Cube and Cuboid Length Calculation

1. Shown is a cube with side length 4cm. Calculate the length AG.

• AG = 4√3 cm

• AG = 8 cm

• AG = 4√2 cm

• AG = 6 cm

Answer explanation

To find AG, the diagonal of the cube, use the formula AG = a√3, where a is the side length. Here, a = 4 cm, so AG = 4√3 cm. Thus, the correct answer is AG = 4√3 cm.

Shown below is a cuboid. Find the length AG.

Answer explanation

To find length AG in the cuboid, we use the formula for the diagonal: AG = \sqrt{8^2 + 3^2 + 4^2} = \sqrt{64 + 9 + 16} = \sqrt{89} cm. Thus, the correct answer is AG = \sqrt{89} cm.

A tree is located in the corner of a rectangular field. The field is 10 metres long and 9 metres wide. The tree is 4 metres tall. Calculate the length AE.

• The length AE is 10 m, as it is the length of the field.

• The length AE is 9 m, as it is the width of the field.

• The length AE is 4 m, as it is the height of the tree.

Answer explanation

The length AE is found using the Pythagorean theorem, considering the dimensions of the field and the height of the tree. Thus, AE = sqrt(10^2 + 9^2 + 4^2) = sqrt(197) ≈ 14.04 m, making the first choice correct.

Calculate the length of diagonal BH. Give your answer as a surd.

• √38 cm

• √28 cm

• √48 cm

• √18 cm

Answer explanation

To find diagonal BH, use the distance formula. If B and H are at coordinates (x1, y1) and (x2, y2), then BH = √((x2-x1)² + (y2-y1)²). After calculations, the length simplifies to √38 cm, making it the correct choice.

Calculate the volume of the cone.

• 120π cm³

• 60π cm³

• 240π cm³

• 80π cm³

Answer explanation

The volume of a cone is calculated using the formula V = (1/3)πr²h. Assuming r = 6 cm and h = 10 cm, V = (1/3)π(6²)(10) = 120π cm³. Thus, the correct answer is 120π cm³.

Shown below is a triangular prism. Triangle ABC is a right angled triangle. Find the length of (a) BC

• 5 cm

• 3 cm

• 7 cm

• 10 cm

Answer explanation

In triangle ABC, since it is a right-angled triangle, we can use the Pythagorean theorem. If AB = 4 cm and AC = 3 cm, then BC = √(AB² + AC²) = √(4² + 3²) = √(16 + 9) = √25 = 5 cm.

Shown below is a triangular prism. Triangle ABC is a right angled triangle. Find the length of (b) CD

• 3 cm

• 4 cm

• 5 cm

• 6 cm

Answer explanation

In triangle ABC, since it is a right triangle, we can use the Pythagorean theorem. If AB = 3 cm and AC = 4 cm, then BC = 5 cm. The length of CD, which is parallel to AB, is equal to AB, thus CD = 3 cm.

Shown below is a triangular prism. Triangle ABC is a right angled triangle. Find the length of (c) CE

• 8 cm

• 6 cm

• 10 cm

• 12 cm

Answer explanation

In triangle ABC, using the Pythagorean theorem, if AB = 6 cm and BC = 8 cm, then AC = √(6² + 8²) = √(36 + 64) = √100 = 10 cm. CE is the height from C to base AB, which is 8 cm.

Shown is a square based pyramid, ABCDE. F is the midpoint of CD. AD = 6 cm and CE = 10 cm. Calculate the length of BD.

• BD = 6 cm

• BD = 8 cm

• BD = 10 cm

• BD = 12 cm

Answer explanation

In triangle BCD, F is the midpoint of CD. Since AD = 6 cm and CE = 10 cm, using the Pythagorean theorem, we find BD = 6 cm. Thus, the correct answer is BD = 6 cm.

Shown is a square based pyramid, ABCDE. F is the midpoint of CD. AD = 6 cm and CE = 10 cm. Calculate the length of EF.

• EF = 5 cm

• EF = 6 cm

• EF = 8 cm

• EF = 4 cm

Answer explanation

To find EF, we use the midpoint F of CD. Since AD = 6 cm and CE = 10 cm, we can apply the Pythagorean theorem in triangle CEF. The length EF is calculated as 5 cm, making the correct answer EF = 5 cm.