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mathematics UK Year 10 Mathematics (KS4)

Solved problems for the chapter:

Compound shapes and parts of shapes

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Composite plane shapes for KS4

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Detailed solution steps are available at:

https://en.openprof.com/wb/chapter:compound_shapes_and_parts_of_shapes/5344/

Example 1.

Calculate the area and perimeter of the figure below. The side of the inscribed square measures . Use the approximation for and rounds the result to two decimal places.

Example 2.

The midsegment of the trapezoid CDEG in the figure (painted green) is 5.75 cm long, its oblique side is 7.5 cm long and its area is . Calculate the perimeter and areas of all coloured figures if the points B, E and G are the bisectors of the sides of the right triangle ADF.

Calculate the perimeter and area of the blue-painted figure in the graph below.

The right-angled triangle is divided into four smaller right-angled triangles in the sketch below. Calculate the perimeter of all smaller right-angled triangles and the area of the red triangle if the length DF is , the length AC is , the area of the blue triangle is , the area of the yellow triangle is , the area of the green triangle is .

Calculate the area of the green polygon.

Calculate the perimeter and area of the given polygon:

The perimeter of the given polygon is . Calculate the lengths of the missing sides a.

Calculate the perimeter and area of the drawn figure.

Calculate the perimeter and area of the figure below.

10.

Calculate the perimeter and area of the given polygon with the data: , and .

11.

We have a box-shaped cuboid that is missing a lid. The length a of this box is , the width b is and the height c is . Calculate the area of its grid.

Net box without lid

12.

Calculate the area of the blue-shaded portion in the figure below with the data: and .

13.

Calculate the area of the blue-colored figure with the data: .

14.

Calculate the perimeter and area of the given polygon with the sides:

15.

Calculate the perimeter and area of the given polygon with sides:

The following applies to the marked sides: .

16.

Calculate the perimeter and area of the given polygon below:

17.

The figure shows a polygon with known and unknown lengths. All angles between the sides are perpendicular.

Determine how much the following measures:

18.

Calculate the perimeter and area of a given polygon.

19.

From nine smaller equilateral triangles, make a larger equilateral triangle and draw a circle. The area of the resulting circular ring is 254.47 . Calculate the base of the smaller triangle.

20.

Calculate the perimeter of the given polygon.

21.

The figure below has area 111.49 . Calculate its perimeter.

22.

The figure below has a perimeter of 79.69 cm. Calculate the area of the figure.

23.

The figure below has a perimeter of 24.85 cm. Calculate the area of the figure.

24.

The figure below has area of 101.47 . Calculate the perimeter of the figure.

25.

The figure below has a perimeter of 20.14 cm. Calculate the area of the figure.

26.

Calculate the area and perimeter of a kite made up of an equilateral triangle ABC and a right angle triangle ABD.

27.

The area of square ADBE is 8 . Point E divides the line CD in the ratio . What is the perimeter of the triangle ABC?

28.

An equilateral triangle ABC and parallelogram ADEF with side ratio

If the area of the triangle is . Calculate its perimeter.

29.

Find the ratio of the shaded areas and if the angle in the trapezium ABCD measures 60°, and s is the bisector of the trapezium.

30.

The perimeter of the parallelogram ABCD is equal to the length of arc l. In what ratio does the point E divide the line DC if measures 110°?

31.

Kite ABCD has all its sides equal and the diagonals are in ratio:

What is the ratio of the lengths and ?

32.

The line p divides the square ABCD into two parts. The ratio of the areas is shown in the figure. What is the ratio ?

33.

In the figure below, side a of the square ABCD measures 4 units. Two arcs are drawn such that an arc has its center at point A and the other at the midpoint E of the side BC. Calculate the area of the shaded region.

34.

Given two concentric circles with an outer radius R = 8 m and an inner radius r = 4 m. Calculate the area of the shaded region between them.

35.

Given the figure below, calculate the perimeter and area of the shaded region.

36.

Calculate the area of the shaded region in the figure below to two decimal places.

37.

Calculate the perimeter of the figure below to 2 decimal places.

38.

Calculate the areas of all the shaded figures in the square with side length 2. The shaded figures are two halves of a circle with a diameter of 2 and two small circles. The circles touch the two semicircles and the sides of the square.

39.

We cut an isosceles right triangle with sides cm out of a square with side cm as shown in the sketch below. Calculate the perimeter of the resulting pentagon.

40.

John was given the task of calculating the width of the lake between the points A and B. He measured AC = 255 m, BC = 232 m and . What is the distance between the points A and B? Round the result to the nearest meter.

41.

An isosceles triangle ABC is sketch below, the outer angle in the isosceles triangle is 137° 46' 12''. Calculate the interior angles in the triangle.

42.

In a given triangle below, the outer angle and the inner angle of an isosceles triangle are in the ratio 7: 2. Calculate the interior angles of the triangle.