# Incircle, circumcircle, circumference and area · Grade 11 · 4 units

> Explanation, worked examples and free practice following the Israeli curriculum.

In Incircle, circumcircle, circumference and area (Grade 11 · 4 units) you practise 6 skills: The centre of the inscribed circle · A circle inscribed in a triangle with a right angle · The circumference of a circle · The area of a disc · The radius of the circle through the corners · The area of a parallelogram.

A circle inscribed in a triangle touches all three of its sides, and its centre is where the angle bisectors meet. The circle through the corners of a triangle passes through all three corners, and its centre is where the perpendicular bisectors of the sides meet. Every triangle has both. The radius of the circle through the corners comes from the sine rule: a side over the sine of the angle opposite it equals the diameter.

The two tangents to a circle from the same point are equal, so each side of a triangle drawn round a circle is made of two tangent pieces. Joining the centre of the inscribed circle to the corners splits the triangle into three triangles, each with the radius as its height, so the area is the radius times half the perimeter.

The circumference of a circle is two times π times the radius, and the area of a disc is π times the square of the radius. The area of a parallelogram is the product of two adjacent sides and the sine of the angle between them.

A model proof that the angle bisectors of a triangle meet at one point: every point on an angle bisector is the same distance from the two arms of the angle. The bisectors of two angles of the triangle meet at a point, which is the same distance from the two sides at the first corner, and from the two sides at the second corner. So it is the same distance from all three sides, in particular from the two sides at the third corner, so it lies on the third bisector too. The circle round that point with that distance as its radius touches all three sides.

**Tip:** The centre of the inscribed circle is the same distance from the sides, and the centre of the circle through the corners is the same distance from the corners.

## What you learn

### The centre of the inscribed circle

- Example: (A triangle with a circle inside it that touches all three sides.) In triangle ABC the point O is the centre of the inscribed circle, and ∢A = 32° . What is ∢BOC ?
- Answer: 106 °
- Solution: The angles ∢B and ∢C add up to 180° − 32° = 148° , and half of that is 74° . That is the sum of ∢OBC and ∢OCB , so ∢BOC = 180° − 74° = 106° .

### A circle inscribed in a triangle with a right angle

- Example: (A triangle with a circle inside it that touches all three sides.) In triangle ABC the angle ∢C is a right angle, and its sides, in centimetres, are AC = 9, BC = 12, AB = 15 . A circle is inscribed in the triangle. What is the radius of the circle?
- Answer: 3 cm
- Solution: The tangent pieces at C equal the radius, and the tangent pieces from A and B make up the hypotenuse AB . So twice the radius is AC + BC − AB , and the radius is (9 + 12 − 15) ÷ 2 = 3 .

### The circumference of a circle

- Example: The radius of a circle is 2 cm. What is its circumference, in cm? Give the answer as a multiple of π .
- Answer: 4π
- Solution: The circumference is 2π × 2 = 4π .

### The area of a disc

- Example: The radius of a disc is 2 cm. What is its area, in cm²? Give the answer as a multiple of π .
- Answer: 4π
- Solution: The area is π × 2² = 4π .

### The radius of the circle through the corners

- Example: (A triangle inside the circle through its corners.) In triangle ABC , BC = 3 cm and ∢A = 150° . What is the radius of the circle through the corners of the triangle?
- Answer: 3 cm
- Solution: The sine of 150° is 1/2 , so 2R = 3 / (1/2) . The radius is half of that: R = 3 .

### The area of a parallelogram

- Example: (A parallelogram with two adjacent sides and the angle between them.) In the parallelogram ABCD , AB = 4, AD = 12 in centimetres and ∢A = 30° . What is its area, in cm²?
- Answer: 24 cm²
- Solution: The area is 4 × 12 × sin 30° . The sine of 30° is 1/2 , so the area is 24 .

## Not on the site yet

- Proofs of the theorems on the inscribed circle and the circle through the corners
- Arc length and the area of a sector
- Enrichment: the history of the number π

The exercises are made by rules and the answers are worked out in code. The topic follows the Israeli Ministry of Education’s high school maths curriculum at 4 units (grades 10 to 12).

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