Worked Examples: Geometry & Trigonometry

Example 1: Volume Application (Easy–Medium)

A cylindrical water tank has a diameter of 8 feet and a height of 12 feet. What is the volume, in cubic feet, of water the tank can hold?

Solution: Radius = diameter / 2 = 4 feet.

$V = \pi r^2 h = \pi(4)^2(12) = 192\pi \approx 603.2$ cubic feet.

SAT trap: Using diameter instead of radius gives $\pi(8)^2(12) = 768\pi$ — always a distractor.

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Example 2: SOH-CAH-TOA in Context (Medium)

From a point on the ground 50 meters from the base of a building, the angle of elevation to the top of the building is $38°$. What is the height of the building to the nearest meter?

Solution: The height is opposite the angle, and 50 m is adjacent.

$\tan(38°) = \frac{h}{50}$

$h = 50 \tan(38°) \approx 50(0.7813) \approx 39 \text{ meters}$

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Example 3: Circle Equation (Medium–Hard)

A circle in the $xy$-plane has the equation $x^2 + y^2 + 10x - 2y = 10$. What is the radius?

Solution: Complete the square for both variables:

• $x^2 + 10x + 25 = (x + 5)^2$ (add 25)
• $y^2 - 2y + 1 = (y - 1)^2$ (add 1)

$(x + 5)^2 + (y - 1)^2 = 10 + 25 + 1 = 36$

Radius $= \sqrt{36} = 6$.

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Example 4: Arc Length (Medium)

A circle has radius 10 cm. What is the length of an arc subtended by a central angle of $72°$?

Solution: $\text{Arc length} = \frac{72}{360} \times 2\pi(10) = \frac{1}{5} \times 20\pi = 4\pi \approx 12.57 \text{ cm}$

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Example 5: Coordinate Geometry (Hard)

Points $A(1, 3)$ and $B(7, 11)$ are endpoints of a diameter of a circle. What is the equation of the circle?

Solution: Center = midpoint of $AB$: $\left(\frac{1+7}{2}, \frac{3+11}{2}\right) = (4, 7)$

Radius = half the distance: $d = \sqrt{(7-1)^2 + (11-3)^2} = \sqrt{36 + 64} = \sqrt{100} = 10$, so $r = 5$.

Equation: $(x - 4)^2 + (y - 7)^2 = 25$