Key Terms and Methods to Solve Circle Problems
There are a lot of things that are circle-shaped around us. Like for example, the sun, pizza, coins, etc. are in the shape of a circle (two dimensional). It is quite different as compared to other shapes and figures in geometry.
However, before we move on with the problems on circles, first let’s recapitulate some essential points and features related to this shape.
What is Meant by a Circle?
In Mathematics, circles can be defined as a round-shaped figure with no edges or corners. Plus, lines drawn from the centre of a circle to the boundary are equidistant. Sketching a circle with hands is quite challenging, and so a compass is usually used for the same.
What are the Terms Associated to a Circle?
Here, we have organised a table representing the essential expression related to a circle. Take a look!
Formulas Required in Solving Circle Area and Circumference Word Problems
Area of a Circle
The expression to find an area of a circle is:
Area = π x r2, here r = radius
Perimeter of a Circle
The expression to find the perimeter of a circle is:
Perimeter = 2 x π x r, here also r = radius
Diameter of a Circle
The expression to find the diameter of a circle is:
Diameter = 2 x r
Area of a Semicircle
The formula to determine the area of a semicircle is:
Area of a semi circle = (π x r2) / 2
Perimeter of a Semi-circle
The expression to calculate the perimeter of a semicircle is:
Perimeter of a semi circle = π x r + 2 x r = (π + 2)r
Note: You can use 3.14 or 22/7 (the value of pi) as per your convenience unless mentioned in the problem.
Problems on Circles with Solutions
Problems on Circles: Problem 1
A circle has a diameter 142.8 mm. Find its radius.
Solution: Diameter of a circle = 142.8 mm
Therefore, putting the value in the equation to find the radius, we get:
d = 2 x r
142.8 = 2 x r
r = 142.8 / 2
r = 71.4 mm
Problems on Circles: Problem 2
What will be the radius of a circle having an area 200.96 sq Ft?
Solution: Area of the circle = 200.96 sq. Ft
Putting the value in the required equation we get:
Area = pi x r2
200.96 = 22/7 x r2
r2 = (200.96 / 3.14)
r2 = 64
r = 8 ft
Problems on Circles: Problem 3
When the diameter of a circular figure is 9 cm, find the radius and perimeter.
Solution: Diameter = 9 cm
Therefore, radius = 9/2 = 4.5 cm
Area = pi x r x r
Area = 3.14 x 4.5 x 4.5 = 63.585 cm sq.
Perimeter = 2 x pi x r
Perimeter = 2 x 3.14 x 4.5
Perimeter = 28.26 cm
Problems on Circles: Problem 4
Evaluate the perimeter and area of a semi-circle having a radius of 7 cm. Please use 22/7 as the value of pi.
Solution: Substituting the values in both the equations we get,
Perimeter of a semi circle = (π + 2)r = 22/7 x 7 + 2 x 7 = 22 + 14 = 36 cm
Area of a semi circle = (π x r2) / 2 = (22/7 x 49) / 2 = 77 cm sq.
Circle Math Problems: Do It Yourself
1. What will be the circumference of a circle whose area is 616 cm sq.?
(a) 88 cm (b) 89 cm (c) 84 cm (d) 80 cm
2. Evaluate the area of the circle inside the square, having each side measuring 20 cm. Refer to the image given below.
[Image will be Uploaded Soon]
(a) 341.2 cm sq. (b) 324.2 cm sq. (c) 314.2 cm sq. (d) 342.2 cm sq.
3. Calculate the perimeter of a semicircle with radius 10 cm.
(a) 54.12 cm (b) 51.42 cm (c) 52.41 cm (d) 52.14 cm
Problems on Circles: Answer Key
By going through the solved examples and working out the given sums, students will be able to predict the types of problems they can expect during exams. Furthermore, if you want to get enlightened with more circle geometry problems and solutions, why don’t you download the Vedantu app?
Make sure to download the same and get access to lots of study materials and online tutorials.
FAQs on Problems on Circles: Definitions, Formulas & Examples
1. What is the precise mathematical definition of a circle and what are its essential parts?
In mathematics, a circle is defined as the set of all points in a plane that are at a fixed distance from a fixed point. The fixed point is called the center, and the fixed distance is called the radius (r). Key parts of a circle include:
- Radius (r): The distance from the center to any point on the circle.
