Area and perimeter

The shaded shape below is made from a right-angled triangle, ABC, and a semicircle. The length of AB is 20 cm and the diameter of the semicircle, BC, is of length 12 cm.

Calculate the perimeter of the shaded shape. You must show your working.

The diagram shows a right-angled triangle with base 9 cm and height 4 cm.

Work out the area of the triangle.

The diagram shows a shape formed from a rectangle measuring 8 cm by 17 cm with a semicircle removed from one side.

Calculate the perimeter of the shape.

The diagram shows a trapezium.

Calculate the area of the trapezium. You must show your working.

The diagram shows a shape formed by a semicircle joined to a sector of a circle, BAC, along a common length AB.

The area of the sector BAC is 16π cm².

Work out the total area of the shape. Give your answer in terms of π. You must show your working.

The diagram shows a quadrilateral ABCD. DB is a diagonal of the quadrilateral.

AD = 5 cm, AB = 7 cm and BC = 14 cm. Angle DAB = x ° and angle DBC = 45°.

cos x ° = 0.8 and sin x ° = 0.6.

Work out the area of triangle BDC. You must show your working.

In this question, all lengths are in centimetres.

The diagram shows an isosceles triangle ABC.

Find the perimeter of the triangle. You must show your working.

The diagram shows a square drawn on a one-centimetre square grid.

Write down the coordinates of point A.

Find the perimeter of the square.

The diagram shows the cross-section of a triangular prism.

Work out the area of the cross-section.

This diagram shows the triangular prism.

Work out the total surface area of the triangular prism. You must show your working.

Two of these triangular prisms are joined to make a new prism.

Give a reason why the total surface area of this prism is not two times your answer in part (b).

AC is a diagonal of the quadrilateral ABCD.

AD = 72 m. Angle ABC = 108 °, angle BCA = 3 8 ° and angle BAC = 34 °. Angle ACD = 90 °, angle CD A = 65 ° and angle CAD = 25 °.

Find the area of ABCD. You must show your working.

A housing estate is built on a triangular piece of land.

There are 3951 people living on the estate.

Work out an estimate of the population density of the estate in people per km².

In the diagram below, the shaded region is a parallelogram. The parallelogram can be identified by four inequalities. Two of the inequalities are y ⩾ 6 and y ⩾ x.

Write down the other two inequalities that identify the parallelogram.

Work out the area of the parallelogram. You must show your working.

(b)........ square units

Measure angle x.

The angles of a triangle are 27°, 126° and 27°.

Explain how you know the triangle is isosceles.

Complete the drawing of the parallelogram.

Include notation to show that the drawing is a parallelogram.

Work out the area of the parallelogram.

The diagram shows a triangle and a trapezium.

Show that they have the same area.

The diagram shows an equilateral triangle and a rectangle.

The equilateral triangle has the same perimeter as the rectangle.

Find the value of a.

In this question all measurements are in centimetres.

The shaded shape is made by cutting a square from the corner of a rectangle.

The width of the rectangle is k. The length of the rectangle is 2 k. Each side of the square is g.

Write down the relationship between the length and the width of the rectangle.

Find an expression for the area of the shaded shape. Give your answer in its simplest form.

Find an expression for the perimeter of the shaded shape. Give your answer in its simplest form.

Find the value of k when the perimeter of the shaded shape is 62.4.

The diagram shows a regular pentagon made using ten congruent right-angled triangles. The length of one side of the pentagon is 12 cm.

Show that the area of the pentagon is 247.75 cm², correct to 2 decimal places.

The regular pentagon is the base of a pyramid.

The pyramid has volume 450 cm³. The perpendicular height of the pyramid is h cm.

Calculate the value of h. [The volume of a pyramid is × area of base × perpendicular height.]

The diagram shows a triangle and a trapezium.

Show that they have the same area.

The diagram shows a circle, centre O and radius 52 cm.

AB is a chord of the circle. The line through T is a tangent to the circle.

The chord is parallel to the tangent. The perpendicular distance between the chord and the tangent is 72 cm.

Calculate the area of the shaded segment. You must show your working.

The diagram shows an equilateral triangle and a rectangle.

The equilateral triangle has the same perimeter as the rectangle.

Find the value of a.

The diagram shows a regular pentagon made using ten congruent right-angled triangles. The length of one side of the pentagon is 12 cm.

Show that the area of the pentagon is 247.75 cm², correct to 2 decimal places.

The regular pentagon is the base of a pyramid.

The pyramid has volume 450 cm³. The perpendicular height of the pyramid is h cm.

