Triangle Area: Sides 5, 12, 13
Calculate the area of a triangle with sides 5, 12, and 13 using Heron's formula.
Area:
Area = 30 square units
Step-by-Step Solution
Step 1
Given Sides
a = 5
b = 12
c = 13
Step 2
Heron's Formula
Area = √[s(s-a)(s-b)(s-c)]
where s = (a + b + c) / 2
Step 3
Calculate Semi-perimeter
s = (a + b + c) / 2
s = (5 + 12 + 13) / 2
s = 15
Step 4
Calculate Factors
s - a = 15 - 5 = 10
s - b = 15 - 12 = 3
s - c = 15 - 13 = 2
Step 5
Calculate Product
s(s-a)(s-b)(s-c)
= 15 × 10 × 3 × 2
= 900
Step 6
Calculate Area
Area = √900
Area = 30
Step 7
Additional Properties
Perimeter = 30
Inradius (r) = Area / s = 2
Circumradius (R) = abc / (4·Area) = 6.5
Step 8
Final Answer
Area = 30
Heron's Formula
For a triangle with sides a, b, and c:
s = (a + b + c) / 2 (semi-perimeter)
Area = √[s(s-a)(s-b)(s-c)]
For this triangle: s = 15