Worksheet: Finding the Area of Triangle Using Trigonometry

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In this worksheet, we will practice finding the area of a triangle without having to calculate its perpendicular height.

Q1:

𝐴 𝐡 𝐢 is a triangle, where 𝐡 𝐢 = 1 5 c m , 𝐴 𝐢 = 2 5 c m , and π‘š ∠ 𝐢 = 4 1 ∘ . Find the area of 𝐴 𝐡 𝐢 , giving your answer to three decimal places.

Q2:

Find the area of the figure below giving the answer to three decimal places.

Q3:

𝐴 𝐡 𝐢 is a triangle where 𝐴 𝐡 = 5 c m , 𝐴 𝐢 = 1 1 c m , and π‘š ∠ 𝐴 = 1 0 7 ∘ . Find the area of the triangle giving the answer to two decimal places.

Q4:

In the given figure, work out the area of the triangle to 2 decimal places.

Q5:

Maged’s house is on a corner lot. Find the area of the front yard given that the edges measure 40 ft and 56 ft, as shown in the diagram.

Q6:

The figure shows a triangular field with sides 670 m, 510 m and 330 m. Find the area of the field giving the answer to the nearest square metre.

Q7:

Which of the following is a formula that can be used to find the area of a triangle?

  • A 1 3 π‘Ž 𝑏 𝐢 s i n
  • B 1 4 π‘Ž 𝑏 𝐢 s i n
  • C 1 2 π‘Ž 𝑏 𝐢 c o s
  • D 1 2 π‘Ž 𝑏 𝐢 s i n
  • E 1 4 π‘Ž 𝑏 𝐢 c o s

Q8:

Find the area of a triangle whose two side lengths are 7 cm and 23 cm, and the angle included between them is 1 0 1 ∘ .Give the answer to three decimal places.

Q9:

An isosceles triangle has a side length of 49 cm and a base angle of 4 4 ∘ . Find the area of the triangle giving the answer to three decimal places.

Q10:

𝐴 𝐡 𝐢 is an isosceles triangle where 𝐴 𝐡 = 𝐴 𝐢 = 1 1 c m and π‘š ∠ 𝐴 is 1 5 8 ∘ . Find the area of 𝐴 𝐡 𝐢 giving the answer correct to two decimal places.

Q11:

𝐴 𝐡 𝐢 is a triangle, where ∠ 𝐴 𝐢 𝐡 is obtuse, π‘Ž = 2 0 c m , 𝑏 = 1 4 c m , and the area is 106 cm2. Find the length of 𝐴 𝐡 , giving the answer to two decimal places.

Q12:

𝐴 𝐡 𝐢 𝐷 is a parallelogram where 𝐴 𝐡 = 9 . 3 c m , π‘š ∠ 𝐢 𝐴 𝐡 = 2 6 4 8 β€² ∘ and π‘š ∠ 𝐷 𝐡 𝐴 = 6 6 3 0 β€² ∘ . Find the area giving the answer to two decimal places.

Q13:

𝐴 𝐡 𝐢 is a triangle where π‘Ž = 2 3 c m , 𝑏 = 2 1 c m , the area is 167 cm2 and the radius of the circumcircle is 17 cm. Find length 𝑐 giving the answer to two decimal places.

Q14:

𝑋 π‘Œ 𝑍 is a triangle where π‘š ∠ 𝑋 = 3 5 ∘ , π‘š ∠ π‘Œ = 6 2 ∘ and the area is 480 cm2. Find the length of π‘Œ 𝑍 giving the answer to the nearest centimetre.

Q15:

𝐴 𝐡 𝐢 is an acute-angled triangle where 𝐴 𝐢 = 3 0 c m , s i n 𝐴 = 0 . 7 and the area is 315 cm2. Find π‘š ∠ 𝐡 , giving the answer to the nearest minute.

