Circles & Angle Relationships
Circle Parts
Draw a line to match each term to its definition.
Radius and Diameter
Circle the correct answer.
Diameter = 10 cm. Radius = ?
Radius = 7 m. Diameter = ?
Diameter = 24 mm. Radius = ?
Circumference
Use C = πd ≈ 3.14 × d. Circle the correct answer (round to 1 decimal place).
Diameter = 5 cm
Radius = 4 m
Diameter = 10 cm
Angle Relationships
Draw a line from each angle relationship to its description.
Find the Missing Angle in a Triangle
Angles in a triangle add to 180°. Find the missing angle.
Angles: 60°, 70°, ?
Angles: 90°, 45°, ?
Angles: 30°, 110°, ?
Isosceles triangle, one angle = 40°. Other base angle = ?
Angles on Lines
Find the missing angle.
Angles on a straight line: 55° + ? = 180°
Vertically opposite to 72°
Complementary to 38°
Geometry Problems
Show your working for each problem.
A circular running track has diameter 80 m. How far do you run in one lap? Use π ≈ 3.14.
A triangle has angles in ratio 1:2:3. Find all three angles.
Two angles on a straight line are in ratio 2:1. Find each angle.
Parallel Lines — Alternate and Co-interior Angles
When two parallel lines are cut by a transversal: alternate angles are equal; co-interior angles add to 180°.
Alternate angle to 65° (parallel lines)
Co-interior angle paired with 110° (parallel lines)
Corresponding angle to 48° (parallel lines)
Co-interior angle paired with 73°
Classify Angle Types
Sort each angle into the correct category.
Exterior Angles
An exterior angle of a triangle equals the sum of the two non-adjacent interior angles.
Interior angles 50° and 70°. Exterior angle = ?
Interior angles 35° and 85°. Exterior angle = ?
Exterior angle = 140°. One interior angle = 60°. Other interior angle = ?
Arc Length
Arc length is a fraction of the full circumference. Arc length = (angle ÷ 360) × π × d.
A circle has diameter 20 cm. Find the arc length for a sector with angle 90°. Use π ≈ 3.14.
A circle has radius 6 m. Find the arc length for a sector with angle 120°. Use π ≈ 3.14.
Circles and Angles in Real Life
Explore circles and angle relationships in your environment.
- 1Find 3 circular objects at home (e.g., a plate, a lid, a coin). Measure each diameter and calculate its circumference using C = π × d.
- 2Draw a set of parallel lines on paper with a ruler. Draw a transversal across them. Measure alternate and co-interior angles with a protractor to check the rules.
- 3Look at a clock face. Calculate the arc length the minute hand traces in 15 minutes if its length is 6 cm.
- 4Find a triangle-shaped object. Measure all three angles with a protractor. Do they sum to 180°?
Angle Problems — Triangles
Solve each angle problem. Show your reasoning.
An isosceles triangle has a vertex angle of 100°. What are the two base angles?
A right-angled triangle has one other angle of 35°. What is the third angle?
A triangle has angles in ratio 2:3:5. Find each angle.
Angle Proofs — Straight Line and Triangle
Prove or justify each angle fact.
Explain why all angles on one side of a straight line add to 180°. Use the idea of a half-rotation.
Two lines cross, forming angles a, b, c, d. Angles a and c are vertically opposite. Prove that a = c using the fact that a + b = 180° and b + c = 180°.
Angle Types and Relationships
Circle the correct relationship.
Two angles that add to 90° are called:
Two angles that add to 180° are called:
Angles formed by two intersecting lines and equal to each other are:
Arc Length Calculations
Calculate the arc length for each sector.
A circle has diameter 20 cm. Find the arc length of a 90° sector. Use π ≈ 3.14.
A circle has radius 6 m. Find the arc length of a 120° sector. Use π ≈ 3.14.
A clock has a minute hand of length 10 cm. How far does the tip of the minute hand travel in 20 minutes? Use π ≈ 3.14.
Sort Angle Relationships
Sort each statement into the correct column.
Angles in Polygons
Use the exterior angle theorem and polygon angle sums.
Find the sum of the interior angles of a quadrilateral. Then find the missing angle if three angles are 95°, 80°, and 110°.
Find the sum of the interior angles of a pentagon. If the pentagon is regular, what is each interior angle?
Multi-Step Angle Problems
Solve each multi-step angle problem. Justify each step with a rule.
A diagram shows two parallel lines. A transversal creates angles of 3x + 10° and 2x + 30°. These are alternate angles. Find x and both angles.
In a triangle, two angles are 4x° and 5x°. The third angle is 2x + 10°. Find x and all three angles.
Exterior Angle Theorem
Apply the exterior angle theorem to each problem.
In a triangle, two interior angles are 55° and 65°. Find the exterior angle at the third vertex.
An exterior angle of a triangle is 135°. One of the non-adjacent interior angles is 70°. Find the other non-adjacent interior angle.
