Calculate the area and perimeter of triangles, quadrilaterals, and circles, and combine formulas for composite shapes.
When all three sides of a triangle are known but not the height, Heron's formula finds the area: Area=√[s(s−a)(s−b)(s−c)], where s is the semi-perimeter.
Find the area of a triangle with sides 5, 6, 7 cm using Heron's formula.
s=(5+6+7)/2=9; area=√[9×4×3×2]=√216≈14.70 cm²
Find the area of a triangle with sides 3, 4, 5 cm.
s=6; area=√[6×3×2×1]=√36=6 cm²
The area of composite figures is found by breaking them into simpler shapes (triangles, rectangles, circles) and adding or subtracting their areas.
A rectangle 10m×6m has a semicircle of radius 3m attached to one side. Find the total area (π≈3.14).
Rectangle=60; semicircle=½×3.14×9=14.13; total≈74.13 m²
A square field of side 20m has a circular pond of radius 5m removed. Find the remaining area (π≈3.14).
Square=400; circle=3.14×25=78.5; remaining=321.5 m²
1. Find the area of a triangle with sides 8, 15, 17 cm using Heron's formula.
2. Find the perimeter of a triangle with sides 9, 12, 15 cm.
3. Find the circumference of a circle with radius 7 cm (use π=22/7).
4. A rectangular garden 15m×8m has a triangular flowerbed (base 6m, height 4m) added. Find the total area.
5. Find the area of a triangle with sides 6, 8, 10 cm using Heron's formula.