Mathematics
Grade 10
15 min
Area of parallelograms and triangles
Area of parallelograms and triangles
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1
Introduction & Learning Objectives
Learning Objectives
Derive the area formulas for parallelograms and triangles.
Calculate the area of any parallelogram given its base and corresponding height.
Calculate the area of any triangle given its base and corresponding height.
Apply trigonometric principles to find the area of a parallelogram or triangle when the height is not directly given.
Algebraically solve for a missing dimension (base or height) of a parallelogram or triangle when the area is known.
Solve multi-step problems involving composite figures made of parallelograms and triangles.
Ever wonder how architects calculate the amount of glass needed for a slanted skyscraper facade or how engineers determine forces in a bridge truss? šļø It all starts with mastering the area of parallelograms and triangles!...
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Key Concepts & Vocabulary
TermDefinitionExample
ParallelogramA quadrilateral with two pairs of parallel opposite sides. The opposite sides and opposite angles are equal.A standard rhombus, rectangle, or square. Imagine a rectangle pushed slightly to one side; the resulting shape is a parallelogram.
BaseAny side of a polygon, but typically a side that is oriented horizontally or one to which a height is drawn.In a triangle resting on a flat surface, the side touching the surface is commonly considered the base.
Altitude (Height)A line segment that is perpendicular to the base, drawn from the opposite vertex (for a triangle) or the opposite side (for a parallelogram). Its length is the height.The vertical distance from the peak of a triangular roof straight down to the floor.
AreaThe measure of the two-dimensional s...
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Core Formulas
Area of a Parallelogram
A = b \cdot h
The area (A) of a parallelogram is the product of its base (b) and its corresponding perpendicular height (h). Use this when you know the length of a base and the height that forms a right angle with that base.
Area of a Triangle
A = \frac{1}{2} \cdot b \cdot h
The area (A) of a triangle is one-half the product of its base (b) and its corresponding perpendicular height (h). This formula is derived from the fact that a diagonal divides a parallelogram into two congruent triangles.
Area of a Triangle using Trigonometry
A = \frac{1}{2}ab \cdot \sin(C)
The area (A) of a triangle is one-half the product of the lengths of two sides (a and b) and the sine of their included angle (C). This is extremely useful when the perpendicular heigh...
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Challenging
The area of a triangle is 96 cmĀ². The lengths of two of its sides are 16 cm and 20 cm. What is the sine of the included angle between these two sides?
A.0.5
B.0.8
C.0.6
D.1.2
Challenging
A rhombus is a type of parallelogram with four equal sides. If a rhombus has a side length of 10 units and one of its interior angles is 45Ā°, what is its area?
A.100 unitsĀ²
B.50ā2 unitsĀ²
C.50 unitsĀ²
D.100ā2 unitsĀ²
Challenging
Why is the area of a parallelogram calculated as base times PERPENDICULAR height (b*h) and not base times slant height (b*s)?
A.Because the Pythagorean theorem only works with perpendicular lines.
B.Because it is a convention that is easier to remember.
C.Because a parallelogram can be rearranged into a rectangle with sides 'b' and 'h', and the area of that rectangle is b*h.
D.Because the slant height is always longer, which would make the area too large.
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