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neurologyYou need to know

• The area of a compound shape can be found by splitting it into simpler shapes, finding each area, then adding the areas or subtracting unwanted areas.
• The perimeter of a compound shape is the total length around the outside boundary, including straight sides and any curved edges.
• The formula for the area of a rectangle is `A = lw`, where `l` is length and `w` is width.
• The formula for the area of a triangle is `A = \frac{1}{2}bh`, where `b` is the base and `h` is the perpendicular height.
• The formula for the area of a circle is `A = \pi r^2`, where `r` is the radius.
• The formula for the circumference of a circle is `C = 2\pi r` or `C = \pi d`, where `d` is the diameter.
• A semicircle is half of a circle, so its area is `\frac{1}{2}\pi r^2` and its curved edge is `\pi r`.
• A quarter-circle is one quarter of a circle, so its area is `\frac{1}{4}\pi r^2` and its curved edge is `\frac{1}{2}\pi r`.
• The volume of any prism is `\text{volume} = \text{area of cross-section} \times \text{length}`.
• A triangular prism has a triangular cross-section, so its volume is `\frac{1}{2}bh \times \text{length}`.
• The volume of a cylinder is `V = \pi r^2h`, where `r` is the radius of the circular cross-section and `h` is the height or length of the cylinder.
• The surface area of a prism is the total area of all its faces, including the two identical end faces and the rectangular side faces.
• The curved surface area of a cylinder is `2\pi rh`, and the total surface area of a closed cylinder is `2\pi rh + 2\pi r^2`.
• The length of an arc with angle `\theta` degrees is `\frac{\theta}{360} \times 2\pi r`.
• The area of a sector with angle `\theta` degrees is `\frac{\theta}{360} \times \pi r^2`.
• A segment of a circle is the region between a chord and an arc, and its area is found by subtracting the area of the triangle from the area of the sector.
• The volume of a pyramid is `V = \frac{1}{3} \times \text{area of base} \times \text{perpendicular height}`.
• The volume of a cone is `V = \frac{1}{3}\pi r^2h`, and the volume of a sphere is `V = \frac{4}{3}\pi r^3`.
• The surface area of a cone is `\pi r^2 + \pi rl`, where `l` is the slant height, and the surface area of a sphere is `4\pi r^2`.
• A frustum is made by removing a smaller similar cone or pyramid from a larger cone or pyramid, so its volume is the larger volume minus the smaller volume.

• Calculate the area of a compound shape by decomposing it into rectangles, triangles, circles or part-circles and combining the areas accurately.
• Calculate the perimeter of shapes involving semicircles, quarter-circles or sectors by adding straight lengths and curved arc lengths only on the boundary.
• Calculate the volume of prisms, including triangular prisms and cylinders, by identifying the cross-section area and multiplying by the length or height.
• Calculate the surface area of cuboids, triangular prisms and cylinders by finding every face area once and adding them systematically.
• Calculate the arc length and sector area for any angle by using the fraction `\frac{\theta}{360}` of the full circumference or full circle area.
• Calculate the area of a circular segment by finding the sector area, finding the associated triangle area, and subtracting the triangle from the sector.
• Solve problems involving prisms, pyramids, cones and spheres by selecting the correct formula, substituting measurements with consistent units, and giving answers to an appropriate degree of accuracy.
• Solve surface area and volume problems for compound 3D shapes by splitting them into standard solids or subtracting removed parts such as holes or missing sections.
• Calculate the volume of a frustum by using similar shapes or by subtracting the volume of the smaller removed cone or pyramid from the original larger solid.