Year 10 Maths: Surface Area & Volume Cheat Sheet

This cheat sheet covers essential Year 10 Australian Mathematics concepts related to surface area and volume of various 3D shapes, including prisms, pyramids, cones, spheres, and composite solids, alongside algebraic simplification and expansion.

Core Principles

• Surface area calculations for prisms, pyramids, cones, spheres.
• Volume calculations for prisms, cylinders, pyramids, cones, spheres.
• Solving problems involving composite solids (combinations of shapes).
• Simplifying and expanding algebraic expressions.
• Working with algebraic fractions and indices.

Action Steps

• Identify all component shapes in a composite solid.
• Calculate the surface area/volume of each component shape.
• Adjust for overlapping surfaces (subtracting areas that are internal).
• Sum the relevant surface areas or volumes.
• Simplify algebraic fractions by finding common factors.
• Expand algebraic expressions using distributive property or FOIL.
• Apply index laws for simplification.

Formulas

• Surface Area of a Sphere: $A = 4\pi r^2$
• Volume of a Sphere: $V = \frac{4}{3}\pi r^3$
• Surface Area of a Cone: $A = \pi r(r + \sqrt{h^2 + r^2})$
• Volume of a Cone: $V = \frac{1}{3}\pi r^2 h$
• Volume of a Pyramid: $V = \frac{1}{3} \times \text{Base Area} \times h$
• Volume of a Cylinder: $V = \pi r^2 h$
• Surface Area of a Cylinder: $A = 2\pi r(r + h)$

Key Terms

• Composite Solid: A 3D shape made up of two or more simpler 3D shapes.
• Right Pyramid: A pyramid where the apex is directly above the centroid of the base.
• Slant Height: The height of a triangular face of a pyramid or the distance from the apex to a point on the circumference of the base of a cone.
• Algebraic Fraction: A fraction where the numerator and/or denominator contain algebraic expressions.
• Indices: Exponents indicating the number of times a base number is multiplied by itself.

Real World Examples

• Calculating the amount of paint needed for a building shaped like a cylinder topped with a cone.: Surface area of composite solids.
• Determining the capacity of a silo composed of a cylinder and a hemisphere.: Volume of composite solids.
• Simplifying expressions in physics formulas or engineering calculations.: Algebraic manipulation.

People

• Pythagoras: Associated with the Pythagorean theorem ($a^2 + b^2 = c^2$), crucial for finding slant heights in pyramids and cones.