Geometry
Shapes, graphs and trigonometry, constructing figures, plotting graphs, and calculating angles and coordinates.
Trapezium & Composite Areas
Volume & Surface Area of a Cylinder
Cone, Pyramid & Sphere Volumes
Pythagoras in Two and Three Dimensions
Sine, Cosine & Tangent
The Unit Circle & Angles Beyond 90 Degrees
Rectangular & Polar Coordinates
Geometry, summary notes
- Trigonometry relates the angles of a right-angled triangle to the ratios of its sides, remembered as SOH-CAH-TOA.
- Pythagoras' theorem links the three sides of a right-angled triangle and is used constantly for resultant forces and diagonal distances.
- Graphs of equations and functions show the relationship between variables; the gradient of a straight line is its rate of change.
- Rectangular (x, y) and polar (r, θ) coordinates describe the same point two ways. Polar form is natural for anything expressed as a magnitude and a direction, a force, a wind vector, an AC phasor.
- Angles convert between degrees and radians; radians are the natural unit for rotational and phase calculations.
- Trigonometric ratios
- sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj
- Pythagoras
- a² + b² = c² (c = hypotenuse)
- Rectangular → polar
- r = √(x² + y²), θ = tan⁻¹(y / x)
- Polar → rectangular
- x = r cos θ, y = r sin θ
- Degrees ↔ radians
- radians = degrees × π/180
- Circle
- circumference = 2πr, area = πr²
A right-angled triangle has legs of 6 cm and 8 cm. Find the hypotenuse.
c² = 6² + 8² = 36 + 64 = 100, so c = √100 = 10 cm.
A 5 m ladder leans against a wall at 60° to the ground. How far up the wall does it reach?
The height is the opposite side: sin 60° = height / 5, so height = 5 × 0.866 = 4.33 m.
Convert the rectangular coordinate (6, 8) to polar form.
r = √(6² + 8²) = √100 = 10. θ = tan⁻¹(8/6) = tan⁻¹(1.333) = 53.13°. So (10, 53.13°).
Geometry concept map
Geometry
Geometry quiz
Geometry
1. In the triangle shown, the two shorter sides are 3 cm and 4 cm. The length of the side marked ? is:
5 cm7 cm6 cm2. In the right-angled triangle shown, the hypotenuse is 10 cm and the marked angle is 30°. The side opposite that angle, marked ?, is:
5 cm8.66 cm5.77 cm3. In the circle shown, which line is the chord?
CAD4. The L-shaped plate shown is 8 units across the top and 5 down the left side, with a 4 by 2 step cut from the lower right. Its area is:
32 square units40 square units28 square units5. The straight line plotted passes through (1,2) and (3,6). Its gradient is:
240.56. The cylinder shown has a diameter of 10 cm and a height of 20 cm. Its volume, to the nearest 10 cm³, is:
1570 cm³6280 cm³3140 cm³7. The triangle shown is isosceles with an apex angle of 40°. Each base angle x is:
70°50°40°8. In the figure the two horizontal lines are parallel and one angle is 65°. The angle marked y is:
65°115°25°9. The shaded sector shown has a radius of 6 cm and subtends 60° at the centre. Its area, to one decimal place, is:
18.8 cm²113.1 cm²6.3 cm²10. The rectangle shown is 8 units long and 6 units high. Its diagonal is:
10 units14 units48 units11. The two triangles shown are similar. The smaller has sides 3 and 4, and the larger has the corresponding side 6. The side marked ? is:
871212. The trapezium shown has parallel sides of 10 and 6 units and a perpendicular height of 4 units. Its area is:
32 square units60 square units40 square units13. The cone shown has a base radius of 3 cm and a perpendicular height of 4 cm. Its volume, to one decimal place, is:
37.7 cm³113.1 cm³12.6 cm³14. The sphere shown has a radius of 3 cm. Its volume, to one decimal place, is:
113.1 cm³36.0 cm³28.3 cm³15. In the right-angled triangle shown the side opposite θ is 5 and the adjacent side is 12. The angle θ, to one decimal place, is:
22.6°67.4°24.6°
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