Calculate the value of h. [The volume of a pyramid is × area of base × perpendicular height.]

The diagram shows a shaded sector inside a parallelogram. The sector has an angle of 30°. The parallelogram, ABCD, has length BC = 20 cm and AB = 12 cm. The perpendicular distance between BC and AD is 6 cm.

Show that the area of the sector is 37.7 cm², correct to 3 significant figures.

Work out the percentage of the parallelogram that is not shaded.

Work out the perimeter of this rectangle.

Finley draws a rectangle and says

The perimeter is 20 cm and the length is 10 cm.

Can Finley be correct? Show how you decide.

The diagram shows a pentagon ABCDE.

Find the area of the pentagon. You must show your working.

In the diagram,

• ABC is a right-angled triangle

• ACD is the sector of a circle with centre A.

Show that the area of the sector ACD is m².

Work out the total area of the shape ABCD.

Here is a right-angled triangle.

The area of the triangle is 100cm², correct to the nearest 10cm². The length of the base of the triangle is 8 cm, correct to the nearest cm.

Calculate the largest possible height of the triangle.

This shape is formed from a rectangle and two sectors of circles.

Points A, B and C lie on a straight line. Angle CB D = 3 5 °. DE = 5 t and EF = 2 t.

Explain why BC = 2 t. Give a reason for each step of your explanation.

Show that the perimeter of the shape is t + 14t.

The diagram shows a quadrilateral and an equilateral triangle. The perimeter of the quadrilateral is equal to the perimeter of the equilateral triangle.

Find an expression for the length of one side of the equilateral triangle. Give your answer in terms of x in its simplest form.

The diagram shows a circle inside a square of side 12 cm.

Work out the percentage of the square that is shaded. You must show your working.

The diagram shows a prism of length 10 cm.

The cross-section of the prism is a right-angled triangle. The base, b cm, is 2 cm longer than the height, h cm. The volume of the prism is 240 cm³.

A student is explaining how they worked out the value of b.

They say

b is 6 because that means h is 4 and 6 × 4 × 10 = 240.

Describe the student’s error and find the correct value of b.

The diagram shows a circle inside a square of side 12 cm.

Work out the percentage of the square that is shaded. You must show your working.

The sides of this triangle are given in centimetres. The perimeter of the triangle is 80 cm.

Find the length of each side of the triangle. You must show your working.

(a)........ cm,........ cm and........ cm

Is the triangle above a right-angled triangle? Use calculations to show how you decide.

The diagram shows triangle ABC.

AB = 10.6 cm, BC = 8.2 cm and AC = 12.5 cm.

Show that angle BAC = 40.5°, correct to 1 decimal place.

Work out the area of triangle ABC.

The diagram shows a prism of length 10 cm.

The cross-section of the prism is a right-angled triangle. The base, b cm, is 2 cm longer than the height, h cm. The volume of the prism is 240 cm³.

A student is explaining how they worked out the value of b.

They say

b is 6 because that means h is 4 and 6 × 4 × 10 = 240.

Describe the student’s error and find the correct value of b.

The diagram shows a quadrilateral, PQRS.

PS = 10 cm. Angle QPS = Angle PSR = 90°.

SR is 6 cm longer than PQ. The area of quadrilateral PQRS is A cm².

Write a simplified expression for the length PQ in terms of A. You must show your working.

The diagram shows a right-angled triangle and a rectangle.

The triangle and rectangle have the same area.

Calculate the length, d cm, of the diagonal of the rectangle. You must show your working.

The parallelogram and the trapezium have the same area.

The ratio of x: y is 3: 5.

Find the value of x and the value of y. You must show your working.

The diagram shows triangle ABC.

Find the area of the triangle.

The diagram shows Kai’s garden. The garden is a rectangle, 16 m by 10 m.

It has a lawn and a flowerbed. The flowerbed is a circle of diameter 6 m.

Work out the area of Kai’s lawn.

The diagram shows Kai’s garden. It is in the shape of a right-angled triangle.

Kai is going to spread grass seed on the garden.

• A bag of grass seed covers an area of 35 m².

• Each bag of grass seed costs £8.99.

Kai can only buy whole bags of grass seed. Kai buys the least number of bags needed for the garden.

Calculate the cost of buying the bags of grass seed that Kai needs. You must show your working.

In this question, all lengths are in centimetres.

The diagram shows an equilateral triangle and a square.

The perimeter of each shape is 36 cm.

Find the value of b.

The diagram shows a triangular prism and a cube. The ends of the prism are right-angled triangles with base 16 cm and height 12 cm. The prism is 18 cm long.

