Perimeter and area of 2D shapes

Geometry and measures • Mensuration and calculation

Flashcards

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Area formula for an equilateral triangle given side s?

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Area = (sqrt(3)/4) × s² because height equals (sqrt(3)/2) × s.

Key concepts

What you'll likely be quizzed about

Area of a triangle

A triangle's area depends on a base and the perpendicular height from that base to the opposite vertex. The formula is area = 1/2 × base × height, because a triangle equals half of a parallelogram with the same base and height. The height must be perpendicular to the chosen base; if the given length is not perpendicular, calculation requires trigonometry or rearrangement into a right-angled component. Units for area are square units and must be stated.

Area of a parallelogram

A parallelogram's area equals base × perpendicular height because opposite sides are parallel and the shape can be rearranged into a rectangle of the same base and height. The height is the perpendicular distance between the parallel sides; slanted side length is not interchangeable with height. Accurate identification of the perpendicular height is essential when sides are given at an angle. Units remain square units.

Area of a trapezium (trapezoid)

A trapezium's area depends on the two parallel sides (often called bases) and the perpendicular height between them. The formula is area = 1/2 × (sum of parallel sides) × height because the shape averages the two base lengths and multiplies by the height. The height must be perpendicular to the parallel sides; sloping distances between non-parallel edges do not count. Clear labelling of the parallel sides prevents formula misuse.

Circumference and area of a circle

A circle's circumference measures its perimeter and follows the formula circumference = 2πr = πd where r is the radius and d is the diameter, because diameter equals twice the radius. A circle's area follows the formula area = πr² because the circle area derives from integrating rings or rearranging sectors into an approximate rectangle of height r and length proportional to the circumference. Use the same unit system for radius and diameter; area units are square units and use the same unit squared.

Perimeters, composite shapes and strategies

Perimeter calculations require summing all boundary lengths. For polygons, add the side lengths. For circles, use the circumference formula. Composite shapes require decomposition into simple components; perimeters may require careful consideration of shared boundaries that do not contribute to the external perimeter. Area calculations for composite shapes require partitioning into familiar shapes, calculating each area, and combining results with addition or subtraction depending on overlap or holes. Careful diagram labelling and unit consistency prevent errors.

Key notes

Important points to keep in mind

Always use the perpendicular height in area formulas; slanted lengths do not substitute.

State and keep consistent units; convert units before calculating area or perimeter.

For triangles, identify the base that pairs with the perpendicular height used.

For trapezia, use the two parallel sides in the average inside the formula.

For circles, use circumference = 2πr = πd and area = πr²; substitute r = d/2 when needed.

Decompose composite shapes into standard shapes; calculate areas separately then add or subtract.

Do not include internal shared edges when calculating the external perimeter of composite shapes.

Use exact values with π where required, and give decimal approximations only when instructed.

Label diagrams clearly with which lengths are bases, heights, radii and diameters.

Check final answers by estimating with simple bounding shapes to confirm scale.