Area of a Trapezium and a Polygon
ICSE Class 8 · Mathematics · Chapter 20
What this chapter covers
- Area: the shapes you already know
- The trapezium and its parts
- Finding the area of a trapezium on a square grid
- The formula for the area of a trapezium
- Working backwards with the trapezium formula
- Polygons and the idea of dividing a shape
- Dividing a quadrilateral using a diagonal
- Areas of other polygons by division
- Area of a semicircle
- Choosing a method
Key points
- Area is the surface enclosed by a closed figure, measured in square units.
- A trapezium has exactly one pair of parallel sides; its height is the perpendicular distance between them.
- Area of a trapezium = (1)/(2) × (sum of parallel sides) × height, or A = (1)/(2)(a + b)h.
- Two identical trapeziums make a parallelogram; cutting and moving corner triangles turns a trapezium into a rectangle of equal area.
- If three of a, b, h and A are known, the formula gives the fourth.
- The area of a polygon is found by dividing it into triangles, rectangles and trapeziums and adding their areas, or by subtracting from a larger shape.
- A quadrilateral with diagonal d and perpendiculars h1 and h2 has area (1)/(2)d(h1 + h2); a rhombus or kite has area (1)/(2)d1 d2.
- Area of a semicircle = (1)/(2)π r2; perimeter of a semicircle = π r + 2r.
- Keep all lengths in one unit, and check that the answer is sensible.
Questions answered
Define a trapezium. Which measurement of a trapezium is called its height?
A trapezium is a quadrilateral with exactly one pair of parallel sides. Its height is the perpendicular distance between the two parallel sides.
The parallel sides of a trapezium are 14 cm and 10 cm, and the distance between them is 7 cm. Find its area.
(1)/(2)(14 + 10) × 7 = 12 × 7 = 84 cm2.
Why does placing two identical trapeziums together, one turned upside down, help you find the area of one trapezium?
The two pieces form a parallelogram with base (a + b) and the same height h. Its area is (a + b)h, and one trapezium is half of it.
A trapezium has area 150 m2 and parallel sides 18 m and 12 m. Find the distance between the parallel sides.
150 = (1)/(2)(18 + 12)h = 15h, so h = 10 m.
A student uses the slanting side of 13 cm as the height of a trapezium whose true height is 12 cm. Will the answer be too large or too small? Explain.
Too large, because the slanting side is longer than the perpendicular distance between the parallel sides.
A quadrilateral field has a diagonal of 30 m. The perpendiculars to it from the other two corners are 9 m and 7 m. Find its area.
(1)/(2) × 30 × (9 + 7) = 15 × 16 = 240 m2.
The diagonals of a rhombus are 16 cm and 12 cm. Find its area.
(1)/(2) × 16 × 12 = 96 cm2.
Find the area and the perimeter of a semicircle of radius 14 cm. Use π = (22)/(7).
Area = (1)/(2) × (22)/(7) × 14 × 14 = 308 cm2. Perimeter = (22)/(7) × 14 + 28 = 44 + 28 = 72 cm.