The volume of the prism is equal to the volume of the cube.

Find the surface area of the cube. You must show your working.

Charlie is making some wooden frames. Charlie has a strip of wood 1.6 m long and 2 cm wide.

Not to scale

Each frame will be made from four pieces of wood cut from the strip to form a rectangle, as shown below.

The width of each frame is x cm. The length of each frame is 2 x cm. The area enclosed by each frame must be 96 cm².

Work out the maximum number of frames Charlie can make from the 1.6 m length of wood. You must show your working.

The diagram shows a triangular prism and a cube. The ends of the prism are right-angled triangles with base 16 cm and height 12 cm. The prism is 18 cm long.

The volume of the prism is equal to the volume of the cube.

Find the surface area of the cube. Y ou must show your working.

The graph of y = 2 x + 1 is drawn on this one centimetre grid.

The region R satisfies these inequalities.

y ⩽ 2 x + 1 y ⩾ 5 x + y ⩽ 13

Show that the area of region R is 12 cm².

A rectangle, 6 cm by 3 cm, and a semi-circle are joined to make this shape.

Work out the area of the shape.

The shape below is formed by a rectangle of width 4 cm and a regular pentagon. For the rectangle, the ratio of the width to the length is 2: 5.

Work out the perimeter of the shape.

Calculate the area of this trapezium.

ABC is an isosceles triangle. The sides of the triangle ABC are all tangents to a circle of radius 6 cm, centre O.

Angle BAC = 70° and BA = BC.

Show that length BO is 17.54 cm, correct to 2 decimal places.

Find the area of triangle ABC. You must show your working.

The diagram shows a pyramid ABCDE.

The pyramid has a square horizontal base ABCD with side 5.6 cm.

The vertex E is vertically above the centre O of the base. The height OE of the pyramid is 6.8 cm.

Calculate the surface area of the pyramid. You must show your working.

The diagram shows Jane’s lawn. It is in the shape of a square of side 36 m and three semi-circles.

She is going to spread fertiliser on the lawn at a rate of 30 g per square metre. The fertiliser is only sold in 10 kg bags costing £15.80 each.

Calculate the cost of buying the bags of fertiliser for her lawn. You must show all your working.

The diagram shows a square.

(4 x – 10) cm
Not to scale

(11 – 2 x) cm

By setting up and solving an equation, show that the perimeter of the square is numerically equal to the area of the square.

A rectangle is three times as long as it is wide. It has a perimeter of 44 cm.

Find the length of the rectangle.

The area of the triangle is equal to the area of the trapezium.

Calculate the height, h cm, of the trapezium.

The diagram shows Jane’s lawn. It is in the shape of a square of side 36 m and three semi-circles.

She is going to spread fertiliser on the lawn at a rate of 30 g per square metre. The fertiliser is only sold in 10 kg bags costing £15.80 each.

Calculate the cost of buying the bags of fertiliser for her lawn. You must show all your working.

The diagram shows a triangular field PQR which is used to grow organic carrots.

PQ = 100 m, PR = 80 m and angle RPQ = 30°.

In recent years, an average of 2.5 kg of carrots has been harvested from each square metre of the field.

Use this information to work out the total mass of carrots that might have been harvested from the field in 2019.

Why might the answer to part (a) be unreliable?

ABCD is a trapezium.

The perimeter of the trapezium is 56 cm. The ratio AD: AB: DC: BC = 5: 12: 6: 5.

Calculate the area of the trapezium. Show your working.

The area of the parallelogram is four times the area of the triangle.

Calculate the length, L, of the parallelogram.

The midpoints of the sides of a rectangle are joined by straight lines as shown.

Work out the percentage of the rectangle that is shaded.

The diagram shows two rectangles, A and B.

Rectangle A has a width of 25 cm and a height of 12 cm. The width of rectangle B is three times the height of rectangle B.

The area of rectangle A is equal to the area of rectangle B.

Find the perimeter of rectangle B.

The diagram shows two rectangles, A and B.

Rectangle A has a width of 25 cm and a height of 12 cm. The width of rectangle B is three times the height of rectangle B.

The area of rectangle A is equal to the area of rectangle B.

Find the perimeter of rectangle B.

A shape is drawn on a one-centimetre grid.

Find the perimeter of the shape.

How many lines of symmetry does the shape have?

The diagram shows a regular hexagon made from six equilateral triangles. Each side is 10 cm. The angle ACB is a right angle.

Show that AC = 8.66 cm, correct to 3 significant figures.

