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    • ▾7th grade
      • ▸Integers
        • •Adding integers
        • •Subtracting integers
        • •Negative Fibonacci numbers
        • •Distance between two numbers
        • •Multiplying integers
        • •Dividing integers
        • •Equivalent expressions with negative numbers
      • ▸Fractions, decimals, and percentages
        • •Multi-step problems
        • •Percents
        • •Negative signs in fractions
        • •Simplifying complex fractions without variables
        • •Dividing by zero
        • •Converting fractions to decimals
      • ▸Rates and proportional relationships
        • •Rates with fractions
        • •Interpreting graphs of proportional relationships
        • •Constants of proportionality
        • •Writing proportional equations
        • •Solving proportions using equivalent fractions
        • •Solving proportions using inverse operations and by cross multiplying
      • ▸Expressions, equations, and inequalities
        • •Interpreting linear expressions
        • •Solving two-step equations
        • •Mentally solving certain two-step equations
        • •Solving and graphing two-step inequalities
        • •Distributing, factoring, and combining like terms
      • ▾Geometry (grade 7)
        • •Angles around a point
        • •Complementary and supplementary angles
        • •Vertical angles theorem
        • •Finding the midpoint on the number line
        • •Intro to circles and \(\pi\)
        • •Area of an obtuse triangle
        • •Area of a regular polygon
        • •Area, volume, and surface area
        • •Circumference and area of a circle
        • •Volume of extruded figures
        • •Cross sections
        • •Platonic solids
        • •Precursor to the triangle inequality
        • •Scale drawings
      • ▸Statistics and probability (grade 7)
        • •Intro to probability
        • •Simple events
        • •Compound probability of independent events
        • •Comparing distributions
        • •Probability models
        • •Theoretical and experimental probability
        • •Representative samples
     › 7th grade › Geometry (grade 7)

    Circumference and area of a circle

    Start by posing the following questions to your students:

    If each of the squares below is a unit square, what are the bounds for the area of the circle? What affect does increasing or decreasing the size of each square have on the accuracy of the bounds?

    Solution: Count the number of squares which are completely shaded for a lower bound. Count the number of squares which are completely or partially shaded for an upper bound. As the size of each square increases, accuracy decreases, and vice-versa.

    The idea for these questions came from here.

    Next, students will see three ways of deriving the area formula, for circles, from the circumference formula. The first way is to inscribe a regular polygon in the circle, as seen here. The greater the number of sides, the more closely the area of the regular polygon will match the area of the circle. The second way is to cut slices, then rearrange into a rectangle, as seen here. The third way is to cut out concentric rings, then arrange into a triangle, as seen here. Then students will practice using the formula.

    Area of a circle by Khan Academy

    After that, students will learn how to find the area of composite figures involving semicircles.

    Khan Academy includes only one of these derivations.

    Watch these Khan Academy videos:

    • Partial circle area and arc length
    • Area of a circle
    • Radius, diameter, circumference & π
    • Relating circumference and area
    • Finding circumference of a circle when given the area
    • Area of a shaded region
    • Impact of increasing the radius
    • Radius & diameter from circumference

    Next, give your students this challenge:

    The perimeter of the square below is 8cm. Mentally determine the perimeter of the shaded region.

    Here's the solution.

    Conclude by leading this investigation:

    Graphene Trampoline (logic & probability)
    by MathPickle

    7.G.B.4: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

    Lessons and practice problems