Pythagoras' theorem

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 square-based pyramid ABCDE. O is the centre of the base.

The length of the base is 10.7 cm. Angle EAO is 67°.

Calculate the length of AE. You must show your working.

The diagram shows two right-angled triangles, ABC and BDC, joined at BC.

Work out angle BCA. You must show your working.

The diagram shows a rectangle, OPQR, and a circle, centre O, which passes through Q. OR = 6 cm and RQ = 8 cm.

Find the circumference of the circle. Give your answer in terms of π.

AOB is a sector of a circle, centre O and radius 8 cm. Angle AOB = x °. The arc, AB, has length 2 cm. π

Find the area of the sector. Give your answer in terms of π.

The diagram shows a trapezium.

Calculate the value of x.

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 sketch shows the coordinate axes and the point (0, 10).

The distance from the point (0, 10) to a point (12, t) is 12.5 units.

Work out the two possible values of t. You must show your working and you may use the sketch to help.

A triangle has sides of length 6 cm, 10 cm and 12 cm.

Is this a right-angled triangle? Show how you decide.

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.

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 a square-based pyramid ABCDE. O is the centre of the base.

The pyramid has base length 20 cm and each sloping edge has length 14.5 cm.

Draw the plan view of the pyramid on the one-centimetre grid below.

Scale: 1 cm represents 4 cm.

Calculate the volume of the pyramid. You must show your working. [The volume of a pyramid is × area of base × perpendicular height]

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 diagram shows two right-angled triangles that are joined together.

All measurements are given accurate to 2 significant figures.

8.3 cm Not to scale

Work out the value of x. Give your answer correct to an appropriate degree of accuracy. You must show your working.

Kai has a box in the shape of a cuboid. The internal dimensions of the box are 10 cm by 4 cm by 6 cm.

Kai is given a pencil of length 13 cm.

Show that the pencil does not fit completely inside the box.

Points A, B and C lie on the circumference of a circle, centre O.

Angle ACB = 90°, AC = 6.4 m and BC = 3.7 m.

Work out the circumference of the circle. You must show your working.

ABC is an equilateral triangle of side length 2 cm. M is the midpoint of AC.

Using this diagram, show that tan 30° = .

The diagram shows rectangle ABCD.

DB = 13 cm and BC = 5 cm.

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

The diagram shows a quadrilateral ABCD.

AD = 10 cm, angle BAD = 30° and angle ABD = angle BDC = 90°.

The ratio of length BD to length DC is 1: 2.4.

Work out length BC. 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.

A triangle has sides of length 14.1 cm, 14.8 cm and 19.5 cm.

Is this a right-angled triangle? Show how you decide.

The diagram shows a right-angled triangular prism ABCDEF.

Length AD = 11 cm, length CD = 10 cm and length CF = 6 cm.

Calculate the volume of the prism.

Use trigonometry to show that angle FDC = 31°, correct to the nearest degree.

Calculate the exact length of AF.

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.

OAB is a sector of a circle, centre O. OA = 6 cm and AX is perpendicular to OB.

The area of sector OAB is 6π cm².

The length of the longest diagonal of a cube is 25 cm.

Calculate the total surface area of the cube.

Here is a right-angled triangle.

Work out the value of x.

Here is Mario’s answer to a question.

Explain the error in Mario’s method.

The diagram shows a right-angled triangular prism ABCDEF.

Calculate angle AFB.

In the diagram, ABC is a right-angled triangle. P is a point on AB. BC = 40 m, AP = 20 m and angle ABC = 30 °.

Show that AC = 20 m.

Find the length of PB.

Here is a right-angled triangle.

Work out the value of x.

The diagram shows two right-angled triangles ABD and BCD, sharing a common side BD. AD = 10 cm, BC = 12 cm and angle DBC = 60°.

Work out the length of AB.

The diagram shows a straight line that passes through points A and B, and a curve that passes through points P and Q.

Find the equation of the straight line.

The equation of the curve is y = x² + kx + 8.

Find the value of k.

Diann draws line BQ. She says

Triangle ABQ is isosceles.

Is Diann correct? You must show all your working.

The diagram below shows two right-angled triangles.

Prove that triangles PQS and QRS are similar.

In the diagram below, ABCD is a trapezium. Length AE is 37.5 cm. DE = CF

Find the value of angle x.

The diagram shows a cuboid ABCDEFGH.

Calculate angle BHF.

Triangles P and Q are right-angled.

Show that the two shorter sides in triangle P have the same lengths as the two shorter sides

in triangle Q.

Explain why the two triangles are congruent.

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.

Alvin has a crate in the shape of a cuboid. The crate is open at the top. The internal dimensions of the crate are 46 cm long by 46 cm wide by 55 cm high.

Alvin has a stick of length 95 cm. Alvin places the stick in the crate so that the shortest possible length extends out above the top of the crate.

Calculate the length of the stick that extends out of the crate.

Calculate the angle the stick makes with the base of the crate.