Circles: arcs, angles, circumference, area, and sectors (Geometry) · lesson

Circumference and area

💡 Circumference is 2πr and area is πr², and both start from the radius.

The radius goes from the center to the edge. The diameter goes all the way across, so it is twice the radius. If a problem gives the diameter, halve it first.

Circumference, the distance around, is C = 2πr, which is also πd. Area is A = πr².

Some questions want the answer as a number times π. Then leave π out and type only the number in front.

🧠 Let's work it out

A circle has radius 4. Find its circumference and area.

Circumference: C = 2 × π × 4 = 8π.
Area: A = π × 4² = 16π.
As decimals, 8π ≈ 25.1 and 16π ≈ 50.3.

What do you think comes next?

radius: the distance from the center of a circle to its edge
diameter: the distance across a circle through its center
circumference: the distance around a circle

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Inscribed angles are half the arc

💡 An inscribed angle is half of the arc it cuts off, so the arc is twice the angle.

An inscribed angle has its corner on the circle. Its two sides cut off a piece of the circle called the intercepted arc.

The angle is always half of that arc. A central angle, with its corner at the center, equals its arc instead.

🧠 Let's work it out

An inscribed angle cuts off an arc of 84°. How big is the angle?

The angle is half the arc.
84 ÷ 2 = 42.
Another example, going the other way: an inscribed angle of 29° cuts off an arc of 2 × 29 = 58°.

What do you think comes next?

inscribed angle: an angle with its corner on the circle
intercepted arc: the part of the circle inside an angle
central angle: an angle with its corner at the center

Arcs and sectors are fractions of the circle

💡 An arc or sector is the fraction angle ÷ 360 of the whole circumference or area.

An arc is a piece of the edge. A sector is a slice, like a slice of pizza. Both come from a central angle.

Arc length = (angle ÷ 360) × 2πr. Sector area = (angle ÷ 360) × πr².

If you know the sector area and want the radius, set the formula equal to the area and solve for r.

🧠 Let's work it out

A circle has radius 16 and a central angle of 45°. Find the arc length and the sector area.

45 ÷ 360 = 1/8 of the circle.
Arc = (1/8) × 2π × 16 = 4π ≈ 12.6.
Sector = (1/8) × π × 256 = 32π ≈ 100.5.

What do you think comes next?

arc: a piece of a circle's edge
sector: a pie-slice part of a circle

Sectors with radians

💡 With the angle in radians, sector area is (1/2)r²θ and arc length is rθ.

A radian is another unit for angles. A full turn is 2π radians, the same as 360°.

With radians, the formulas get shorter. Arc length = rθ. Sector area = (1/2)r²θ. Here θ is the angle in radians.

🧠 Let's work it out

A sector has radius 6 and an angle of 1.5 radians. Find its area.

Area = (1/2) × 6² × 1.5.
(1/2) × 36 × 1.5 = 27.
If asked for the arc length instead: 6 × 1.5 = 9.

What do you think comes next?

radian: an angle unit; a full turn is 2π radians
θ: theta, a letter often used for an angle

Tangent lines and crossing chords

💡 A tangent meets the radius at a right angle, and crossing chords have equal products of their pieces.

A tangent line touches a circle at one point. At that point, it is perpendicular to the radius. So the radius, the tangent piece, and the line to the center make a right triangle.

A chord is a segment with both ends on the circle. When two chords cross inside, each is cut into two pieces. The product of one chord's pieces equals the product of the other's.

🧠 Let's work it out

Two chords cross. One is cut into 2 and 10. The other is cut into 5 and x. Find x.

The parts of each chord multiply to the same number: 2 × 10 = 5 × x.
20 = 5x, so x = 4.
Another example: a tangent from a point 17 units from the center of a circle with radius 8.
Use the right triangle: √(17² − 8²) = √225 = 15.

What do you think comes next?

tangent line: a line that touches a circle at exactly one point
chord: a segment with both ends on a circle
perpendicular: meeting at a right angle