Circles and Squares

This page focuses on problems involving circles in relation to squares.

Question 13 presents a circle inscribed in a square:

• The square has sides of 8 cm
• Students must find the area of the shaded region (the area between the circle and the square)
• This involves subtracting the area of the circle from the area of the square

Example: For Question 13, the steps would be:

1. Calculate the square area: 8² = 64 cm²
2. Calculate the circle area: π(4)² ≈ 50.3 cm²
3. Subtract: 64 - 50.3 ≈ 13.7 cm²

Highlight: This type of problem tests students' ability to work with multiple shapes and use subtraction to find areas of irregular regions.

Advanced Circle Relationships

The final page presents a complex problem involving the relationship between two circles.

Question 14 requires students to prove a relationship between a semi-circle and a smaller circle:

• The semi-circle has a radius of 12 cm
• Its area is 8 times the area of the smaller circle
• Students must show that the radius of the smaller circle is 3 cm

This problem involves:

• Using the area formulas for both shapes
• Setting up an equation based on the given relationship
• Solving the equation to find the radius of the smaller circle

Highlight: This question tests students' ability to work algebraically with circle formulas and demonstrate mathematical proof skills.

Example: The key steps in the solution are:

1. Express the area of the semi-circle: (1/2)π(12)² = 72π
2. Set up the equation: 72π = 8πr², where r is the radius of the smaller circle
3. Solve to show that r = 3 cm

This final question serves as a challenging culmination of the concepts covered throughout the worksheet.

Circle Basics and Terminology

This page introduces fundamental concepts related to circles.

Question 1 asks students to draw and identify key parts of a circle:

• Drawing a radius
• Drawing and shading a sector

Question 2 tests students' knowledge of circle terminology:

• Identifying a tangent (a line that touches the circle at a single point)
• Identifying a diameter (a line segment that passes through the center of the circle and has its endpoints on the circle)

Definition: A radius is a line segment from the center of a circle to any point on its circumference.

Definition: A sector is a region of a circle enclosed by two radii and an arc.

Advanced Circle Problems

This page presents more complex problems involving circles and semi-circles.

Question 7 is a multi-step problem involving a semi-circle:

• Given the area of a semi-circle (50 m²), students must find its perimeter
• This requires working backwards from the area formula to find the radius, then calculating the circumference

Question 8 is a real-world application problem:

• A circular field with a diameter of 32 meters
• Students must calculate the cost of fencing the entire circumference at £15.95 per meter

Example: For Question 7, the steps would be:

1. Use the semi-circle area formula: 50 = (πr²)/2
2. Solve for r: r ≈ 5.64 m
3. Calculate the perimeter: πr + 2r ≈ 29.0 m

Highlight: These problems demonstrate how circle geometry concepts apply to real-world situations.