Area of a Triangle - Mathematics Form 2 Notes

• Area of a Triangle Given Two Sides and an Included Angle
• Area of the triangle, given the three sides.
• Past KCSE Questions on the Topic.

Area of a Triangle Given Two Sides and an Included Angle

The area of a triangle is given by A = ¹/₂bh but sometimes we use other formulas too as follows.

If the length of two sides and an included angle of a triangle are given, the area of the triangle is given by A = ¹/₂ absinθ

Example

In the figure above PQ is 5 cm and PR is 7 cm angle QPR is 50⁰. Find the area of the the triangle.

Solution

Using the formulae by A = ¹/₂absinθ a= 5 cm b = 7 cm and θ = 50⁰
Area = ¹/₂ x 5 x 7 sin 50⁰
=2.5 x 7 x 0.7660
=13.40 cm²

Area of the Triangle, Given the Three Sides.

Example

Find the area of a triangle ABC in which AB = 5 cm, BC = 6 cm and AC =7 cm.

Solution

When only three sides are given us the formulae

A = √[s(s−a)(s−b)(s−c)] Hero’s formulae
S = ¹/₂ of the perimeter of the triangle
= ¹/₂(a + b + c) a, b, c are the lengths of the sides of the triangle.

= ¹/₂(6+7+5) = 9 And A =√[9(9−6)(9−7)(9−5)]
=√(9 x 3 x 2 x 4)
= √216
= 14.70 cm²

Past KCSE Questions on the Topic.

1. The sides of a triangle are in the ratio 3:5:6. If its perimeter is 56 cm, use the Heroes formula to find its area (4mks)
2. The figure below is a triangle XYZ. ZY = 13.4cm, XY = 5cm and angle xyz = 57.7^o
   
   
   1. Calculate
      
      1. Length XZ. (3mks)
      2. Angle XZY. (2 mks)
   2. If a perpendicular is dropped from point X to cut ZY at M, Find the ratio MY:ZM.(3 mks)
   3. Find the area of triangle XYZ. (2 mks)