The circumference and the circle are fundamental concepts in geometry, essential both in the theoretical study of mathematics and in practical applications. The circumference is defined as the set of points that are at a fixed distance, called the radius, from a central point. The circle, on the other hand, is the region bounded by the circumference and contains all points that are at a distance less than or equal to the radius.

Both the circle and the circumference are found in nature and in everyday life, from the shape of planets and orbits to architectural structures, wheels, and many precision mechanisms. For this reason, understanding their properties and learning to calculate measurements such as length, area, diameters, and arcs is very valuable for developing a solid understanding of geometry.

Score:

Find the perimeter of a circumference whose radius measures 2 in.

Solution

1. Recall that the formula for the perimeter (or length) of a circumference is:

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2. We substitute the known data and obtain:

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Calculate the area of a circle with a radius of 4 in.

Solution

1. Recall that the formula for the area of a circle is:

$\text{[math]}$

2. We substitute the known data and obtain:

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Find the length of a 100° arc in a circumference with a radius of 3.5 in.

Solution

1. Recall that the formula for arc length of a circumference is:

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2. We substitute the known data and obtain:

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In a circumference with a radius of 4 in, an arc measures 6 in. What is the central angle subtended by the arc?

Solution

1. Recall that the formula for arc length of a circumference is:

$\text{[math]}$

2. We substitute the known data, solve for the angle, and obtain:

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Find the area of a circular sector with a central angle of 60° and radius of 2.8 in.

Solution

1. Recall that the formula for the area of a sector is:

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2. We substitute the known data and obtain:

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If the perimeter of a circumference is 27.6 in, find its diameter.

Solution

1. Recall that the formula for the perimeter of a circumference is:

$\text{[math]}$

2. We substitute the known data, solve for the radius, and obtain:

$\text{[math]}$

Thus, the diameter is $\text{[math]}$

If the area of a circle is $\text{[math]}$ , find its diameter.

Solution

1. Recall that the formula for the area of a circle is:

$\text{[math]}$

2. We substitute the known data, solve for the radius, and obtain:

$\text{[math]}$

Thus, the diameter is $\text{[math]}$

In a circumference with a radius of 6.7 in, two points A and B are separated by an angle of 80° with respect to the center. What is the distance between A and B measured along the circumference?

Solution

1. Recall that the formula for angular length is:

$\text{[math]}$

2. We substitute the known data and obtain:

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Thus, the distance from A to B along the circumference is $\text{[math]}$

Find the distance from the point (7,2) to the circumference with center at (0,0) and radius 3.

Solution

1. We calculate the distance between the point and the center of the circumference:

$\text{[math]}$

2. Then the distance between the point and the circumference is:

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The arms of a swing measure 5.9 ft long and can describe a maximum angle of 146°. Calculate the distance traveled by the swing seat when the angle described in its swing is at maximum.

Solution

1. We calculate the path for $\text{[math]}$ :

$\text{[math]}$

The wheel of a truck has a 35.4 inch radius. How far has the truck traveled when the wheel has made 100 revolutions?

Solution

1. We convert the radius to feet:

$\text{[math]}$

2. We calculate the distance for one revolution:

$\text{[math]}$

3. We calculate the distance for 100 revolutions:

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A lighthouse sweeps a flat angle of 128° with its light. If the maximum range of the lighthouse is 7 miles, what is the maximum length in feet of the corresponding arc?

Solution

1. We convert the radius to feet, knowing that one mile equals 5,280 feet:

$\text{[math]}$

2. We calculate the path for $\text{[math]}$ :

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The length of a circumference is 17.3 in. What is the area of the circle?

Solution

1. We calculate the radius of the circumference:

$\text{[math]}$

2. We calculate the area of the circle:

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The area of a circular sector of 90° is 4π square inches. Calculate the radius of the circle to which it belongs and the length of the circumference.

Solution

1. We calculate the radius of the sector:

$\text{[math]}$

2. We calculate the length of the circumference:

$\text{[math]}$

On a carousel, Anna got on the horse that is 11.5 ft from the center of a rotating platform and her friend Laura got on the lion that was 6.6 ft from the center. Calculate the distance traveled by each one when the platform has made 50 revolutions.

Solution

1. We calculate the distance for one revolution with $\text{[math]}$ :

$\text{[math]}$

2. The total distance is obtained by multiplying one revolution by 50:

$\text{[math]}$

Find the area of a circular sector whose chord is the side of the inscribed equilateral triangle, with the circle having a radius of 0.8 in.

Solution

1. The requested sector corresponds to one-third of the total area of the circle, that is, a sector of $\text{[math]}$ .

2. We calculate the area for $\text{[math]}$ :

$\text{[math]}$

Given two concentric circumferences with radii of 3.1 and 2 in respectively, the radii OA and OB are drawn, forming a 60° angle. Calculate the area of the circular trapezoid formed.

