Geometry & Statistics
Scale Drawings and Circles: Area & Circumference
Grade 7
- ✓Interpret and use scale drawings to determine actual lengths and distances.
- ✓Calculate the circumference of a circle given its radius or diameter.
- ✓Calculate the area of a circle given its radius or diameter.
- ✓Solve real-world problems involving scale drawings, circumference, and area of circles.
Key concepts
A scale drawing is a drawing that shows a real object with accurate sizes reduced or enlarged by a certain amount (the scale). It maintains the proportions of the actual object.
The scale factor is the ratio of a length on a scale drawing to the corresponding actual length. It can be expressed as a ratio (e.g., 1 inch : 10 feet) or a fraction. It tells you how many units on the drawing represent how many units in real life.
The radius of a circle is the distance from the center of the circle to any point on its edge.
The diameter of a circle is the distance across the circle passing through its center. It is always twice the length of the radius.
The circumference of a circle is the distance around the edge of the circle. It is similar to the perimeter of a polygon.
Pi (π) is a mathematical constant that represents the ratio of a circle's circumference to its diameter. It is an irrational number, meaning its decimal representation never ends and never repeats. Common approximations for π are 3.14 or 22/7.
The area of a circle is the amount of surface enclosed by the circle. It measures the space inside the circle.
Key facts to remember
- 1A scale drawing is a proportional representation of a real object.
- 2The scale factor is the ratio of a length on the drawing to the actual length.
- 3The circumference of a circle is the distance around it, calculated by C = πd or C = 2πr.
- 4The area of a circle is the space it covers, calculated by A = πr².
- 5Pi (π) is an irrational constant, approximately 3.14 or 22/7.
- 6The diameter (d) of a circle is twice its radius (r): d = 2r.
Worked examples
Example 1
A map has a scale of 1 inch : 25 miles. If two cities are 4.5 inches apart on the map, what is the actual distance between them?
Answer
The actual distance between the two cities is 112.5 miles.
Ensure units cancel out correctly to leave the desired unit for the answer.
Example 2
A circular fountain has a radius of 3.5 feet. What is its circumference? Use π ≈ 22/7.
Answer
The circumference of the fountain is 22 feet.
Using the fraction 22/7 for π can simplify calculations when the radius or diameter is a multiple of 7 or 0.5.
Example 3
A circular rug has a diameter of 8 feet. What is the area of the rug? Use π ≈ 3.14.
Answer
The area of the rug is 50.24 square feet (ft²).
Remember to square the radius, not the diameter, and to use square units for area.
Common mistakes
- ✗Confusing radius and diameter: Using the diameter in formulas that require the radius (e.g., A = πd² instead of A = πr²).
- ✗Using the wrong formula: Applying the circumference formula when the area is needed, or vice-versa.
- ✗Incorrectly applying the scale factor: Multiplying when division is needed, or vice-versa, when converting between drawing and actual measurements.
- ✗Forgetting to square the radius for area: Calculating A = πr instead of A = πr².
- ✗Using incorrect units: Forgetting to include units or using linear units (e.g., feet) for area instead of square units (e.g., square feet).
Exam tips
- ★Always read the problem carefully to determine whether you need to find circumference, area, or work with a scale drawing.
- ★Identify whether the problem gives you the radius or the diameter of a circle, and convert if necessary before applying formulas.
- ★Pay close attention to the units given in the problem and ensure your final answer has the correct units (e.g., miles for distance, feet for circumference, square feet for area).
- ★If a problem specifies an approximation for π (e.g., 3.14 or 22/7), use that value. Otherwise, use the π button on your calculator for greater accuracy.
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