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Areas of Parallelograms & Triangles

Areas of a Parallelogram and a Triangle

What are the Formulas for the Area of a Triangle and a Parallelogram?

To calculate the exact surface area of geometric shapes without counting grids, we use specific mathematical formulas based on two crucial measurements: the base and the height.

  • The base is any chosen side of the shape on which it appears to stand.
  • The height (also known as the altitude) is the shortest perpendicular straight line drawn from the opposite corner down to that chosen base line, meeting it at a right angle (90°).

Let us break down the formulas for our two key shapes:

  1. Area of a Parallelogram: A parallelogram is a four-sided shape where opposite sides run parallel to each other. The area is simply found by multiplying the base by the corresponding perpendicular height.
Formula: Area = base × height
  1. Area of a Triangle: A triangle can be thought of as exactly half of a parallelogram. If you draw a diagonal line through a parallelogram, it splits it into two congruent triangles. Because it is half the size, its area formula includes a fraction of a half.
Formula: Area = 0.5 × base × height

A crucial rule to remember is that the height must always match its corresponding base. If you use a different side as your base, you must use the specific altitude that drops vertically onto that new side.

Triangles on Same Base & Between Same Parallelsl₁l₂Common base BCADBCTheorem: Triangles on the same base BC and betweenthe same parallels l₁ and l₂ have EQUAL areas.ar(△ABC) = ar(△DBC) — same base, same height!
Example 1: Parallelogram base 8 cm, height 5 cm. Area?
8 × 5 = 40 cm².
Example 2: Triangle base 10 cm, height 6 cm. Area?
½ × 10 × 6 = 30 cm².
Quick recap
  • Parallelogram area = base × height.
  • Triangle area = ½ × base × height (perpendicular height).
✓ Quick check
If the diagonals of a quadrilateral divide it into four triangles of equal area, then the quadrilateral must be a:
A quadrilateral in which diagonals divide it into four triangles of equal area is necessarily a parallelogram.
If a regular hexagon is divided into 6 equilateral triangles by its diagonals, and the area of one triangle is 10 cm², the area of the hexagon is:
A regular hexagon is made of 6 identical equilateral triangles. Area = 6 × 10 = 60 cm².

Same Base, Same Parallels

What are the Area Relationships Between Shapes on the Same Base and Parallels?

Mathematicians discovered a fascinating rule when shapes share the exact same base line and sit nestled between the same pair of parallel lines.

Let us explore these key relationship rules:

  1. Parallelograms on the Same Base: If two or more parallelograms share the same base line and lie inside the same parallel lines, their surface areas will be perfectly equal. Why? Because they share the exact same base length, and the distance between the parallel lines ensures their vertical heights are completely identical!
  2. Triangles on the Same Base: Similarly, two triangles sharing the same base line and lying between the same parallel lines will have completely equal areas.
  3. The Triangle and Parallelogram Mix: If a triangle and a parallelogram sit on the same base line and lie between the same parallel lines, the area of the triangle will be exactly half of the area of the parallelogram.
Formula: Area(Triangle) = 0.5 × Area(Parallelogram)

These theorems are incredibly helpful because they allow you to instantly determine the area of a highly distorted, slanted shape simply by comparing it to a simpler straight shape next to it!

Area Theorems — Key ResultsMedian divides △ into 2 equal area trianglesIf AD is median of △ABC, ar(△ABD) = ar(△ACD)1Diagonal of parallelogram bisects areaDiagonal AC divides ▱ABCD into 2 equal △s2Same base, same parallels → equal areaar(△ABP) = ar(△ABQ) if P, Q on same parallel3Converse: Equal area + same base → same parallelsIf equal area on same base → vertices are collinear4Remember: Area of rhombus = ½ × d₁ × d₂ (product of diagonals)Parallelogram and rectangle on same base and height have equal area
Example 1: A triangle and a parallelogram share a base and lie between the same parallels; the parallelogram's area is 48 cm². The triangle's?
Half of 48 = 24 cm².
Example 2: Two parallelograms, same base and same parallels. Compare areas.
They are equal in area.
Quick recap
  • Parallelograms on same base + parallels are equal in area.
  • A triangle is half such a parallelogram.
✓ Quick check
A parallelogram and a square have the same area. If the side of the square is 12 cm and the base of the parallelogram is 18 cm, the height of the parallelogram is:
Area of square = 12 × 12 = 144 cm². Area of parallelogram = 18 × h. So, 18h = 144. h = 8 cm.
A triangle and a parallelogram are on the same base and between the same parallels. If the area of the parallelogram is 64 cm², what is the area of the triangle?
ABCDEcommon base AB
Area of triangle = ½ × Area of parallelogram = ½ × 64 = 32 cm².

Medians and Equal Areas

A median of a triangle joins a vertex to the midpoint of the opposite side, and it divides the triangle into two triangles of equal area.

The three medians meet at the centroid, which splits the triangle into six smaller triangles of equal area and divides each median in the ratio 2 : 1.

Example 1: A median divides a 36 cm² triangle. Area of each part?
Equal halves, so 18 cm² each.
Example 2: In what ratio does the centroid divide a median?
2 : 1, measured from the vertex.
Quick recap
  • A median splits a triangle into two equal areas.
  • The centroid divides each median 2 : 1.
✓ Quick check
A school decides to plant trees in a triangular area of the playground. The base is 30 m and altitude is 10 m. If 1 tree needs 3 m² of space, how many trees can be planted?
Area of triangular area = ½ × 30 × 10 = 150 m². Number of trees = 150 / 3 = 50.
An architect is designing a roof truss in the shape of a triangle. The medians of the triangle intersect at the centroid G. If the total area of the triangular truss is 36 m², what is the area of the triangle formed by any two vertices and the centroid?
The centroid divides a triangle into 3 smaller triangles of equal area. Area = 36 / 3 = 12 m².
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