Solution

circulo circunscrito grafica

1. We calculate the area of the two sectors and then their difference:

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In a circular park with a radius of 2,297 ft, there is a fountain in the center, also circular, with a radius of 16.4 ft. Calculate the area of the walking zone.

Solution

dibujo de fuente circular en parque circular grafica

1. We calculate the area of the two circles and then their difference:

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The surface of a table consists of a central square part with a 3.3 ft side and two semicircles attached on two opposite sides. Calculate the area.

Solution

grafica de una mesa

1. We calculate the area of the square and the circle formed by the two semicircles, and then their sum:

$\text{[math]}$

Find the area of the circular sector whose chord is the side of the inscribed square, with the circle having a radius of 1.6 in.

Solution

1. We calculate the area for the sector of $\text{[math]}$ :

$\text{[math]}$

Calculate the shaded area, knowing that the side of the square is 2.4 in and the radius of the circle measures 1.2 in.

Solution

circulo circunscrito cuadrado grafica

1. We calculate the area of the square and the circle, and then their difference:

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In a circular plaza with a radius of 820 ft, 7 lampposts with circular bases of 3.3 ft radius will be placed; the rest of the plaza will be used to plant grass. Calculate the area of the grass.

Solution

grafica de plaza circular y farolas problema mates

1. We calculate the area of the larger circle, the lampposts, and then their difference:

$\text{[math]}$

Calculate the area of the shaded part, if the radius of the larger circle measures 2.4 in and the radius of the small circles measure 0.8 in.

Solution

Ejercicio de circunferencia grafica circulo

1. We calculate the area of the larger circle, the smaller circles, and then their difference:

$\text{[math]}$

Calculate the area of the shaded part, where AB = 3.9 in, ABCD is a square, and APC and AQC are arcs of circumferences with centers B and D.

Solution

Ejercicio de circunferencia 11

1. The shaded part is composed of two circular segments.

Ejercicio de circunferencia 12 Ejercicio de circunferencia 13

2. We calculate the area of the circular segment and multiply it by 2:

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A regular hexagon with a 1.6 in side has a circle inscribed in it and another circumscribed around it. Find the area of the circular ring thus formed.

Solution

Ejercicio de circunferencia 14

1. We calculate the radius of the inner circle, which coincides with the apothem of the hexagon:

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2. We calculate the area of each circle and then their difference:

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In a circumference, a chord of 18.9 in is 2.8 in away from the center. Calculate the area of the circle.

Solution

1. We calculate the radius of the circle by applying the Pythagorean theorem:

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2. We calculate the area of the circle:

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The legs of a triangle inscribed in a circumference measure 8.7 in and 11.7 in respectively. Calculate the area of the circle.

Solution

Ejercicio de circunferencia 16

1. We calculate the hypotenuse, which is the diameter of the circumference:

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2. We calculate the area of the circle:

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On a circle with a 1.6 in radius, a central angle of 60° is drawn. Find the area of the circular segment between the chord that joins the ends of the two radii and its corresponding arc.

Solution

Ejercicio de circunferencia 17

1. We calculate the area of the sector:

$\text{[math]}$

2. We calculate the height of the triangle and its area:

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3. We calculate the difference in areas:

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Given an equilateral triangle with a 19.7 ft side, find the area of one of the sectors determined by the circumscribed circle and by the radii passing through the vertices.

Solution

1. The center of the circle is the centroid. Therefore $\text{[math]}$ .

Ejercicio de circunferencia 18

2. We calculate the height of the triangle and the radius:

$\text{[math]}$

3. We calculate the area of the sector:

$\text{[math]}$

Calculate the area of the circular ring determined by the circles inscribed and circumscribed around a square with a 26.2 ft diagonal.

Solution

Ejercicio de circunferencia 19

1. We calculate the radius of the outer circle, whose diameter is the diagonal of the square, and the radius of the inner circle using the Pythagorean theorem:

$\text{[math]}$

2. We calculate the difference in areas:

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Agostina Babbo

Agostina Babbo is an English and Italian to Spanish translator and writer, specializing in product localization, legal content for tech, and team sports—particularly handball and e-sports. With a degree in Public Translation from the University of Buenos Aires and a Master's in Translation and New Technologies from ISTRAD/Universidad de Madrid, she brings both linguistic expertise and technical insight to her work.

Circle and Circumference Problems and Exercises

The Thales of Miletus Theorem

Applications of the Pythagorean Theorem

Area and Volume Exercises and Problems

Area III problems

Problems and Exercises Involving Circumference and Circles

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Circle and Circumference Problems and Exercises

Theory

The Thales of Miletus Theorem

Applications of the Pythagorean Theorem

Other Exercises

Area and Volume Exercises and Problems

Area III problems

Problems and Exercises Involving Circumference and Circles

Cancel reply