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Geometry - Properties of Circles

Grade 8IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Parts of a Circle: Understanding radius, diameter, chord, tangent, arc, sector, and segment.

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Angle in a Semi-circle: The angle subtended by a diameter at the circumference is always 90∘90^\circ.

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Angle at the Center: The angle subtended by an arc at the center is twice the angle subtended at the circumference.

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Angles in the Same Segment: Angles subtended by the same arc (or chord) at the circumference are equal.

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Cyclic Quadrilaterals: Opposite angles in a cyclic quadrilateral (a four-sided shape where all vertices touch the circle) sum to 180∘180^\circ.

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Tangent-Radius Property: A tangent at any point on a circle is perpendicular (90∘90^\circ) to the radius through the point of contact.

📐Formulae

Circumference = 2πr2\pi r or πd\pi d

Area = πr2\pi r^2

Arc Length = θ360×2πr\frac{\theta}{360} \times 2\pi r

Sector Area = θ360×πr2\frac{\theta}{360} \times \pi r^2

💡Examples

Problem 1:

A circle has a radius of 7 cm. Calculate the length of an arc that subtends an angle of 60∘60^\circ at the center. (Use π≈3.142\pi \approx 3.142)

Solution:

Arc Length = 60360×2×3.142×7≈7.33\frac{60}{360} \times 2 \times 3.142 \times 7 \approx 7.33 cm

Explanation:

Apply the arc length formula by substituting θ=60\theta = 60 and r=7r = 7. Simplify the fraction 60360\frac{60}{360} to 16\frac{1}{6} and multiply.

Problem 2:

In a cyclic quadrilateral ABCDABCD, angle A=85∘A = 85^\circ. Find the size of the opposite angle CC.

Solution:

180∘−85∘=95∘180^\circ - 85^\circ = 95^\circ

Explanation:

According to the property of cyclic quadrilaterals, opposite angles are supplementary, meaning they add up to 180∘180^\circ.

Problem 3:

A triangle is drawn inside a circle where one side is the diameter. If one of the other angles is 35∘35^\circ, find the third angle.

Solution:

180∘−(90∘+35∘)=55∘180^\circ - (90^\circ + 35^\circ) = 55^\circ

Explanation:

The property 'angle in a semi-circle' states the angle opposite the diameter is 90∘90^\circ. Since the sum of angles in a triangle is 180∘180^\circ, we subtract the known angles from 180∘180^\circ.