Show that the area of triangle ACB is 21.7 cm², correct to 3 significant figures.

Find the area of the hexagon, giving your answer to an appropriate degree of accuracy.

The diagram shows a shape ABCDE. The shape is made from a rectangle, a right-angled triangle and a quarter of a circle.

F is the mid-point of BD. AE = 18 m and the perpendicular distance from C to AE is 41 m.

Work out the perimeter of the shape ABCDE.

The perimeter of the pentagon is equal to the perimeter of the square.

Not to scale

Find an expression for the length of one side of the square. Give your answer in terms of x in its simplest form.

The diagram shows a square with four identical corners shaded.

The length of each side of the square is 3 x cm. The length of each shaded corner is x cm.

Show all your working.

In the diagram, the square and the trapezium share a common side of length x cm.

The area of the square is equal to the area of the trapezium.

Work out the value of x.

The shape below is formed from two semicircles and a straight line.

The radius of the large semicircle is 8 cm. The radius of the small semicircle is t cm.

Find an expression, in terms of t, for the exact perimeter of the shaded shape.

In this triangle:

• AB = 9 cm

• AC = 10 cm

• BC > 5 cm

• angle BCA = 60°

• angle ABC < 90°.

Calculate the area of triangle ABC.

A shape is drawn on a one-centimetre grid.

Ring the mathematical name of the shape.

Pentagon Square Octagon Rhombus

How many lines of symmetry does the shape have?

Work out the area of the shape.

Here is the floor plan of a rectangular room.

Tim buys carpet tiles for this room.

Each tile is a square measuring 50 cm by 50 cm. The tiles are only sold in packs of ten. Each pack costs £20. Tim pays for fitting at a rate of £7.50 per square metre, with any fraction of a square metre rounded up.

Work out the total cost of the tiles and fitting.

This shape consists of three semicircles.

The length of PQ is 4 x cm.

Show that the area, in cm², of the whole shape is 3x².

In this question, assume all dimensions are in centimetres.

Jess joins three tiles together to make a larger rectangle, as shown.

Write an expression for the perimeter of her rectangle. Give your answer in its simplest form.

An expression for the area of her rectangle is 6ab + 6b².

Factorise this expression fully.

Pete joins some tiles together to make a different rectangle. The area of his rectangle is 8ab + 8b².

Draw a possible arrangement of tiles for Pete’s rectangle. Write down expressions for the length and for the width of the rectangle.

Calculate the area of this triangle.

The diagram represents a rectangular garden of length 10 m and width 8 m. The flower bed is a triangle and the patio is a trapezium. The rest of the garden is lawn.

Work out the area of the lawn.

ABCD and PQRS are rectangles. O is the centre of both rectangles.

AC is a straight line passing through P, O and R. BD is a straight line passing through Q, O and S.

PQ = 6 cm and QR = 2 cm. The perimeter of rectangle ABCD is 40 cm.

Work out the length and width of rectangle ABCD.

PQRS is a square. G is the midpoint of PQ and H is the midpoint of QR. Triangle GHS is shaded.

Find the ratio shaded area: area of square in its simplest form. Show all your working.

This rectangle has length (4 x – 5) cm and width (x + 3) cm.

The perimeter of the rectangle is 46 cm.

Calculate the area of the rectangle.

The design below is made from two sectors of circles, centre O.

Calculate the perimeter of the shaded part. Give your answer correct to 3 significant figures.

The diagram shows a semi-circle inside a rectangle of length 120 m. The semi-circle touches the rectangle at A, B and C.

Calculate the perimeter of the shaded region. Give your answer correct to 3 significant figures.

A farmer has a field that is in the shape of a trapezium. He measures the field so that he can work out the area. He puts his measurements on this diagram of the field.

The farmer has rounded his measurements to two significant figures.

Give a reason why he may have done this.

The field produces 6400 kilograms of wheat per hectare. One hectare is 10 000 m².

Work out how many kilograms of wheat the field produces.

The shape below is formed from a rectangle measuring 12 cm by 15 cm from which a rectangle of length 7.4 cm has been removed.

Work out the perimeter of the shape.

The diagram shows a semi-circle inside a rectangle of length 120 m. The semi-circle touches the rectangle at A, B and C.

Calculate the perimeter of the shaded region. Give your answer correct to 3 significant figures.

The diagram shows some land in the shape of a quadrilateral, ABCD.

AB = 3 km, AD = 5 km, CD = 12 km and angle BAC = 30°.

The land is sold for £10 million per square kilometre.

Calculate the total cost of the land.