Prove why the exterior angle of a triangle equals the sum of the two non-adjacent interior angles. (Hint: angles in a triangle sum to 180° and angles on a straight line sum to 180°.)
Area of a Circle — Introduction
Calculate the area of each circle using A = πr².
A circular garden has radius 4 m. What is its area? Use π ≈ 3.14.
A circular coin has diameter 2.5 cm. Find its area. Use π ≈ 3.14.
A circular pizza has circumference about 75.36 cm. What is its radius, and then its area? Use π ≈ 3.14.
Circle Area — Circle Correct
Circle the correct area. Use π ≈ 3.14.
Circle with radius 6 cm
Circle with diameter 10 m
Circle with radius 2.5 cm
Mixed Geometry Problems
Solve each problem using circles, triangles, and parallel lines.
A semicircular archway has a diameter of 4 m. What is the perimeter of the arch? (Include the diameter as the base.) Use π ≈ 3.14.
A diagram shows two parallel lines cut by a transversal. An alternate angle is labelled as (2x + 5)° and its pair as (x + 35)°. Find x and both angles.
Circumference Sequence
A circle's radius increases by 1 cm each step. Fill in the missing circumferences (use C = 2π r ≈ 6.28r).
Composite Shape — Circle and Rectangle
Find the area and perimeter of each composite shape.
A shape consists of a rectangle 8 cm × 6 cm with a semicircle attached to one short end (diameter = 6 cm). Find the total area and the perimeter. Use π ≈ 3.14.
A circular swimming pool of radius 5 m has a rectangular deck 3 m wide surrounding it on all sides. What is the area of the deck alone? Use π ≈ 3.14.
Sector Area
Calculate the area of each sector.
A sector of a circle with radius 8 cm has an angle of 90°. Find its area. Use π ≈ 3.14.
A sector of radius 6 m has an angle of 120°. Find its area. Use π ≈ 3.14.
Design Challenge — Circular Garden
Apply circles and angles to a garden design problem.
A circular garden of radius 6 m is divided into 4 equal sectors. One sector will be lawn, one will be flowers, one will be vegetables, and one will be paving. Find the area and arc length of each sector.
The paving sector has a path of width 0.5 m running along its two straight edges. What is the total length of paving edging needed?
Proof: Angles in a Triangle Sum to 180°
Construct a geometric proof using parallel lines.
Draw triangle ABC. Through vertex A, draw a line parallel to side BC. Label the angles formed at A as p (alternate to angle B) and q (alternate to angle C). Write a proof explaining why angle A + angle B + angle C = 180°.
Circles in Architecture and Engineering
Apply circle and angle knowledge to real-world contexts.
The Sydney Harbour Bridge arch has a span of approximately 503 m and a rise of about 134 m. Estimate the radius of the circular arc forming the arch. (Hint: use the chord-sagitta formula R ≈ (L²/8h) + h/2 where L = span, h = rise.)
A circular radar system sweeps through 360° in 5 seconds. How many degrees does it sweep per second? If the radar has range (radius) 50 km, what arc length does it sweep per second? Use π ≈ 3.14.
Trigonometry Preview — Right-Angled Triangles
Explore the connection between angles and side ratios in right-angled triangles.
In a right-angled triangle with angles 30°, 60°, and 90° and hypotenuse 10 cm, the sides are 5 cm and 8.66 cm. Calculate the ratio: opposite ÷ hypotenuse for the 30° angle. What do you notice?
Research: what is 'sine' of an angle? How is it related to a right-angled triangle?
Extending — Annulus Area
An annulus is the region between two concentric circles.
A circular path surrounds a pond. The pond has radius 8 m, and the path is 2 m wide. Find the area of the path (annulus). Use π ≈ 3.14.
A metal washer has outer radius 3 cm and inner radius 1 cm. Find its area. Use π ≈ 3.14.
Extending — Bearings and Angles
Bearings are angles measured clockwise from north, used in navigation.
A ship travels on a bearing of 120° for 50 km. Sketch the path and mark the angle from north. What compass direction is closest to this bearing?
If a hiker walks on a bearing of 040° for 3 km, what is the back-bearing (the direction to return to start)?
Two towns are 60 km apart. Town B is due east of Town A. A plane flies from Town A to Town B. What is the bearing?
Extending — Angle Algebra
Write and solve equations to find unknown angles.
In a quadrilateral, the angles are x°, (x + 20)°, (2x − 10)°, and 90°. Find x and all angles.
Two co-interior angles between parallel lines are (5x + 10)° and (4x + 20)°. Find x and both angles.
The base angles of an isosceles triangle are each (3x + 15)° and the apex angle is (x + 10)°. Find x and all three angles.
Extending — Radian Measure Introduction
Radians are another unit for measuring angles used in senior mathematics.
Convert 180° to radians. Convert 90° to radians. Convert π/3 radians to degrees.
A circle has radius 5 cm. An arc subtends an angle of π/2 radians. Find the arc length using: arc = r × θ (where θ is in radians).