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7.11 Finding the Area of Any Triangle

EDEXCEL 1MA1 · Calculator allowed · about 75 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Which of these is the correct formula for finding the area of a triangle when you know two sides, a and b, and the included angle C between them?

A) Area = a x b x sin C
B) Area = 1/2 x a x b x sin C
C) Area = 1/2 x a x b x cos C
D) Area = a x b x cos C

(Total for Question 1 is 1 mark)

Triangle ABC has a right angle at B. AB = 9.2 cm and BC = 6 cm. Calculate the area of triangle ABC.

Diagram NOT accurately drawn

(Total for Question 2 is 2 marks)

A triangular plot of land has a base of 14.5 m and a perpendicular height of 3.6 m. Calculate the area of the plot.

Diagram NOT accurately drawn

(Total for Question 3 is 2 marks)

Triangle A has a base of 10 cm and a perpendicular height of 8 cm. Triangle B has the same area as Triangle A and a base of 16 cm. Work out the perpendicular height of Triangle B.

(Total for Question 4 is 3 marks)

In triangle DEF, DE = 7 cm, DF = 10 cm and angle EDF = 55 degrees. Calculate the area of triangle DEF. Give your answer correct to 3 significant figures.

Diagram NOT accurately drawn

(Total for Question 5 is 3 marks)

In triangle GHI, GH = 9 cm, HI = 13 cm and angle GHI = 140 degrees. Calculate the area of triangle GHI. Give your answer correct to 3 significant figures.

Diagram NOT accurately drawn

(Total for Question 6 is 3 marks)

Triangle PQR has an area of 48 cm^2. Triangle STU is similar to triangle PQR, and each side of STU is 1.5 times the length of the corresponding side of PQR. Work out the area of triangle STU.

(Total for Question 7 is 3 marks)

The area of triangle JKL is 34 cm^2. JK = 9 cm and angle JKL = 40 degrees. Calculate the length of KL. Give your answer correct to 3 significant figures.

Diagram NOT accurately drawn

(Total for Question 8 is 3 marks)

The area of triangle MNO is 60 cm^2. MN = 15 cm and NO = 10 cm. Angle MNO is acute. Calculate angle MNO. Give your answer correct to 1 decimal place.

Diagram NOT accurately drawn

(Total for Question 9 is 3 marks)

An isosceles triangle has two equal sides of length 14 cm, with an apex angle of 38 degrees between them. Calculate the area of the triangle. Give your answer correct to 3 significant figures.

Diagram NOT accurately drawn

(Total for Question 10 is 3 marks)

A triangular flower bed has two sides of length 6.5 m and 8.2 m, with an included angle of 72 degrees between them. Turf costs 12.50 pounds per square metre. Work out the total cost of turfing the flower bed. Give your answer to the nearest penny.

Diagram NOT accurately drawn

(Total for Question 11 is 4 marks)

A rectangular lawn ABCD has AB = 12 m and BC = 7 m. An isosceles triangular flower bed is attached to side AB, outside the rectangle, with the two equal slant sides of the triangle each measuring 7.5 m. Work out the total area of the lawn and the flower bed together.

Diagram NOT accurately drawn

(Total for Question 12 is 5 marks)

An equilateral triangle has sides of length 2*√3 cm and each interior angle is 60 degrees. Show that the area of the triangle is 3*√3 cm^2.

(Total for Question 13 is 3 marks)

Triangle UVW has UV = 6 cm, VW = 8 cm and angle UVW = 60 degrees. Work out the exact area of triangle UVW, giving your answer in the form k*√3, where k is an integer.

Diagram NOT accurately drawn

(Total for Question 14 is 3 marks)

In triangle ABC, angle BAC = 48 degrees, angle ABC = 67 degrees and BC = 10.4 cm (BC is opposite angle A). Calculate the area of triangle ABC. Give your answer correct to 3 significant figures.

Diagram NOT accurately drawn

(Total for Question 15 is 5 marks)

A pyramid VABCD has a square base ABCD with side length 10 cm. The apex V is directly above the centre of the base, and the vertical height of the pyramid is 12 cm. Calculate the area of the triangular face VBC.

Diagram NOT accurately drawn

(Total for Question 16 is 5 marks)

The area of triangle ABC is 51 cm^2. AB = y cm, AC = (y + 5) cm, and angle BAC = 30 degrees.

Diagram NOT accurately drawn

(a)Show that y² + 5y - 204 = 0(3)

(b)Hence find the length of AC.(3)

(Total for Question 17 is 6 marks)

A sector of a circle with centre O has radius 9 cm and an angle of 100 degrees at the centre. Calculate the area of the shaded segment (the region between the chord and the arc). Give your answer correct to 3 significant figures.

Diagram NOT accurately drawn

(Total for Question 18 is 5 marks)

The area of triangle PQR is 90 cm^2. PQ = 20 cm and QR = 15 cm. Angle PQR is obtuse.

