What is the definition of a circle?

A circle is a set of points in a plane that are equidistant from a fixed center point. The distance from the center to any point on the circle is called the radius.

How do you calculate the circumference of a circle?

The circumference of a circle can be calculated using the formula C = 2πr, where r is the radius of the circle. Alternatively, if you know the diameter (d), you can use C = πd.

What is the area of a circle and how is it calculated?

The area of a circle is calculated using the formula A = πr², where r is the radius of the circle. This gives the total space enclosed within the circle.

If a circle has a radius of 4 cm, what is its area?

Hint: Use the area formula A = πr².

Using the formula A = πr²: A = π(4)² = π(16) = 16π cm². The area is approximately 50.27 cm² when using π ≈ 3.14.

A circle has a circumference of 31.4 cm. What is the radius?

Hint: Use the circumference formula C = 2πr.

Using the formula C = 2πr, we can solve for r: 31.4 = 2πr. Dividing both sides by 2π gives r = 31.4 / (2π) ≈ 5 cm.

What is the relationship between the radius and diameter of a circle?

The diameter of a circle is twice the length of the radius. This can be expressed as D = 2r or r = D/2.

What is a chord in a circle?

A chord is a line segment whose endpoints both lie on the circle. The longest chord in a circle is the diameter.

What is the formula for the area of a sector of a circle?

Hint: The central angle is involved in the formula.

The area of a sector is given by A = (θ/360) * πr², where θ is the central angle in degrees and r is the radius.

If a sector has a central angle of 90° and a radius of 6 cm, what is its area?

Hint: Use the area of a sector formula.

Using A = (θ/360) * πr²: A = (90/360) * π(6)² = (1/4) * π(36) = 9π cm².

What is an inscribed angle in a circle?

An inscribed angle is formed by two chords that share an endpoint on the circle. The measure of an inscribed angle is half the measure of the central angle that subtends the same arc.

How do you find the length of an arc?

Hint: The central angle is needed for this calculation.

The length of an arc is calculated using the formula L = (θ/360) * 2πr, where θ is the central angle in degrees and r is the radius.

A circle has a radius of 10 cm. What is the circumference?

Hint: Use the formula C = 2πr.

Using the formula C = 2πr: C = 2π(10) = 20π cm, which is approximately 62.83 cm.

If a circle has a diameter of 14 cm, what is its radius?

Hint: Use the relationship between radius and diameter.

The radius is half the diameter: r = D/2 = 14/2 = 7 cm.

In a circle, what is the relationship between a tangent and a radius?

Hint: Consider the point of contact.

A tangent to a circle is perpendicular to the radius at the point of contact.

What is the formula for the length of an arc if the angle is in radians?

Hint: Consider how radians relate to the circle's radius.

The length of an arc in radians is given by L = θr, where θ is the angle in radians and r is the radius.

If the area of a circle is 36π cm², what is its radius?

Hint: Use the area formula A = πr².

Setting 36π = πr², we divide by π to get 36 = r², so r = √36 = 6 cm.

What is the area of a semicircle with a radius of 4 cm?

Hint: Remember the area of a circle and divide by two.

The area of a semicircle is A = (1/2) * πr² = (1/2) * π(4)² = (1/2) * π(16) = 8π cm².

If an inscribed angle is 40°, what is the measure of the central angle subtended by the same arc?

Hint: Use the property of inscribed angles.

The central angle is twice the inscribed angle: Central angle = 2 * 40° = 80°.

What is the circumference of a semicircle with a diameter of 10 cm?

Hint: Calculate the circumference of the full circle first, then divide.

The full circumference is C = πd = π(10) = 10π cm. The semicircle's circumference is (10π / 2) + 10 = 5π + 10 cm.