Introduction
Circles are perfectly round shapes where every point on the edge is the same distance from the center. We can calculate the area of a circle using its diameter or circumference.
The diameter is the distance across the circle through its center, while the circumference is the distance around the circle. The relationship between these measurements helps us find the area.
Key formulas we know:
\[ \text{Circumference} (C) = \pi \times \text{diameter} (d) = 2\pi r \] \[ \text{Area} (A) = \pi r^2 \]Derive the Formula
Visual Derivation
We can derive the area formula by dividing a circle into sectors and rearranging them:
- Divide the circle into 16 or more equal sectors (like pizza slices)
- Cut out the sectors and arrange them alternately to form a parallelogram-like shape
- As we increase the number of sectors, the shape becomes more like a rectangle
Mathematical Derivation
Using the relationship between diameter and circumference:
Step 1: Relate circumference to diameter
\[ C = \pi d = 2\pi r \]Step 2: The rearranged sectors form a rectangle with:
Length = Half the circumference = \( \frac{C}{2} = \pi r \)
Width = Radius = \( r \)
Step 3: Calculate area of the rectangle
\[ \text{Area} = \text{Length} \times \text{Width} = \pi r \times r = \pi r^2 \]This gives us the familiar area formula for a circle.
Practice Exercise
Question 1
A circle has diameter 14 cm. Calculate its area using π = 22/7.
Solution:
First find the radius: \( r = \frac{d}{2} = \frac{14}{2} = 7 \) cm
\[ A = \pi r^2 = \frac{22}{7} \times 7 \times 7 = 154 \text{ cm}^2 \]Question 2
The circumference of a circle is 44 cm. Find its area (use π = 22/7).
Solution:
First find radius from circumference: \( C = 2\pi r \)
\[ 44 = 2 \times \frac{22}{7} \times r \] \[ r = \frac{44 \times 7}{2 \times 22} = 7 \text{ cm} \] \[ A = \pi r^2 = \frac{22}{7} \times 7 \times 7 = 154 \text{ cm}^2 \]Question 3
If the area of a circle is 616 cm², find its diameter (π = 22/7).
Solution:
\[ A = \pi r^2 \] \[ 616 = \frac{22}{7} \times r^2 \] \[ r^2 = \frac{616 \times 7}{22} = 196 \] \[ r = \sqrt{196} = 14 \text{ cm} \] \[ d = 2r = 28 \text{ cm} \]Ready for more challenges?
Full Practice SetSolve Problems
Word Problems
Now let's apply the area formula to solve real-world problems involving circles.
Problem 1
A circular garden has diameter 21 meters. Calculate the cost of paving the garden at GH₵50 per square meter (π = 22/7).
Solution:
First find the area:
\[ r = \frac{21}{2} = 10.5 \text{ m} \] \[ A = \frac{22}{7} \times 10.5 \times 10.5 = 346.5 \text{ m}^2 \]Calculate cost:
\[ \text{Cost} = 346.5 \times 50 = \text{GH₵}17,325 \]Problem 2
The circumference of a circular table is 3.14 meters. Find the area of the table (use π = 3.14).
Solution:
First find radius:
\[ C = 2\pi r \] \[ 3.14 = 2 \times 3.14 \times r \] \[ r = 0.5 \text{ m} \]Now calculate area:
\[ A = 3.14 \times 0.5 \times 0.5 = 0.785 \text{ m}^2 \]Problem 3
A circular path 2 meters wide runs around a circular garden. If the diameter of the garden is 14 meters, find the area of the path (π = 22/7).
Solution:
Garden radius = 7 m, Total radius (garden + path) = 7 + 2 = 9 m
Area of garden:
\[ A_{\text{garden}} = \frac{22}{7} \times 7 \times 7 = 154 \text{ m}^2 \]Total area:
\[ A_{\text{total}} = \frac{22}{7} \times 9 \times 9 = 254.57 \text{ m}^2 \]Area of path:
\[ A_{\text{path}} = 254.57 - 154 = 100.57 \text{ m}^2 \]Math Challenge
Timed Test in 7 Levels
Test your math skills through progressively challenging levels