Plane Geometry

Q: Two angles of a triangle are 50° and 70°. What is the third angle?

60° — 180 − 50 − 70 = 60°.

Q: What do the interior angles of a hexagon add up to?

720° — 180 × (6 − 2) = 180 × 4 = 720°.

Q: When a line cuts across two parallel lines, what is true of a pair of alternate angles?

They are equal — Alternate angles formed by parallel lines are equal.

Q: What is the area of a circle with a radius of 5 cm? (Use π ≈ 3.14)

78.5 cm² — π × 5² = 3.14 × 25 = 78.5 cm².

Q: What is the circumference of a circle with a diameter of 12 cm? (Use π ≈ 3.14)

37.68 cm — 2π × 6 = 3.14 × 12 = 37.68 cm.

Q: An exterior angle of a triangle is 110°. What do the two interior angles that are not next to it add up to?

110° — An exterior angle equals the sum of the two interior angles not next to it, so the sum is 110°.

Q: What is one interior angle of a regular octagon?

135° — The interior angles add to 180 × (8 − 2) = 1080°, so one angle is 1080 ÷ 8 = 135°.

Q: When two lines cross, what is true of a pair of vertical angles?

They are equal — Vertical angles are always equal.

Where this is used

Building plans, game graphics, the way Google Maps measures roads — none of it works without plane geometry. How angles behave, how the angles inside a polygon add up, how to measure a circle: these are everyday tools for engineers and designers alike. The rule "the angles inside an n-sided shape add to 180(n−2)°" in particular is exactly what computer graphics uses when it works with polygons.

Key points

Vertical, corresponding, and alternate angles

When two lines cross, the angles opposite each other are called vertical angles, and they are equal. When a third line cuts across two parallel lines, angles in matching positions are corresponding angles (equal), and angles on opposite sides of the crossing line are alternate angles (also equal).

The sum of the interior angles of a polygon

The interior angles of a shape with n sides add up to 180(n−2)°. A triangle gives 180°, a quadrilateral 360°, and a pentagon 540°.

Interior and exterior angles of a triangle

The three angles inside a triangle add up to 180°. Also, an exterior angle of a triangle equals the sum of the two interior angles that are not next to it.

Properties of a circle

The distance around a circle (its circumference) is 2πr, and its area is πr². The diameter is d = 2r. The number π is about 3.14159…

The exterior angles of a regular polygon

For any convex polygon, the exterior angles always add up to 360°. In a regular n-sided polygon each exterior angle is 360°/n. That is exactly how much a robot turns at each corner as it walks once around the shape.

Check yourself 10questions

Tap an option to check

Q1 Two angles of a triangle are 50° and 70°. What is the third angle?

Q2 What do the interior angles of a hexagon add up to?

Q3 When a line cuts across two parallel lines, what is true of a pair of alternate angles?

Q4 What is the area of a circle with a radius of 5 cm? (Use π ≈ 3.14)

Q5 What is the circumference of a circle with a diameter of 12 cm? (Use π ≈ 3.14)

Q6 An exterior angle of a triangle is 110°. What do the two interior angles that are not next to it add up to?

Q7 What is one interior angle of a regular octagon?

Q8 When two lines cross, what is true of a pair of vertical angles?

Q9 What is one exterior angle of a regular hexagon?

Q10 What do the interior angles of a pentagon add up to?

Jobs that use this

Architect $80k

Computer graphics engineer (games and film) $100k

Land surveyor $70k

Architect$80k

Works out angles, areas, and curves in building plans to design structures that are both safe and good to look at.

Maths used: Plane geometryTrigonometric ratiosArea calculationsSolid geometry

Computer graphics engineer (games and film)$100k

Builds 3D models out of polygons (collections of triangles) and computes the angles and surface normals that make them look right.

Maths used: Plane geometryVectorsMatrix transformationsTrigonometric functions

Land surveyor$70k

Measures the area, boundaries, and height changes of land precisely, using calculations of angles and distances.

Maths used: Plane geometryTrigonometric ratiosCoordinate geometryError analysis

Where this is used

Key points

Check yourself 10questions

Jobs that use this

Comments