Diagram NOT accurately drawn

(a)Calculate angle PQR. Give your answer correct to 1 decimal place.(3)

(b)Calculate the length of PR. Give your answer correct to 3 significant figures.(3)

(Total for Question 19 is 6 marks)

Mark scheme · 7.11 Finding the Area of Any Triangle

Question 1

B1 B oe
Answer: B) Area = 1/2 x a x b x sin C

Question 2

M1 1/2 x 9.2 x 6 oe
A1 27.6 cao (cm²)
Answer: 27.6 cm²

Question 3

M1 1/2 x 14.5 x 3.6 oe
A1 26.1 cao (m²)
Answer: 26.1 m²

Question 4

M1 area of A = 1/2 x 10 x 8 (=40)
M1 40 = 1/2 x 16 x h oe
A1 h = 5 cm cao
Answer: 5 cm

Question 5

M1 1/2 x 7 x 10 x sin55 oe
A1 35 sin55 (=24.6 to 28.7 range value seen)
A1 28.7 cm² awrt
Answer: 28.7 cm² (3 sf)

Question 6

M1 1/2 x 9 x 13 x sin140 oe
A1 58.5 sin140 seen
A1 37.6 cm² awrt
Answer: 37.6 cm² (3 sf)

Question 7

B1 area scale factor = 1.5² (=2.25) identified
M1 48 x 2.25
A1 108 cm² cao
Answer: 108 cm²

Question 8

M1 34 = 1/2 x 9 x KL x sin40 oe
M1 KL = 34 / (4.5 x sin40)
A1 11.8 cm awrt
Answer: 11.8 cm (3 sf)

Question 9

M1 60 = 1/2 x 15 x 10 x sin(MNO) oe
M1 sin(MNO) = 60/75 (=0.8)
A1 53.1 degrees awrt (acute value only)
Answer: 53.1 degrees (1 dp)

Question 10

M1 1/2 x 14 x 14 x sin38 oe
M1 98 sin38 seen
A1 60.3 cm² awrt
Answer: 60.3 cm² (3 sf)

Question 11

M1 1/2 x 6.5 x 8.2 x sin72 oe
A1 25.3 m² awrt (area)
M1 area x 12.50
A1 316.82 pounds awrt (ft from their area)
Answer: GBP 316.82

Question 12

M1 half base = 6 m identified
M1 height² = 7.5² - 6² (Pythagoras)
A1 height = 4.5 m cao
M1 triangle area = 1/2 x 12 x 4.5 (=27) and rectangle area = 12 x 7 (=84)
A1 111 m² cao
Answer: 111 m²

Question 13

M1 1/2 x 2sqrt(3) x 2sqrt(3) x sin60 oe
M1 1/2 x 12 x sin60 (=6sin60), sin60 = √3/2 used
A1 3sqrt(3) cm² cso (all steps shown, answer given)
Answer: 3*√3 cm² (answer given)

Question 14

M1 1/2 x 6 x 8 x sin60 oe
M1 sin60 = √3/2 used, 24 x √3/2
A1 12sqrt(3) cm² oe exact
Answer: 12*√3 cm²

Question 15

M1 angle C = 180 - 48 - 67 (=65 degrees)
M1 AC / sin67 = 10.4 / sin48 (sine rule) oe
A1 AC = 12.9 cm awrt
M1 1/2 x 10.4 x AC x sin65 oe (ft their AC)
A1 60.7 cm² awrt
Answer: 60.7 cm² (3 sf)

Question 16

M1 identifies right-angled triangle formed by the vertical height, half the base edge (5 cm) and the slant height
M1 slant² = 12² + 5² (Pythagoras)
A1 slant height = 13 cm cao
M1 1/2 x 10 x 13
A1 65 cm² cao
Answer: 65 cm²

Question 17

(a) M1 51 = 1/2 x y x (y+5) x sin30 oe
(a) M1 sin30 = 0.5 used, 51 = 0.25y(y+5)
(a) A1 y² + 5y - 204 = 0 cso (answer given, all steps shown)
(a) Answer: y² + 5y - 204 = 0 (answer given)
(b) M1 (y-12)(y+17) = 0 oe, or use of the quadratic formula
(b) A1 y = 12 (rejecting y = -17 as a length cannot be negative)
(b) A1 AC = 17 cm cao
(b) Answer: AC = 17 cm

Question 18

M1 sector area = 100/360 x π x 9² oe
A1 70.7 cm² awrt (sector area)
M1 triangle area = 1/2 x 9 x 9 x sin100 oe
A1 39.9 cm² awrt (triangle area)
A1 30.8 cm² awrt (segment = sector - triangle, ft)
Answer: 30.8 cm² (3 sf)

Question 19

(a) M1 90 = 1/2 x 20 x 15 x sin(PQR) oe
(a) M1 sin(PQR) = 90/150 (=0.6)
(a) A1 143.1 degrees awrt (obtuse value selected)
(a) Answer: 143.1 degrees (1 dp)
(b) M1 PR² = 20² + 15² - 2 x 20 x 15 x cos(PQR) oe (ft their angle)
(b) M1 cos(143.1) = -0.8 used, PR² = 625 - 600 x (-0.8) (=1105)
(b) A1 33.2 cm awrt
(b) Answer: 33.2 cm (3 sf)

Finding the Area of Any Triangle - Worksheets, Questions and Revision

7.11 Finding the Area of Any Triangle

Question 1

Question 2

Question 3

Question 4

Question 5

Question 6

Question 7

Question 8

Question 9

Question 10

Question 11

Question 12

Question 13

Question 14

Question 15

Question 16

Question 17

Question 18

Question 19