A diagram or formula becomes useful when you connect it to the quantities asked for. Mark radii, perpendicular lengths and corresponding sides. Distinguish length, area and volume: they scale differently and use different units.
A reliable method
Sketch and label
A radius goes from the centre to a point on a circle. A triangle’s height is perpendicular to its base. In a right triangle, the hypotenuse is opposite the right angle.
Choose the relationship
Use a² + b² = c² for right triangles. Use sin θ = opposite/hypotenuse and cos θ = adjacent/hypotenuse. The circle equation (x − h)² + (y − k)² = r² gives centre (h,k) and radius r.
Track scale and angle units
A length factor k gives area factor k² and volume factor k³. Arc length is rθ when θ is in radians, or the fraction θ/360 of the circumference when θ is in degrees.
Check a physical constraint
Lengths and volumes are positive. The hypotenuse is the longest side. A sector cannot exceed the whole circle when its angle is between 0° and 360°.
Worked examples
A cone has radius 3 cm and height 4 cm. A similar cone has 27 times its volume. What is the larger cone’s height?
- For similar solids, volume changes by the cube of the length factor.
- k³ = 27 gives k = 3.
- Multiply the original height by 3: 4 × 3 = 12 cm.
Answer: 12 cm
The circle x² + y² − 6x + 4y − 12 = 0 has what radius?
- Group and complete both squares: (x − 3)² − 9 + (y + 2)² − 4 = 12.
- Rearrange: (x − 3)² + (y + 2)² = 25.
- The right side is r², so r = 5. The centre is (3,−2).
Answer: 5
Now use the method yourself.
Mark this guide as read, then use the method in topic practice before the checkpoint.