  • A 2 2 1 3 β€² ∘
  • B 4 4 2 6 β€² ∘
  • C 4 5 3 4 β€² ∘
  • D 6 7 4 7 β€² ∘

Q16:

𝐴 𝐡 𝐢 is a triangle where π‘Ž = 2 4 . 9 c m , 𝑏 = 2 7 . 9 c m , 𝑐 = 2 2 . 5 c m and the radius of the circumcircle is 22.4 cm. Find the area of the triangle giving the answer to one decimal place.

Q17:

𝐴 𝐡 𝐢 is a triangle where π‘š ∠ 𝐴 = 6 4 ∘ , π‘š ∠ 𝐢 = 7 1 ∘ and the perimeter is 59.34 cm. Find the area of the triangle giving the answer to two decimal places.

Q18:

𝐴 𝐡 𝐢 is a triangle where π‘š ∠ 𝐴 = 4 7 ∘ , π‘š ∠ 𝐡 = 6 6 ∘ and the area is 197 cm2. Find the perimeter of 𝐴 𝐡 𝐢 and the circumference of the circumcircle giving the answer to two decimal places.

  • A perimeter is 64.96 cm and circumference is 39.75 cm.
  • B perimeter is 41.65 cm and circumference is 79.5 cm.
  • C perimeter is 41.65 cm and circumference is 39.75 cm.
  • D perimeter is 64.96 cm and circumference is 79.5 cm.

Q19:

Given that 𝐴 𝐡 𝐢 is a triangle where π‘š ∠ 𝐡 = 6 8 ∘ and π‘š ∠ 𝐢 = 4 0 ∘ . The point 𝐷 lies on 𝐡 𝐢 where 𝐴 𝐷 = 2 6 c m and bisects ∠ 𝐴 . Find the area of β–³ 𝐴 𝐡 𝐢 giving the answer to two decimal places.

Q20:

The area of the triangle 𝐴 𝐡 𝐢 is π‘₯ and the radius of the circumcircle is π‘Ÿ . Find the value of 4 π‘Ÿ π‘₯ π‘Ž 𝑏 𝑐 .

  • A2
  • B4
  • C 1 2
  • D1

Q21:

𝐴 𝐡 𝐢 is a triangle where π‘š ∠ 𝐡 = 6 6 ∘ , π‘š ∠ 𝐢 = 6 3 ∘ and the diameter of the circumcircle is 27 cm. Find the area of the triangle 𝐴 𝐡 𝐢 giving the answer to two decimal places.

Q22:

𝐴 𝐡 𝐢 is a triangle where π‘š ∠ 𝐡 = 3 6 ∘ , π‘š ∠ 𝐢 = 6 0 ∘ and the area of 𝐴 𝐡 𝐢 is 153 cm2. Find π‘Ž giving the answer to two decimal places.

Q23:

𝐴 𝐡 𝐢 is an isosceles triangle where 𝑏 = 𝑐 , π‘š ∠ 𝐴 = 4 2 ∘ and the perimeter is 31.31 cm. Find the area of the triangle giving the answer to two decimal places.

Q24:

In the given figure.

Find an expression for the height β„Ž in terms of π‘Ž , 𝑏 , and 𝐢 .

  • A 𝑏 𝐢 t a n
  • B 𝑏 𝐢 c o s
  • C π‘Ž 𝐢 s i n
  • D 𝑏 𝐢 s i n
  • E π‘Ž 𝐢 c o s

Find an expression for the area of the triangle in terms of π‘Ž , 𝑏 , and 𝐢 .

  • A 1 2 π‘Ž 𝑏 𝐢 s i n
  • B 1 2 π‘Ž 𝑏 𝐢 c o s
  • C 1 4 π‘Ž 𝑏 𝐢 s i n
  • D π‘Ž 𝑏 𝐢 s i n
  • E π‘Ž 𝑏 𝐢 c o s

Q25:

The Bermuda triangle is a region of the Atlantic Ocean whose boundary is made up of straight lines connecting Bermuda, Florida, and Puerto Rico. Given that Florida is 1 0 3 0 miles from Bermuda in the direction 24 degrees south of west and Puerto Rico is 980 miles from Bermuda in the direction 4 degrees west of south, find the area of the Bermuda triangle.

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