Epsom College headteacher Emma Pattison was shot in the chest and her daughter Lettie died of an shotgun wound to the head, an inquest heard today.
Both are believed to have been murdered by 39-year-old chartered accountant George Pattison - their husband and father respectively - before he killed himself on February 5.
Tests showed Mrs Pattison, 45, also suffered shock and haemorrhage, Surrey Coroner's Court heard.
The three were found dead at their home within the grounds of the private boarding school in Surrey after Mrs Pattison made a distressed call to her sister.
Coroner Simon Wickens expressed his condolences to those who loved or knew Mrs Pattison and Lettie, whose full name was Ellette Francesca.
After the brief hearing into Mrs Pattison's death, Mr Wickens said: "I would like to offer my condolences to Emma's family and friends and also the wider community she served and also to the students whose lives she no doubt touched."
The coroner added: "I would like to offer my sincerest condolences to Ellette's family and all who have been touched by her life."
Post-mortem examinations were carried out by Dr Ashley Fegan-Earl at East Surrey Hospital four days after the shooting, coroner's officer Kelly Truss said.
Toxicology and histology tests were carried out, while Mrs Pattison and Lettie were identified by their teeth, the court heard.
A pre-inquest review hearing into the deaths has been set for June 27.
Surrey Police had said that a firearm legally registered to Mr Pattison was found at the scene.
Live ammunition is not kept at the school rifle range, it has been reported.
A statement released by Mrs Pattison and Lettie's family after their deaths said: "To see the esteem in which Emma is held by all who knew her is an enormous comfort.
"She was everything one could hope for in a daughter, sister, mother, wife, friend, teacher and so much more. We are an extremely close family and family was at the centre of Emma and Lettie's universe.
"The Epsom College community had become part of that universe for them both.
"Seven-year-old Lettie was Emma's pride and joy: an adorable, vibrant little girl with a compelling curiosity, a heart-melting smile and an intellect beyond her years.
"The two of them were inseparable and we take comfort in that they will remain so."
The family's statement also thanked well-wishers for their messages of support.
The inquest into Mr Pattison's death was opened earlier this week and heard he died of a shotgun wound to the head.
Toxicology and histology reports had been carried out and he was identified by his teeth.
A pre-inquest review hearing relating to his case has also been set for June 27 at the same court.
