Area of a Quarter Circle - Definition, Examples, Quiz, FAQ, Trivia
Learn to calculate the area of a quarter circle with easy explanations and practice activities
What is a Quarter Circle?
A quarter circle is exactly what it sounds like - one-fourth of a full circle! It's formed when a circle is divided into four equal parts.
Imagine cutting a pizza into four equal slices. Each slice is a quarter circle. In math, we call this shape a quadrant.
Here are some important things to know about quarter circles:
- It has a curved edge that is one-fourth of the circle's circumference
- It has two straight edges that are both equal to the radius of the circle
- The angle at the center is exactly 90 degrees (a right angle)
Key Concept
A quarter circle is one of four equal parts of a full circle, with a 90° angle at its center point.
The Formula for Area of a Quarter Circle
To find the area of a quarter circle, we first need to remember the formula for the area of a full circle:
Area of Full Circle
Where π (pi) is approximately 3.1416 and r is the radius
Area of Quarter Circle
This is the same as A = (πr²)/4
Using Diameter
Remember
Always use the radius measurement if possible, as it makes calculations simpler. If you only have the diameter, divide it by 2 to get the radius.
Step-by-Step Examples
Let's practice calculating the area with some examples:
Example 1: Using Radius
A quarter circle has a radius of 6 cm. What is its area?
Solution:
Step 1: Formula = ¼ × π × r²
Step 2: Plug in values = ¼ × 3.1416 × (6)²
Step 3: Calculate = ¼ × 3.1416 × 36
Step 4: = ¼ × 113.0976
Step 5: = 28.2744 cm²
Example 2: Using Diameter
A quarter circle has a diameter of 10 cm. What is its area?
Solution:
Step 1: First find radius: r = d/2 = 10/2 = 5 cm
Step 2: Formula = ¼ × π × r²
Step 3: Plug in values = ¼ × 3.1416 × (5)²
Step 4: Calculate = ¼ × 3.1416 × 25
Step 5: = ¼ × 78.54
Step 6: = 19.635 cm²
Example 3: Word Problem
Maria has a circular pizza with radius 12 inches. She cuts it into 4 equal slices. What is the area of one slice?
Solution:
Step 1: Each slice is a quarter circle
Step 2: Formula = ¼ × π × r²
Step 3: Plug in values = ¼ × 3.1416 × (12)²
Step 4: Calculate = ¼ × 3.1416 × 144
Step 5: = ¼ × 452.3904
Step 6: = 113.0976 in²
Practice Tip
Remember that π is approximately 3.14, but for more precise calculations, use the π button on your calculator or 3.1416.
Practice Quiz
Test your understanding with this 5-question quiz. Choose the correct answer for each question.
Frequently Asked Questions
Here are answers to common questions about quarter circles:
Circle Trivia
Discover interesting facts about circles and mathematics:
The History of π
The symbol π was first used by Welsh mathematician William Jones in 1706. Ancient Babylonians approximated π as 3.125, while Egyptians used 3.1605 around 1650 BC.
Perfect Circles in Nature
While perfect circles are mathematical concepts, some natural phenomena come close: the cross-section of tree trunks, ripples in water, and full moons appear circular.
Circles in Architecture
The Pantheon in Rome features a massive circular dome that was the largest in the world for over 1300 years. Its circular design demonstrates advanced understanding of geometry.
Circular Motion
Circles are fundamental to understanding motion in physics. Planets orbit in elliptical paths that are nearly circular, and wheels use circular motion to make transportation possible.
