How to Solve Problems on Circles Stepwise Methods and Key Formulas
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 with Concepts and Solutions
1. What is a circle in mathematics?
A circle is the set of all points in a plane that are at a fixed distance from a fixed point called the center. The fixed distance is known as the radius.
- The fixed point is the center.
- The fixed distance is the radius (r).
- The distance across the circle through the center is the diameter (d = 2r).
2. What is the formula for the area of a circle?
The area of a circle is given by the formula A = πr², where r is the radius.
- π (pi) is approximately 3.14 or 22/7.
- r is the radius of the circle.
3. What is the formula for the circumference of a circle?
The circumference of a circle is calculated using C = 2πr or C = πd.
- r is the radius.
- d is the diameter.
4. How do you find the radius when the diameter is given?
The radius is half of the diameter, so r = d/2.
- If the diameter is 12 cm, then r = 12/2 = 6 cm.
- This relation is used in most geometry problems on circles.
5. What is a chord of a circle?
A chord is a line segment joining any two points on the circumference of a circle.
- The longest chord of a circle is the diameter.
- Chords equidistant from the center are equal in length.
6. What is a tangent to a circle?
A tangent is a line that touches a circle at exactly one point.
- The point of contact is called the point of tangency.
- The radius drawn to the point of tangency is perpendicular to the tangent.
7. How do you find the area of a sector of a circle?
The area of a sector is given by (θ/360°) × πr², where θ is the central angle in degrees.
- θ is the measure of the central angle.
- r is the radius of the circle.
8. What is the equation of a circle in coordinate geometry?
The standard equation of a circle with center (h, k) and radius r is (x − h)² + (y − k)² = r².
- If the center is at the origin (0,0), the equation becomes x² + y² = r².
- This equation is widely used in coordinate geometry problems on circles.
9. How do you solve problems involving angles in a circle?
Angles in a circle are solved using key circle theorems, especially that the angle at the center is twice the angle at the circumference.
- Central angle = 2 × Inscribed angle (on the same arc).
- Angles in the same segment are equal.
- Angle in a semicircle is 90°.
10. What are common mistakes to avoid in circle problems?
Common mistakes in problems on circles include using the wrong formula and confusing radius with diameter.
- Using r instead of d in circumference formula.
- Forgetting to square the radius in A = πr².
- Mixing degrees and radians in sector formulas.
- Ignoring key circle theorems in angle problems.
