Plane Areas & Trapezoidal Rule

Key ideas

• Rectangle: \(A = lw\)
• Triangle: \(A = \tfrac12 bh\)
• Circle: \(A = \pi r^2\)
• Trapezium: \(A = \dfrac{b_1 + b_2}{2}h\)
• Trapezoidal rule: \[ A \approx \frac{w}{2}\bigl(y_0 + 2(y_1+\dots+y_{n-1}) + y_n\bigr) \]

Example 1 — Field area

The spacing is \(24\text{ m}\). The perpendicular measurements are \(84, 54, 45, 66, 54, 66, 96\). Use the trapezoidal rule to estimate the area of the field.

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Volumes of Prisms & Cylinders

Key ideas

• Prism: \(V = A_{\text{base}}h\)
• Cylinder: \(V = \pi r^2h\)

Example 2 — Triangular prism

The triangular base has area \(24\text{ cm}^2\). The length of the prism is \(15\text{ cm}\). Find the volume of the prism.

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Example 3 — Cylinder

A cylinder has radius \(12\text{ cm}\) and height \(18\text{ cm}\). Find its volume in terms of \(\pi\).

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Spheres, Cones & Composite Solids

Key ideas

• Sphere: \(V = \tfrac{4}{3}\pi r^3\)
• Hemisphere: \(V = \tfrac{2}{3}\pi r^3\)
• Cone: \(V = \tfrac13\pi r^2h\)

Example 4 — Cone & similar cone

A cone has radius \(12\text{ cm}\) and height \(18\text{ cm}\). A smaller similar cone of height \(12\text{ cm}\) is cut from the top. Find the volume of each cone in terms of \(\pi\).

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Example 5 — Hemisphere displacing water

A solid hemisphere of radius \(12\text{ cm}\) is submerged in a cylindrical container of water. The water level rises by \(4\text{ cm}\). Find the radius of the cylinder, correct to two decimal places.

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