part = 12 cm ÷ 4 = 3 cm.
The radius of the small circle corresponds to 1 part, so its radius is 3 cm.
2. **Calculate the area of the large circle:**
Area = πr².
Area_large = π × (12)² = 144π cm².
3. **Calculate the area of the small circle:**
Area = πr².
Area_small = π × (3)² = 9π cm².
4. **Calculate the shaded area:**
Shaded area = Area of large circle - Area of small circle.
Shaded area = 144π - 9π = 135π cm².
Full Answer
135π cm²
To find the shaded area, we need to calculate the area of the large circle and subtract the area of the small (unshaded) circle.
**Step 1: Find the radius of the small circle.**
We are given the ratio of the radii:
Radius_large : Radius_small = 4 : 1
We know the radius of the large circle is 12 cm. This corresponds to the '4' in the ratio.
So, 4 parts = 12 cm.
To find the value of 1 part, we divide by 4:
1 part = 12 cm / 4 = 3 cm.
The radius of the small circle is 1 part, so Radius_small = 3 cm.
**Step 2: Calculate the area of the large circle.**
The formula for the area of a circle is A = πr².
Area_large = π × (12)² = π × 144 = 144π cm².
**Step 3: Calculate the area of the small circle.**
Area_small = π × (3)² = π × 9 = 9π cm².
**Step 4: Calculate the shaded area.**
Shaded Area = Area_large - Area_small
Shaded Area = 144π - 9π = 135π cm².
The answer is required in terms of π, so this is the final answer.
Common mistakes
✗ Using the radius of the large circle for the small one, or vice versa.\n✗ Making an error when squaring the radii (e.g., 12² = 24).\n✗ Adding the areas instead of subtracting.
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