- Diameter (d): A line segment that passes through the center and has its endpoints on the circle. It is the longest chord and is equal to twice the radius (d = 2r).
- Circumference (C): The total distance around the circle.
- Chord: A line segment whose endpoints both lie on the circle.
- Arc: A portion of the circumference of a circle.
- Secant: A line that intersects the circle at two distinct points.
- Tangent: A line that touches the circle at exactly one point, known as the point of tangency.
2. What are the fundamental formulas used for solving problems related to a circle's area and circumference?
To solve problems involving circles, two fundamental formulas are essential. These are based on the circle's radius (r) and the mathematical constant Pi (π ≈ 3.14159 or 22/7).
- Area of a Circle (A): The space enclosed by the circle is calculated using the formula A = πr². The area is measured in square units.
- Circumference of a Circle (C): The perimeter or the length of the boundary of the circle is calculated using the formula C = 2πr. The circumference is measured in linear units.
These formulas are crucial for calculating material requirements, distances, and spatial areas in various applications.
3. What are some important properties of circles related to their chords and tangents?
Understanding the properties of chords and tangents is key to solving complex geometry problems. Some of the most important properties as per the CBSE syllabus include:
- Equal Chords: Equal chords of a circle are equidistant from the center. Conversely, chords that are equidistant from the center are equal in length.
- Perpendicular from Center: A line drawn from the center of a circle perpendicular to a chord bisects the chord.
- Tangent-Radius Property: The tangent at any point of a circle is perpendicular to the radius through the point of contact. This means a 90° angle is formed.
- Tangents from an External Point: The lengths of two tangents drawn from an external point to a circle are equal.
4. How does a secant differ from a tangent, and why is this difference important?
The primary difference between a secant and a tangent lies in how they interact with a circle. A secant is a line that intersects a circle at two distinct points, cutting through its interior. In contrast, a tangent is a line that touches the circle at exactly one point, called the point of tangency, without entering the circle's interior. This distinction is crucial because the properties associated with them are fundamentally different. For example, problems involving tangents often use the property that a tangent is perpendicular to the radius at the point of contact, a property that does not apply to secants.
5. Why is the angle subtended by a diameter at any point on the circumference always a right angle (90°)?
This is a fundamental theorem of circles. The reason the angle in a semicircle is always 90° is based on the theorem that states: 'The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.' A diameter is a straight line, forming an angle of 180° at the center. The arc corresponding to this is a semicircle. Therefore, the angle subtended by this arc (the semicircle) at any point on the circumference will be exactly half of the central angle, which is 180° / 2 = 90°. This property is frequently used to prove geometric relationships and solve for unknown angles in circle problems.
6. How can you apply theorems related to tangents to solve geometry problems?
Theorems related to tangents are powerful tools for solving geometry problems. Here’s how they are typically applied:
- Finding Angles: Since a tangent is always perpendicular to the radius at the point of tangency, you can immediately identify a 90° angle. This helps in problems involving right-angled triangles, allowing the use of the Pythagoras theorem or trigonometric ratios.
- Proving Congruence: When two tangents are drawn from an external point to a circle, the lengths of the tangents are equal. By joining the external point and the points of contact to the center, you can form congruent triangles, which helps in finding unknown lengths and angles.
- Alternate Segment Theorem: This theorem states that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment. This is used to find unknown angles in complex figures involving both circles and triangles.
7. What are some real-world examples where the concepts of circles are applied?
The principles of circles are fundamental to many real-world applications in science, engineering, and daily life. Some examples include:
- Engineering and Mechanics: The invention of the wheel and gears is the most direct application. The design of engines, turbines, and other rotating machinery relies on the properties of circles.
- Physics and Astronomy: The orbits of planets and satellites are often modelled as circles or ellipses. Circular motion is a key concept in physics.
- Architecture and Design: Circles are used to create domes, arches, and circular windows in buildings. Ferris wheels, roundabouts, and athletic tracks are all designed using circle geometry.
- Navigation and GPS: Satellite signals create circular zones of possible locations. The intersection of signals from multiple satellites (triangulation) pinpoints a precise location on Earth.
